habit persistence - arxiv.org

RETIREMENT DECISION AND OPTIMAL CONSUMPTION-INVESTMENT UNDER ADDICTIVE

HABIT PERSISTENCE

Guohui Guana, Zongxia Liangb, Fengyi Yuanb,∗

aSchool of Statistics, Renmin University of China, Beijing 100872, China

bDepartment of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Abstract. This paper studies the retirement decision, optimal investment and consumptionstrategies under habit persistence for an agent with the opportunity to design the retirement time.The optimization problem is formulated as an interconnected optimal stopping and stochasticcontrol problem (Stopping-Control Problem) in a finite time horizon. The problem containsthree state variables: wealth x, habit level h and wage rate w. We aim to derive the retirementboundary of this “wealth-habit-wage” triplet (x, h, w). The complicated dual relation is proposedand proved to convert the original problem to the dual one. We obtain the retirement boundaryof the dual variables based on an obstacle-type free boundary problem. Using dual relationwe find the retirement boundary of primal variables and feed-back forms of optimal strategies.We show that if the so-called “de facto wealth” exceeds a critical proportion of wage, it willbe optimal for the agent to choose to retire immediately. In numerical applications, we showhow “de facto wealth” determines the retirement decisions and optimal strategies. Moreover,we observe discontinuity at retirement boundary: investment proportion always jumps downupon retirement, while consumption may jump up or jump down, depending on the change ofmarginal utility. We also find that the agent with higher standard of life tends to work longer.

JEL Classifications: G11, C61.

2010 Mathematics Subject Classification: 91G10, 93E20, 90B50.

Keywords: Retirement decision; Stopping-control problem; Habit persistence; Wealth-habit-wage triplet; Retirement boundary; Dual transformation.

1. Introduction

Retirement decision-making has always been an important topic in research fields like quan-

titative finance, economics and actuarial sciences. In the scenario of voluntary early retirement,

it becomes a trade-off between more labour income and more leisure gains for the agent facing

the retirement-decision problem. There are more and more works combining voluntary early

retirement, optimal investment and optimal consumption. The agent is faced with two problems

coupled together: the optimal stopping problem and the stochastic control problem (Stopping-

Control Problem, or StopCP for short). The choice of early retirement has been accounted for

many reasons, such as health condition, habit formation, socioeconomic status, financial factors,

∗ Corresponding author.Email: [email protected], [email protected], [email protected]

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etc. We study the retirement decision-making problem with voluntary early retirement, under

linear habit persistence, and aim to give a quantitative and visualized description for retirement

boundaries and optimal strategies for consumption and investment. In other words, we investi-

gate this important problem: how do certain factors, such as wealth, standard of life and wage,

influence the retirement decision, consumption and investment choice?

Optimal consumption and portfolio problems with early retirement option, which are modeled

as StopCP, have attracted much attention in the recent decades. Karatzas and Wang (2000)

develop dual theory to handle the interaction in StopCP. The problem is transformed to a

pure stopping dual problem and then studied by super harmonic characterization or variational

methods. Farhi and Panageas (2007) consider voluntary early retirement model with power utility

and “no-borrowing-against-future-labour-income” constraint. Choi, Shim, and Shin (2008) and

Dybvig and Liu (2010) consider similar problems with no borrowing constraint. More related

works in infinite time horizon with StopCP can refer to Jeanblanc, Lakner, and Kadam (2004),

Choi and Koo (2005) and Liang, Peng, and Guo (2014), etc. Some recent works have also

considered finite horizon problems. In either case, researchers aim at finding thresholds for

wealth (or factors) that push the agent to retire. This topic remains very active recently, see

Yang and Koo (2018), Park and Jeon (2019), Yang, Koo, and Shin (2020), Jang, Park, and Zhao

(2020), Xu and Zheng (2020), etc.

Habit persistence has also been taken into consideration in research of finance and economics

for decades, which reflects the belief that the more people have consumed in the past, the

more they will expect to consume in the future. It has been applied to resolve equity premium

puzzle (Sundaresan (1989), Constantinides (1990) and Campbell and Cochrane (1999)), and to

be coupled with asset pricing theory (Abel (1990) and Schroder and Skiadas (2002)). More

recently, linear habit persistence has been well studied in optimal investment and consumption

problem (Englezos and Karatzas (2009), Yu (2012), Yu (2015) and references therein), as well as

the target-based insurance management (He, Liang, and Yuan (2020)). Therefore, it would be

interesting to investigate how this persistence influences the retirement decision, consumption

and investment.

To study the influence, we provide the retirement boundary in terms of state triplet (x, h, w)

which represent wealth, habit and wage, respectively. Because we consider a finite mandatory

retirement time, the StopCP is formulated in a finite time horizon. There come two main

difficulties in our model. On the one hand, we must establish dual relation to eliminate the

mixed structure of StopCP, which becomes more challenging when habit persistence and finite

mandatory retirement time are considered. This is because the problem we consider is formulated

RETIREMENT DECISION AND OPTIMAL CONSUMPTION-INVESTMENT 3

in a finite horizon, and the dual free boundary problem does not admit explicit form. In existing

literature concerning infinite horizon problem such as Jeanblanc, Lakner, and Kadam (2004),

Choi and Koo (2005), Farhi and Panageas (2007), Choi, Shim, and Shin (2008), Dybvig and Liu

(2010), Liang, Peng, and Guo (2014), the derived variational inequalities are expected to have

closed-form solutions. Therefore, direct verification procedures are sufficient to resolve those

problems, and the proofs of the dual relation are not necessary. To find the dual relation in our

problem, we need a novel habit reduction method involving stopping times (see Lemmas 3.3 and

3.4). The dual relation is then established by investigating the continuity of stopping time with

respect to the dual state variables. Finally we derive an integral equation representation of the

dual retirement boundary. On the other hand, the analysis of dual problem becomes complicated

because of the introduction of habit formation and an additional state variable w. Unlike Yang

and Koo (2018), the dual problem is two dimensional, and a change of measure is applied to

reduce the dimension. Moreover, we introduce the habit persistence and the time-dependent

leisure preference, rather than a negative linear constant penalty as in Yang and Koo (2018). As

such, the derived obstacle-type free boundary problem has an inhomogeneous term which evolves

with time (see (4.8)), and we find an approach to bypass the monotonicity in time. Furthermore,

in our setting, to apply comparison principles, we need growth estimates, which are accomplished

by choosing proper auxiliary functions (see Appendix C).

In this paper we resolve aforementioned challenges and make the following contributions to the

research of early retirement: First, we establish dual theorem in a more complicated model for

general utility. The dual relation is established by investigating the continuity of optimal stopping

times with respect to dual state variable, and a novel habit reduction method involving stopping

times is proposed and used. Second, for power utility we present the retirement boundaries and

optimal strategies by so-called “de facto wealth” and wage level. To the best of our knowledge, we

are the first to give retirement boundaries in multi-dimensional problems with habit persistence.

Although Chen, Hentschel, and Xu (2018) adopt habit formation in retirement problem, the

retirement time is pre-commited and deterministic, and they do not give information about the

wealth and habit thresholds of retirement. Third, we find that the amount x − pT (t)h, called

de facto wealth, has clear economic interpretation and appears as threshold for retirement. It is

defined by the nominal wealth the agent holds, minus the wealth that he can not actually utilize.

In some existing literature where similar quantity has been defined (see Englezos and Karatzas

(2009), Chen, Hentschel, and Xu (2018) and Yang and Yu (2019)), it seems that they do not

propose its formal definition and emphasize its economic meaning. Besides, they do not consider

retirement decision-making for the agent.


Based on the theoretical and numerical results of this paper, we can provide an answer to

the following important questions: how much wealth is enough for the agent to choose early

retirement, and how does this critical wealth level change with different standard of life? The

results in this paper indicate that they can be (at least partially) answered by investigating de

facto wealth x − pT (t)h, which is allowed to be negative (see Remark 3). We prove that if it

is far too low, say x − pT (t)h ≤ −q(t)w, the admissible set is empty. The case x − pT (t)h > 0

means that he actually has some wealth that he can use freely, and this must be satisfied once he

retires (cf. Yu (2015), Yang and Yu (2019)). The retirement boundary is expressed by a linear

relation among wealth, habit and wage, which evolves with time. If x− pT (t)h is large enough,

that is, x − pT (t)h ≥ G∗(t)w, it is optimal for him to retire (see Proposition 5.4). In addition,

we find that, the introduction of habit formation will enlarge the working region of the agent,

which is to say, one may work longer than usual if habit is considered. Moreover, the more he

is influenced by his habit, the longer time he may choose to work. The numerical results show

that the “retirement consumption puzzle” and the “saving for retirement” are implicated in our

financial model.

The rest of this paper is organized as follows. In Section 2 we set up the model and characterize

the admissible set of strategies as well as the allowed region of the state variables, and describe

our main problem. Section 3 establishes the dual theorem in a general setting. In Section 4 we

present a specific example, and rigorous results about dual retirement boundary are obtained. In

Section 5 we obtain the results of retirement boundary in primal coordinate axis and provide a

semi-explicit form of optimal consumption and portfolio. Section 6 concludes the paper. Detailed

proofs are in the appendices.

2. The Financial Model and Optimization Problem

2.1. The financial market and labour income. Consider a complete filtered probability

space (Ω,F ,F,P), where the filtration F = Ft0≤t≤T satisfies usual conditions with FT = F .

T <∞ is the maximal surviving time from the initial time t = 0, and [0, T ] is the time horizon

within which the agent can adjust the strategies.

We suppose that the financial market contains one risk-free asset (bond) S0 = S0(t) : 0 ≤t ≤ T and one risky asset (stock) S1 = S1(t) : 0 ≤ t ≤ T. For simplicity, we assume that the

prices of these two assets are formulated in the Black-Sholes financial market

dS0(t) = rS0(t)dt, S0(0) = s0,

dS1(t) = µS1(t)dt+ σS1(t)dB(t), S1(0) = s1,


where r, µ, σ, s0 and s1 are all positive constants. B = B(t) : 0 ≤ t ≤ T is a standard

Brownian motion on (Ω,F ,F,P) representing market risk. The financial market is complete and

we define as usual the exponential martingale and the pricing density ξ = ξ(t) : 0 ≤ t ≤ T by

Z(t) , exp−κB(t)− 1

2κ2t,

ξ(t) ,Z(t)

S0(t)= exp−κB(t)− (

1

2κ2 + r)t,

where κ = µ−rσ is the market price of risk. The risk-neutral measure in the financial market is

given by Q with dQdP∣∣FT

= Z(T ). Based on Girsanov’s Theorem, B0 = B0(t) = B(t) + κt : t ∈[0, T ] is a standard Brownian motion on (Ω,F ,F,Q).

The agent invests in the financial market with a continuous stochastic labour income before

retirement. We follow Chen, Hentschel, and Xu (2018) to introduce the model of the labour

income. The wage rate is assumed to be a geometric Brownian motion with risk perfectly

correlated to the financial market. The closed form of the wage rate W = W(t) : 0 ≤ t ≤ T is

W(t) , e(µw− 12σ2w)t+σwB(t),

where µw and σw are positive constants representing increasing rate and volatility, respectively.

Throughout this paper, we considerWt,w(·) = wW(·)/W(t) for some wage rate at present w > 0.

For the agent, there is a mandatory retirement time T1 ∈ [0, T ]. We assume that the agent has

an opportunity to choose the retirement time τ ∈ [0, T1]. After retirement, the agent has no

labour income while more leisure gains. The present value of the future labour income is

given by

b(s) =1

ξ(s)Es[∫ T1

sξ(u)Wt,w(u)du] =Wt,w(s)q(s), t ∈ [s, T1]. (2.1)

The expectation operator is defined by Et(·) = E[·|Ft] and q(·) is a deterministic function. In

Dybvig and Liu (2010), Dybvig and Liu (2011), Chen, Hentschel, and Xu (2018) and references

therein, a similar b is defined, while in some papers it is defined by the total discounted human

capital up to τ , not T1. We note that in Choi, Shim, and Shin (2008) and Farhi and Panageas

(2007), under the constant wage rate model, it has the form wlr . In Yang and Koo (2018), it is

g(s) = −ρr (ers−rT1−1)Is<T1 with the constant wage rate ρ. In particular, we have b(t) = q(t)w,

and q(·) can be expressed explicitly

q(s) =

eϑ(T1−s) − 1

ϑ, ϑ 6= 0,

T1 − s, ϑ = 0,

(2.2)

with ϑ , −r + µw − κσw. For theoretical reasons, we assume that

Assumption 1. ϑ < 0.


This assumption requires that µw−rσw

< κ, i.e., the wage rate can not possess log-increase rate

too much. Intuitively, the agent may choose to work the whole life in the case of high increasing

rate of wage. Then τ = T1, and the problem is a standard stochastic control problem, which is

well-studied. Dybvig and Liu (2011) also introduce the same requirement.

2.2. The admissible set of consumption, investment and retirement decision. The

agent chooses the retirement time within [0, T1], and we define

ST1t , τ : τ is an F− stopping time, and t ≤ τ ≤ T1 almost surely.

At time t, the agent we study will choose a stopping time τ ∈ ST1t as retirement time before the

mandatory retirement time T1 < T . The agent is faced with an optimal portfolio problem after

retirement decision is made. The agent will invest in the financial market and consume to satisfy

own needs within [0, T ]. When the consumption rate c = c(s) : t ≤ s ≤ T and the wealth

allocated to risky asset π = π(s) : t ≤ s ≤ T are determined, the wealth process satisfies the

following stochastic differential equation (abbr. SDE):dX(s) = (rX(s) + π(s)(µ− r))ds− c(s)ds+W(s)I0≤s≤τds+ σπ(s)dB(s),

X(t) = x.(2.3)

The agent expects to maintain the standard of living during the working and retirement periods,

and has habit persistence. We formulate the standard of living, or the consumption habit level,

as the weighted average of the consumption in the past time, plus a decaying term. To be precise,

we define the habit level ht,c,h for t ∈ [0, T ] and h > 0 asdht,c,h(s) = (αc(s)− βht,c,h(s))ds,

ht,c,h(t) = h,(2.4)

where α and β are positive constants. The goal of the agent is to determine the optimal in-

vestment and consumption strategies and the retirement time. The strategies are restricted in

the admissible set A(t, x, h, w) for (t, x, h, w) ∈ [0, T1]×R×R2+ defined below. Throughout this

paper, whenever there is no confusion, we will drop all the superscripts.

Definition 2.1. For fixed (t, x, h, w) ∈ [0, T ]× R× R2+, (τ, c, π) ∈ A(t, x, h, w) if and only if

(1) τ ∈ ST1t .

(2) c and π are progressively measurable with respect to F.

(3) (2.3) has a unique (strong) solution, denoted by Xτ,c,π,t,x, with X(s) + b(s)Is<τ ≥ 0 for

s ∈ [t, T ], where b is defined in (2.1). We do not consider bequest motives and assume that

X(T ) = 0.

(4)

X(τ)− pT (τ)h(τ) > 0, almost surely, (2.5)


where pT (·) will be defined in (3.8). pT (·) represents cost of subsistence consumption per

unit of habit.

(5)

c(s)− h(s) ≥ 0, almost surely. (2.6)

(6) ∫ T

tπ2(s)ds <∞,

∫ T

tc(s)ds <∞, almost surely.

Remark 1. The constraint X(t) ≥ −b(t)It<τ means that the agent is allowed to borrow money

and hold a negative wealth level, while the amount of money he borrows can not exceed the total

value of future labour income. Once the agent retires, he is prohibited from any borrowing.

Remark 2. In (4) it is required that X(τ)− pT (τ)h(τ) > 0. Here pT (·) represents the “cost of

subsistence consumption per unit of habit”, as stated in Assumption 2.1 of Yang and Yu

(2019). We emphasize that in the retirement setup, we use the subscript T or τ to distinguish

whether it is calculated up to the retirement τ , or the terminal time T . The requirement

X(τ)− pT (τ)h(τ) > 0 is the necessary condition to ensure the solvability of the control problem

after retirement (where there is no wage income and no retirement option). Similar arguments

are shown in Lemma B.1 of Yang and Yu (2019), Lemma 3.3 of Yu (2015) and Theorem 5.5 of

Englezos and Karatzas (2009). For detailed discussions, refer to Subsection 2.4 and Section 3.

Remark 3. Our optimal problem contains three variables. By the form of x − pT (t)h, the

dimension is reduced. Later we will see that in the expressions of value function or strategies,

(x, h) is usually united to x − pT (t)h. This motivates us to call this variable as “de facto

wealth” of the agent. This reveals that, the same amount of wealth may have different de facto

value for agent with different standard of life. For example, $2 for people with higher standard

of life may have equal value as $1 for people with lower standard of life. (4) requires that de

facto wealth must be positive at retirement.

Remark 4. The requirement in (2.6) is called “addictive” as the consumption is restricted to

be always beyond the standard of living, see Yu (2015). Define c , c − h. Obviously, h(·) can

also be derived from c

dh(t) = (αc(t) + (α− β)h(t))dt. (2.7)

The agent searches the optimal strategies in the admissible set. We define the set of all “addictive

consumption” plans, i.e., all c = c(s) : t ≤ s ≤ T that satisfy (2), (5) and (6), as UTt = UTt (h).

Also, define the set of π = π(s) : t ≤ s ≤ T that satisfy (2) and (6), as VTt .


2.3. Utility functions and preference change. After retirement, the agent loses labour in-

come while achieves more leisure gains. To distinguish the preferences between working and

retirement periods, we define the preferences as follows: Ui(t, x), i = 1, 2 are strictly concave

utility functions of x satisfying the following conditions:

Assumption 2. (1) domUi = [0, T ]× R+, and Ui ∈ C1,2([0, T ]× R+).

(2) ∂xUi(t,+∞) = 0, ∂xUi(t, 0+) = +∞, uniformly in t.

(3) There exist constants K, C > 0 such that

IUi(t, x) ≤ C(1 + x−K), ∀x > 0, (2.8)

where Iφ = (φ′)−1 for any strictly concave function φ here and throughout this paper.

For convenience, we denote U(u, v, x) = U1(u, x)Iu≤v + U2(u, x)Iu>v. U1 represents the

utility before retirement, and U2 is the utility after retirement. It can be directly verified that

most famous and often-studied utility functions satisfy all the assumptions here, including those

in the literature listed above. If U1 < U2, the agent has utility losses after retirement and will

choose to retire at the mandatary retirement time T1, which will be shown in Section 4.

2.4. The allowed region of state variables. In this paper, we consider a stochastic labour

income and habit formation. Therefore, in our system, we have three state variables: current

wealth, habit level and labour income. Before presenting and solving our optimization problem,

we briefly discuss and define the allowed region of the state triplet (x, h, w). Define

Gt , (x, h, w) ∈ R× R2+ : A(t, x, h, w) 6= ∅. (2.9)

Gt is called the allowed region of state variables. For fixed level of wealth w, we denote Gwt =

(x, h) : (x, h, w) ∈ Gt. After the strategies are chosen, (X(s), h(s),W(s)) should be contained

in Gs for s ≥ t.We first consider the agent who retires immediately at t. Plugging τ = t into (2.5), we obtain

x− pT (t)h > 0. Thus we define

Dt , (x, h) : x > 0, h > 0, x− pT (t)h > 0 (2.10)

as the set of positive wealth, habit and de facto wealth. Dt is a natural candidate for Gwt . How-

ever, it is not clear whether Gwt = Dt holds for any w. But it is indeed true that Dt ⊂ Gwtfor any w, because if (x, h) ∈ Dt then A(t, x, h) 6= ∅, where A(t, x, h) = (c, π) : (t, c, π) ∈A(t, x, h, w) for some w > 0 (A(t, x, h, w) dose not depend on w > 0 when τ = t in the defini-

tion, cf. Yang and Yu (2019), Yu (2015) and Englezos and Karatzas (2009)). Moreover, we will

see in Section 3 that when (x, h) ∈ Dt, the dual relation also holds. If (x, h) /∈ Dt, we conclude

that the agent never considers retiring now as an option and will continue working. As such, if


(x, h) ∈ Gwt \Dt, it is in the continuation region (see Section 5 for the definition). In order to

characterize the stopping boundary and show when to retire, we only need to investigate the set

(x, h) ∈ Dt. Besides, in Remark 5 we will prove that Gwt has an upper bound.

Finally, we formulate the preference of the agent with retirement choice, consumption and in-

vestment strategies. The agent needs to search the optimal strategies to maximize the preference

of the whole life. The optimization problem of the agent is a combination of optimal stopping

problem and stochastic control problem. In order to maintain the standard of life, the agent

cares about the consumption over habit.

We define the objective function for the agent as follows:

J(t, x, h, w; τ, c, π),

E[∫ T

te−ρ(s−t)U

(s, τ, c(s)−h(s)

)ds

], if E

[∫ T

tU−(s, τ, c(s)−h(s)

)ds

]<∞,

−∞, otherwise,

where U− = max−U, 0. Then the agent primarily wants to solve the following optimization

problem:

V (t, x, h, w) = sup(τ,c,π)∈A(t,x,h,w)

J(t, x, h, w; τ, c, π). (2.11)

Problem (2.11) contains optimal stopping and stochastic control, which is not easy to be solved.

We will first transform Problem (2.11) into a regular form in Section 3. Then we will formulate

the dual relation between primal value function and a value function of a pure optimal stopping

problem. Value function of the optimal stopping problem will be characterized by an obstacle-

type free boundary problem in Section 4. The optimal retirement time will then be given by the

first entrance time of a so-called retirement (stopping) region of the state triplet (x, h, w). The

dual relation relies on a novel habit reduction method and the continuity of optimal stopping

time.

3. Transformation of the Original Problem

Problem (2.11) is not a standard form of an optimal stopping problem. In this section, we

transform Problem (2.11) into an equivalent problem without strategies after retirement. We

call the equivalent problem a regular form because it is composed of an integrated utility over

the horizon [0, τ ] and a distorted terminal utility at τ . Similar regular form has been studied

in Karatzas and Wang (2000) without habit and wage. This regular form also has three state

variables and two control variables. After establishing the regular form, we first transform

the problem with control variables (c, π) to an equivalent one with control variable c only by

martingale method. Second, the problem contains an objective function depending pathwisely

on c. We reduce it to an equivalent problem with a control variable c and an objective function

that only depends on the value of c(t) at each time t ∈ [0, T ]. Finally, we establish the dual


problem, and disentangle it by a variational inequality and a parabolic equation. We see that

in dual coordinate system, the derived variational inequality and parabolic equation are both

linear.

3.1. Regular form of Problem (2.11). In order to represent the integrated utility after re-

tirement in a simple form, we first consider an auxiliary problem, which is a standard optimal

consumption-portfolio problem with habit formation while without any labour income or retire-

ment choice. Define the following distorted utility function

U(t, x, h) = sup(c,π)∈A(t,x,h)

E[∫ T

te−ρ(s−t)U2

(s, c(s)− h(s)

)ds

]. (3.1)

If there is no voluntary retirement choice (or retirement time is chosen before t) and no labour

income, Problem (2.11) degenerates to Problem (3.1). Using the distorted function U(t, x, h),

we transform the original problem into a regular form without strategies after retirement. The

following results ensure the equivalence of these two problems and present the connection between

U(t, x, h) and strategies after retirement. Similar arguments can refer to Park and Jeon (2019),

Liang, Peng, and Guo (2014), Choi, Shim, and Shin (2008) and Jeanblanc, Lakner, and Kadam

(2004), etc. The proofs are similar to those in the previously mentioned literature, and we omit

them.

Theorem 3.1. Suppose that (x, h, w) ∈ Dt ×R+, t ∈ [0, T1]. Denote U− = max−U , 0. Then:

i) For any τ ∈ ST1t and Fτ -measurable random variables B and C such that (B(ω), C(ω)) ∈Dτ(ω) for almost sure ω ∈ Ω as well as

E[U−(τ,B,C)] < +∞,

there exists a strategy (c, π) ∈ UT1t × VT1t such that the solution X(c,π) of SDE (2.3) with

initial condition X(τ) = B and the solution hc of (2.4) with initial condition h(τ) = C,

satisfy

X(c,π)(s) ≥ 0, ∀s ∈ [τ, T ],

and

E[∫ T

τe−ρ(s−t)U2(s, c(s)− hc(s))ds

]= E

[e−ρ(τ−t)U(τ,B,C)

]. (3.2)

ii) For V defined in Problem (2.11), we have

V (t, x, h, w)

= sup(τ,c,π)∈A(t,x,h,w)

E[∫ τ

te−ρ(s−t)U1(s, c(s)−h(s))ds+e−ρ(τ−t)U(τ,X(τ), h(τ))

].

(3.3)


Eq. (3.2) establishes the connection between the utility function U(t, x, h) and strategies after

retirement. It has an intuitive interpretation that the agent regards the preference after retire-

ment as a standard stochastic control problem because of Markovian property. (3.3) shows that

the original problem is equivalent to the regular form as in Karatzas and Wang (2000). Problem

(2.11) with retirement choice and strategies in the whole life is equivalent to the optimization

problem with retirement choice, strategies before retirement and claims at retirement. By (3.2),

the claims at retirement are correlated with strategies after retirement. We obtain a regular

form here, but the optimization problem is still very complex compared to Karatzas and Wang

(2000). There exist three state variables (x, h, w) here and the involvement of the habit also

brings in difficulties. Next, using habit reduction method and dual method, we solve Problem

(3.3) to derive the strategies for the agent.

3.2. The budget constraints and replication. Before applying dual method, we present the

budget constraints in our financial system. We first define the processes ξt(·) , ξ(·)/ξ(t) and

St0(s) , S0(s)/S0(t). In order to transform the original problem with two control variables (c, π)

into an equivalent one with only one control variable c, we consider the set

B(t, x, h, w) , (τ, c) : ∃ π s.t. (τ, c, π) ∈ A(t, x, h, w).

For simplicity, we drop the index (t, x, h, w) and write B = B(t, x, h, w) if there is no confusion.

Based on standard local martingale argument together with optional stopping theorem ( cf.

Karatzas and Wang (2000)), we get the following budget constraint:

E[

ξt(τ)B

wealth left at retirement

+

∫ τ

tc(u)ξt(u)du

wealth to consume

]≤ x

current wealth

+ b(t)

future labour income

, (3.4)

where B = X(τ) + b(τ). Conversely, for any (τ, c) ∈ ST1t × UTt (for definition of UTt , see Remark

4) and Fτ -measurable random variable B > b(τ) + pT (τ)h(τ) a.s., such that

E[ξt(τ)B +

∫ τ

tc(u)ξt(u)ds

]= x+ b(t), (3.5)

we can find some π such that the triple (τ, c, π) ∈ A based on replication. We introduce the

following notations

B≤ , (τ, c) ∈ ST1t × UTt : There exists Fτ measurable, nonnegative random variable

B > b(τ) + pT (τ)h(τ) a.s., such that (3.4) holds,

B= , (τ, c) ∈ ST1t × UTt : There exists Fτ measurable, nonnegative random variable

B > b(τ) + pT (τ)h(τ) a.s., such that (3.5) holds.

Then we have the following proposition, which shows the relationship between these three sets:


Proposition 3.2.

B= ⊂ B ⊂ B≤.

Besides, for any (τ, c) ∈ B=, there exists a strategy π such that (τ, c, π) ∈ A and

X(s) + b(s) =1

ξt(s)Es[ξt(τ)B +

∫ τ

sξt(u)c(u)du]. (3.6)

Proof. It is straightforward and similar to the proof of Lemma 6.3 of Karatzas and Wang (2000)

with minor modifications, hence omitted.

Proposition 3.2 shows that we only need to search the optimal (τ, c) first. The optimal invest-

ment strategy can be derived by replication. As noted in the expression of budget constraint,

B≤ is the set of all admissible consumption plans whose discounted value does not exceed the

total wealth (cash wealth x plus future labour income b(t)) of the agent).

3.3. Habit reduction method. To characterize the standard of life, habit persistence is in-

troduced as (2.4), which increases the dimension of our financial system. The preference of the

agent also depends on the past consumption path. By the relation of c and c, we can reduce

the original problem to one without habit formation and the path-dependent property. For this

habit reduction framework, see Schroder and Skiadas (2002), Englezos and Karatzas (2009),

Chen, Hentschel, and Xu (2018) and references therein. The core of such transformation is the

relationship between c and H c , c = c−h. The following lemma presents the relationship with

the stopping time τ considered.

Lemma 3.3. For any (τ, c) ∈ ST1t × UTt , and s ∈ [t, τ ], we have

Es[∫ τ

sc(u)ξt(u)du

]= Es

[∫ τ

sc(u)ξτ,t(u)du

]+ h(s)ξt(s)pτ (s), (3.7)

where m(s) = e(α−β)s,

pτ (s) = Es[∫ τsm(u)m(s)

ξt(u)ξt(s) du

],

µτ (s) = 1 + αpτ (s),

ξt,τ (s) = ξt(s) + αEs[∫ τsm(u)m(s) ξ

t(u)du]

= µτ (s)ξt(s).

(3.8)

Proof. See Appendix A.

Lemma 3.3 shows that the integrand in the budget constraint of (3.4) can be replaced by c.

As the agent only cares about the consumption over habit, i.e., c = c − h, we can reduce the


problem to one with control c while without habit h. In particular, if we choose (τ, c) ∈ B≤ and

take s = t in (3.7), the budget constraint (3.4) becomes

E

ξt(τ)(X(τ) + b(τ))

wealth left at retirement

+

∫ τ

tc(u)µτ (u)ξt(u)du

wealth to consume

+ hpτ (t)

wealth needed to satisfy habit

≤ xcurrent wealth

+ b(t)

future labour income

.

(3.9)

We now transform our problem into an equivalent one w.r.t. control c. Define the mapping

H : c 7→ c and the set UTt ,H UTt . First, we define B≤ as the set of all (τ, c) ∈ ST1t ×UTt satisfying

(3.9), with X(τ) + b(τ) replaced by some Fτ measurable random variable B ≥ b(τ) + pT (τ)h(τ)

a.s., and define B= as those (τ, c) satisfying the equality in (3.9). Then we have the natural

inclusions

B= = H B= ⊂ H B ⊂ H B≤ ⊂ B≤,

where the mappings H : c 7→ c and H : (τ, c) 7→ (τ, c = H c) are both linear. We see that it

has a natural inverse map H −1 : c 7→ c.

In Lemma 3.3, when replacing c by c, we have an additional multiplier µτ and an additional

term pτ . Because these two additional terms both depend on τ , the dual problem is hard to be

derived and solved. In order to establish the dual problem, we need to calculate these terms in

more concrete forms. We have the following lemma.

Lemma 3.4. Denote UT1t ,H UT1t . For any (τ, c) ∈ ST1t × UT1t , and s ∈ [t, τ ] we have

Es[∫ τ

sξt(u)µτ (u)c(u)du+ h(s)pτ (s)

]= Es

[∫ τ

sξt(u)µT (u)c(u)du− h(τ)ξt(τ)pT (τ)

]+ Es

[ξt(s)h(s)pT (s)

],

(3.10)

where µT and pT are defined as in Lemma 3.3 with τ replaced by T , respectively.

Proof. See Appendix A.

Remark 5. We can present an upper bound for the allowed region Gt as follows. For definition

of Gt, refer to (2.9). Define

Et = (x, h, w) : x− pT (t)h+ q(t)w > 0, (3.11)

Ewt = (x, h) : (x, h, w) ∈ Et. (3.12)

We claim that Gt ⊂ Et. In fact, combining (3.9) and (3.10), we know that if (x, h, w) ∈ Gt, then

x− pT (t)h+ q(t)w ≥ E[ξt(τ)(X(τ)− pT (τ)h(τ))] > 0


holds for any (τ, c) ∈ B(t, x, h, w) ⊂ B≤(t, x, h, w). Therefore, Gt ⊂ Et. From the proof of

Theorem 3.5 we know that if some further information of τy is available, the lower bound of Gtwill be tighter. If, in addition, we can prove limy→∞ Eb(τy)ξt(τy) = 0, the equality Gt = Et holds.

This upper bound requires that the agent can not let his de facto wealth fall below the negative

value of his future labour income.

We have replaced the budget constraint w.r.t. c by c. The linearity of the budget constraint

ensures the existence of the dual problem. The next subsection will be dedicated to solving

Problem (3.3) based on dual approach.

3.4. The dual problem. Problem (3.3) is still a combination of optimal stopping and stochastic

control problems. In order to separate the stopping time and stochastic controls, we first define

the dual functions of “running utility” U1 and “terminal utility” U , respectively. By the dual

approach, Problem (3.3) can be transformed to a pure optimal stopping problem (see Park and

Jeon (2019), Yang and Koo (2018), Liang, Peng, and Guo (2014), Choi, Shim, and Shin (2008)

or Karatzas and Wang (2000) for such a transformation). After obtaining the optimal stopping

problem, we construct the corresponding variational inequality. Then we derive the retirement

boundary in terms of the dual variables. As such, the relationship between the dual variables

and primal variables induces the retirement boundary w.r.t. the primal variables. Moreover, in

order to obtain the properties of U and its dual function, a parabolic equation will be established.

At last, the verification theorem will be stated and proved in Section 4 based on the optimal

strategies in the dual variables, and the properties of optimal strategies in the original variables

will be listed and proved in Section 5.

Define the dual variable Y t,y(s) = yeρ(s−t)ξt(s), and observe that for y > 0, using Lemmas 3.3

and 3.4 we have

V (t, x, h, w)−y(x+ b(t))

≤ sup(τ,c)∈B≤

E[e−ρ(τ−t)U(τ,X(τ), h(τ))+

∫ τ

te−ρ(u−t)U(u, c(u))du

]−yE

[ξt(τ)(X(τ)+b(τ))+

∫ τ

tc(u)µτ (u)ξt(u)du+ hpτ (t)

]= sup

(τ,c)∈B≤E[e−ρ(τ−t) (U(τ,X(τ), h(τ))−Y t,y(τ)(X(τ)+b(τ)−pT (τ)h(τ))

)]+E

[∫ τ

te−ρ(u−t) (U(u, c(u))−Y t,y(u)c(u)µT (u)

)du

]−hpT (t)

≤ supτ∈ST1t

E[e−ρ(τ−t)V (τ, Y t,y(τ), wt,w(τ))+

∫ τ

te−ρ(u−t)V1(u, Y t,y(u))du

]−hpT (t), (3.13)


where

V (t, y, w) , sup(x,h)∈Dt

U(t, x, h)− y(x− pT (t)h) − yq(t)w, (3.14)

V1(t, y) , supc>0U1(t, c)− yµT (t)c. (3.15)

As such, if we introduce the following optimal stopping problem

W (t, y, w) = supτ∈ST1t

E[e−ρ(τ−t)V (τ, Y t,y(τ), wt,w(τ))+

∫ τ


]

= supτ∈ST1t

Et,y,w[e−ρ(τ−t)V (τ, Y (τ),W(τ)) +

∫ τ

te−ρ(u−t)V1(u, Y (u))du

], (3.16)

we have that ∀t ∈ [0, T1], (x, h) ∈ Dt, y > 0 and w > 0,

V (t, x, h, w)− y(x+ q(t)w − pT (t)h) ≤W (t, y, w). (3.17)

Thus, if we establish the relationship between Problem (3.3) and the dual problem (3.16), the

dual problem contains no controls and is a pure optimal stopping problem, which is easier to be

solved. In fact, we have the following dual theorem to guarantee the equivalence of these two

problems.

Theorem 3.5. Let t ∈ [0, T1), w > 0 and (x, h) ∈ Dt be fixed. Suppose that the optimal stopping

time τy = τt,y,w of Problem (3.16) can be chosen such that τy+ε → τy almost surely as ε → 0±,

then the value function of Problem (2.11) satisfies

V (t, x, h, w) = infy>0W (t, y, w) + y(x− pT (t)h) + yq(t)w. (3.18)

Proof. See Appendix B.

Theorem 3.5 proposes the framework for dual argument with endogenous habit persistence

and general utility. In practice, the conditions are usually verified depending on the form of the

problem (see Section 5 for example). Besides, our argument shows that only the continuity of τy

is needed to establish the dual relation, which remains true under some mild regularities of the

boundary (see the proof of Lemma 5.1).

3.5. Properties of the post-retirement value function U . In Problem (3.3), there exists a

utility function U , which is related to the integral of U2 after retirement. We observe in Problem

(3.16) that we need to introduce the dual function V of the function U . In order to solve the dual

problem, we briefly discuss the properties of U . We list them in the following several lemmas,

of which all proofs are omitted, because they are all classical results and can be found in, e.g.,

Englezos and Karatzas (2009) or Yu (2015).


Lemma 3.6. For t ∈ [0, T1) and (x, h) ∈ Dt, we have

U(t, x, h) = U(t, x− pT (t)h), (3.19)

where U(t, ·) is strictly concave for any fixed t.

Lemma 3.6 shows that the integrated utility after retirement relies on the de facto wealth

x − pT (t)h, and the dimension of state space has been essentially reduced by one. We will see

that the optimal strategies of the agent depend on x, h through x− pT (t)h. By habit reduction

method, Problem (3.1) can be transformed to a standard optimal consumption problem without

habit formation. As such, we apply the dual approach to solve the standard problem. We have

the following lemma of the standard optimal consumption problem.

Lemma 3.7. Define

V (t, y) , E[∫ T


], (3.20)

where

V2(t, y) = supc>0U2(t, c)− yµT (t)c.

Then we have the following dual relation:

V (t, y) = supz>0U(t, z)− yz, (3.21)

U(t, z) = infy>0V (t, y) + yz. (3.22)

Lemma 3.7 shows that V2(t, y) can be derived by the dual relation with U2(t, c). We rely on

the dual relation with V to obtain U . However, V is unknown, but (3.20) establishes the relation

between V and V2. Using Feynman-Kac’s formula, we can obtain the following lemma to get V

from V2.

Lemma 3.8. Let NT , [0, T )× R+ and N T , [0, T ]× R+. Define

L , y(ρ− r)∂y +1

2y2κ2∂yy − ρ.

Then we have

i) V ∈ C1,2(NT ) ∩ C(NT ), and(−∂t − L )V = V2, in NT ,

V (T, ·) = 0.(3.23)

ii) For any fixed t, V (t, y) is strictly convex and strictly decreasing with respect to y. In

addition, there exist constants C and K such that

|∂yV | ≤ C(1 + y−K), ∀(t, y) ∈ NT . (3.24)


Remark 6. In fact, from the proof of Lemma 3.8 we know that for any other measure P and

corresponding expectation operator E, if we have the expression

V (t, z) , E[∫ T

te−ρ(u−t)V2(u, Zt,z(u))du

],

where Z is any diffusion process with drift term µZ and diffusion term σZ under P, then(−∂t −LZ)V = V2, in NT ,

V (T, ·) = 0,(3.25)

with

LZ = zµz∂z +1

2z2∂zz − ρ.

Remark 7. In this section we know that V (t, y, w) = V (t, y)− q(t)yw and V is strictly convex

w.r.t. y. Thus, using the same arguments as in Lemma 8.1 of Karatzas and Wang (2000), we

can prove that W is strictly convex w.r.t. y, which is a necessary condition in Section 5.

4. Retirement Boundary in Terms of Dual Variables

Section 3 presents the theoretical results of the StopCP. In order to obtain the forms of the

retirement boundary and optimal strategies, we specify the preferences by the CRRA utility

function and a time-dependent leisure model. Our analysis relies highly on the homogeneous

structure of the utility function, otherwise the retirement boundary will be rather difficult to

describe. Unlike Yang and Koo (2018), the primal problem contains three variables and the dual

problem contains two variables. As such, more general analysis with other preferences may be

impossible. We consider the following preferences before and after retirement:

U1(t, x) =x1−γ

1− γ, (4.1)

U2(t, x) = K(t)x1−γ

1− γ, (4.2)

where γ > 0, γ 6= 1 is the risk aversion of the agent and K(·) > 0 is a smooth function

which represents the leisure preference. As the retiree always has less leisure gains as time goes

by, we consider a time dependent leisure preference function here. Especially in the case that

K(t)1−γ ≤ 1, i.e., U1 ≥ U2, which means that the agent has leisure losses after retirement, he

will never choose to retirement until mandatory retirement time.

4.1. Measure change. To characterize the retirement boundary in dual variables, we need

to solve the dual optimal stopping problem (3.16) without controls. However, Problem (3.16)

is two dimensional and the infinitesimal operator of Markov process (Y (·), w(·)) is degenerate,

which bring twofold difficulties. First, establishing the existence, uniqueness and regularity of


the associated differential equations and obstacle-type free boundary problems is quite difficult

(although there are indeed some classical results in the field of partial differential equations).

Second, the retirement boundary naturally depends on the pair (y, w). We can hardly guess

and describe the retirement boundary with two state variables. In fact, when there is only

one variable, we can simply conjecture that the stopping region has the form z ≤ z∗(t) or

z ≥ z∗(t) (where z is the reduced variable) and then verify it. In order to obtain the retirement

boundary, we apply a measure change method to reduce these two state variables (y, w) to one

variable.

Lemma 4.1. Consider the exponential martingale

E =

E(t) = exp

(−1

2(σw − κ)2t+ (σw − κ)B(t)

): 0 ≤ t ≤ T1

.

Let dP = E(T1)dP and E be the corresponding expectation operator. Then we have

W (t, y, w)/yw = w(t, z)

with

w(t, z) = supτ∈ST1t

Et,z[∫ τ

teϑ(u−t)V1(u, Z(u))du+ eϑ(τ−t)(V (τ, Z(τ))− q(τ))

], (4.3)

and the change of variable z = y1/(1−γ)wγ/(1−γ). The underlying Markov process Z is defined by

Z = Y 1/(1−γ)Wγ/(1−γ).

Proof. Combining (3.14), (3.19) and (3.21), we have

V (t, y, w) = V (t, y)− ywq(t).

Direct calculations show

Y (t)W(t) = e(ρ+ϑ)tE(t).

Then the Bayesian rules of the measure change (c.f. Lemma 3.5.3 of Karatzas and Shreve (1991))

imply that for any fixed τ ∈ ST1t ,

Et,y,w[e−ρ(τ−t)V (τ, Y (τ),W(τ))

]= Et,y,w

[e−ρ(τ−t)Y (τ)W(τ)(V (τ, Y (τ))/Y (τ)W(τ)− q(τ))

]= Et,z

[e−ρ(τ−t)+(ρ+ϑ)τE(τ)(V (τ, Z(τ))− q(τ))

]= ywEt,z

[eϑ(τ−t)(V (τ, Z(τ))− q(τ))

],

where we have used the homogeneous property

V (t, y)/yw = V (t, y(yw)−γ/(γ−1)).


Based on the same arguments, we also have

Et,y,w[∫ τ

te−ρ(u−t)V1(u, Y (u))du

]= ywEt,z

[∫ τ

teϑ(u−t)V1(u, Z(u))du

].

Combining the above calculations, the results are derived.

Lemma 4.1 shows that the dual problem (3.16) with two state variables (y, w) is equivalent

to the reduced dual problem (4.3) with one state variable z = y1/(1−γ)wγ/(1−γ). Using optimal

stopping theory, we obtain the optimal stopping time of Problem (4.3). We observe in Problem

(4.3) that the discount factor is ϑ, which initially appears in (2.2). Based on Assumption 1 we

have ϑ < 0. It is interesting to further explore this connection.

4.2. The free boundary problem of Problem (4.3). In this subsection we solve the dual

problem (4.3) by deriving the related free boundary problem. We first establish the verification

theorem for Problem (4.3), and propose some regularities that will be necessary later. The

existence of the solution of the associated free boundary problem can be verified based on the

classical theory of free boundary problems. We present two important theorems (Theorems 4.2

and 4.3) in this subsection.

Theorem 4.2. Denote NT1 , [0, T1)× R+. Suppose that there exists a function w ∈ C(NT1) ∩C1,1(NT1) ∩ C1,2(C) ∩ C1,2(NT1\C) satisfying the following conditions:

i) w ≥ V − q,ii) The open subset C = (t, z) : w > V − q ⊂ NT1 has a Lipschitz continuous boundary

∂C,

iii) Z(t) spends almost no time on ∂C, i.e.,∫ T1

tI(u,Z(u))∈∂Cdu = 0, Pt,z − almost surely, for any (t, z) ∈ NT1 , (4.4)

iv) w satisfies the growth condition:

|w(t, z)|+ |∂zw(t, z)| ≤ C(zK + z−K), ∀(t, z) ∈ NT1 , (4.5)

v) w solves the following obstacle-type free boundary problem:(−∂t −LZ)w = V1 in C,

(−∂t −LZ)w ≥ V1 in NT1\C,

w(T1, ·) = V (T1, ·)− q(T1),

(4.6)

where

LZ =σ2z

2z2∂zz + µzz∂z + ϑ,

with µz and σz representing the drift and diffusion coefficients of Z, respectively,

we have w = w in NT1 .


Proof. See Appendix C

The regularities proposed in Theorem 4.2 are sufficient and necessary. wz(t, z) is continuous

while not differentiable at z = z∗(t), which is illustrated in the numerical example of Figure 1.

More details of the parameters and the numerical method can refer to Subsection 4.4. As w

0.6 0.7 0.8 0.9 1 1.1 1.2

-35

-30

-25

-20

-15

(a) γ = 0.5

21 21.5 22 22.5 23 23.5 24 24.5 25 25.5

-1.5

-1.4

-1.3

-1.2

-1.1

-1

-0.9

(b) γ = 1.5

Figure 1. wz(t, ·) and Vz(t, ·) at time t = 8.

is not necessary to have second order derivatives on the boundary ∂C, the inequalities for w in

(4.6) hold in limit sense, i.e.,

(−∂t −LZ)w ≥ V1 in intNT1\∂C

implies that

lim inf(t′, z′) /∈ ∂C

(t′, z′) → (t, z)

(−∂t −LZ)w(t′, z′)− V1(t′, z′) ≥ 0 on ∂C.

The following theorem ensures the existence of w.

Theorem 4.3. The free boundary problem (4.6) admits a solution that satisfies all the conditions

in Theorem 4.2.

Proof. Using classical theory of free boundary problems (or variational inequalities), we can

easily obtain the regularities of w and ∂C. The derivation is similar to that Friedman (1975) and

we omit it here. In Appendix C, we prove Theorem 4.5 to ensure other properties of ∂C, the

growth condition (4.5) and other further results. (4.4) follows naturally from the properties of

∂C and the regularity of the process Z.


Now we solve the free boundary problem and show the optimal retirement time. We mainly

concentrate on the regularity property, the growth condition (4.5) and the differential inequality

(4.6) of w.

4.3. The retirement boundary of Problem (4.3). In this subsection, we characterize the

retirement boundary based on a critical value of the reduced variable z. The boundary condition

in (4.6) has two functions V and q(t). In order to eliminate these two functions, we define a new

function F (t, z) , w(t, z) − V (t, z) + q(t) and study the stopping region S = (t, z) : F = 0and the continuation region C = (t, z) : F > 0. Combining (3.25) and (4.6), F satisfies the

following obstacle-type free boundary problem:(−∂t −LZ)F = V1 − V2 + 1 in C,

(−∂t −LZ)F ≥ V1 − V2 + 1 in S,

F (T1, ·) = 0.

(4.7)

The right hand side of the above equation can be calculated explicitly

V1(t, z)− V2(t, z) =1

γ(µT (t)z)−γ − K(t)1/γ

γ(µT (t)z)−γ

= 4(t)(µT (t)z)−γ ,

where γ = (1 − γ)/γ and 4(t) = 1−K(t)1/γ

γ . Here we have used the fact (−∂t − LZ)q(t) =

−q′(t) − ϑq(t) = 1, which is obvious by the expression (2.2) of q. For convenience, we rewrite

Problem (4.7) as follows: (−∂t −LZ)F = 4(t)(µT (t)z)−γ + 1 in C,

(−∂t −LZ)F ≥ 4(t)(µT (t)z)−γ + 1 in S,

F (T1, ·) = 0.

(4.8)

The sign of 4(t) is an indicator to show whether the agent has leisure gains or losses. Precisely,

4(t) < 0 is equivalent to U1 < U2 in our problem. The following proposition shows that if

U1 ≥ U2, the agent will never retire, which is quite natural.

Proposition 4.4. If (t, z) ∈ (t, z) : 4(t) > −(µT (t)z)γ, then (t, z) ∈ C. In particular, at the

time t < T1 such that 4(t) ≥ 0, the agent will never choose to retire.

Proof. If (t, z) ∈ (t, z) : 4(t) > −(µT (t)z)γ and (t, z) ∈ intS, then, based on (4.8),

we have 4(t)(µT (t)z)−γ + 1 ≤ 0, which is a contradiction. Thus we have intS ⊂ (t, z) : 4(t) ≤−(µT (t)z)γ. As (t, z) : 4(t) ≤ −(µT (t)z)γ is a closed set, we conclude that S ⊂ (t, z) :

4(t) ≤ −(µT (t)z)γ and the proposition is proved.


Proposition 4.4 shows the retirement boundary characterized by the set (t, z) : 4(t) >

−(µT (t)z)γ. In the following, we show the retirement boundary expressed by z. For theoretical

reasons, we need the following assumption.

Assumption 3. −M ≤ 4(t) ≤ −δ < 0, ∀t ∈ [0, T1].

As K(·) is a multiplier of utility function, the boundness of 4(t) is a natural assumption,

which guarantees that Theorem 4.5 holds for the retirement boundary.

Theorem 4.5. If 0 < γ < 1, C = (t, z) : 0 ≤ t < T1, z > z∗(t). If γ > 1, C = (t, z) : 0 ≤t < T1, 0 < z < z∗(t). Here z∗(·) is a (0,∞)-valued function which is bounded above and below

from 0, and is Lipschitz continuous.

Proof. See Appendix C.

Using stochastic calculus involving local time, we obtain the following integral equation of

z∗(·).

Theorem 4.6. If γ > 1, function z∗(·) satisfies the following integral equation:

Et,z∗(t)

[∫ T1

teϑ(u−t)(1 +4(u)(µT (u)Z(u))−γ)IZ(u)<z∗(u)du

]= 0. (4.9)

If 0 < γ < 1, function z∗(·) satisfies the following integral equation:

Et,z∗(t)

[∫ T1

teϑ(u−t)(1 +4(u)(µT (u)Z(u))−γ)IZ(u)>z∗(u)du

]= 0. (4.10)

In either case, we have z∗(T1−) = (−4(T1))1/γµT (T1)−1.

Proof. See Appendix C.

Although we can not obtain the explicit form of the retirement boundary z∗, it can be solved

numerically based on Theorem 4.6.

µ σ r ρ γ α β0.05 0.22 0.01 0.01 0.5/1.5 0.3 0.5µw σw T T1 N ε0

0.01 0.1 20 21 2000 0.06

Table 1. Basic parameters in the numerical illustrations.

4.4. Numerical illustrations of the dual retirement boundary. In this subsection, we

provide numerical examples for the retirement boundary in dual variables as described in Theo-

rems 4.5 and 4.6. The basic parameters are listed in Table 1. We set a smooth leisure function


K(t) = expε0γ(T − t). We discretize the time interval and use recursive method to get the

retirement boundary. The retirement boundary is illustrated in Figure 2. Figure 2 provides a(n)

lower/upper bound for γ > 1 and 0 < γ < 1, respectively, as stated in Remark 9. Although the

retirement boundaries move in the opposite direction with time for γ > 1 and 0 < γ < 1, the

continuation region becomes larger and the retirement region becomes smaller.

0 0.5 1 1.5 2 2.5 3

0

1

2

3

4

5

6

7

8

9

10

(a) γ = 0.5

0 10 20 30 40 50 60 70 80 90

0

1

2

3

4

5

6

7

8

9

10

(b) γ = 1.5

Figure 2. Images of z∗(t).

The retirement boundaries in dual variables are shown in Figure 3. The retirement boundaries

when fixing w ≡ 1 (in one variable y only) are also plotted, see Figure 4. Unlike Yang and Koo

(2018) and other related literature, no monotonicity results can be expected for the y−retirement

boundary, even in the constant wage rate model. As we consider a time depend leisure preference

here, the agent will enjoy less leisure gains when time involves. As such, early retirement is

a better choice, and Figure 4 shows that retirement region gets smaller with time. In the

existing literature, such as Yang and Koo (2018) or Dybvig and Liu (2010), they consider time

independent leisure preference and show that the retirement boundary increases with time. Our

results are consistent with them when setting K(·) ≡ 1.1 for 0 < γ < 1. However, when γ > 1,

the retirement boundary always decreases with time under the choice K(·) ≡ 0.9.

5. Retirement Boundaries and Optimal Strategies in Terms of Primal Variables

In this section, we show the retirement boundary expressed by primal variables (x, h, w), and

provide the optimal investment and consumption strategies. The relation of the primal problem

(3.3) and dual problem (3.16) is revealed in Theorem 3.5. The validity of Theorem 3.5 will be

verified first. Based on the primal-dual relation, the retirement boundary w.r.t. (y, w) in Section

4 can be transformed to be expressed w.r.t. (x, h, w). By replication, we show the feedback


(a) γ = 0.5 (b) γ = 1.5

Figure 3. Retirement boundary in (y, w)-axis.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0

1

2

3

4

5

6

7

8

9

10

(a) γ = 0.5

0.1 0.15 0.2 0.25 0.3 0.35

0

1

2

3

4

5

6

7

8

9

10

(b) γ = 1.5

Figure 4. Retirement boundary in y-axis when w ≡ 1.

forms of optimal consumption and investment strategies, which are functions of the de facto

wealth x− pT (t)h and wage w. Numerical implications of the retirement boundary and optimal

strategies are presented. We find the discontinuities of optimal strategies at retirement boundary

intuitively.

5.1. The primal-dual relation and characterization of Gt. In this section, we present

Lemma 5.1 to show that the condition in Theorem 3.5 holds. The optimal stopping time in

terms of dual variables can also be obtained by the relation of z with (y, w).


Lemma 5.1. Let z∗ be as in Section 4, y > 0, w > 0 and t ∈ [0, T1) be fixed. Define

τy =

inft ≤ T1 : (Y t,y(s))1/(1−γ)(Wt,w(s))γ/(1−γ) ≤ z∗(s), 0 < γ < 1,

inft ≤ T1 : (Y t,y(s))1/(1−γ)(Wt,w(s))γ/(1−γ) ≥ z∗(s), γ > 1.

Then, τy is the optimal stopping time of Problem (3.16), and satisfies the condition that τy+ε → τy

almost surely as ε→ 0±.

Proof. See Appendix D.

In fact, in the case of the utility functions specified in Section 4, the allowed region Gt in

Remark 5 is equal to its upper bound Et. Thus the results in Theorem 3.5 can be extended to

the whole set Gt.

Lemma 5.2. Under the setting of Section 4, we have Gt = Et and the dual relation (3.18) holds

in Gt.


Now we characterize the retirement boundary in the primal variables (x, h, w).

Lemma 5.3. Define the primal map P as

Pt,w(y) = −Wy(t, y, w),

and define the primal and dual continuation regions, respectively, as

Cpt,w = (x, h) : (x, h, w) ∈ Gt, V > U,

Cdt,w = y > 0 : W > V − q(t)yw.

Then Cpt,w = Cpt,w , (x, h) : x− pT (t)h+ q(t)w ∈Pt,wCdt,w.


Remark 8. It follows from Remark 7 that Pt,w is surjective.

Proposition 5.4. The retirement boundary can be expressed in primal variables as (t, x, h, w) :

(x, h) ∈ ∂Cpt,w, where ∂Cpt,w = (x, h) : x− pT (t)h+ q(t)w ∈Pt,w∂Cdt,w. In particular, we have

∂Cpt,w = (x, h) : x− pT (t)h = G∗(t)w,

Cpt,w = (x, h) : x− pT (t)h < G∗(t)w,(5.1)

where G∗(t) = γH(t)z∗(t)−γ, with H defined by the expression V (t, z) = H(t)z−γ based on

(3.20).



We see that x and h appear in a whole part as de facto wealth in the retirement boundary.

This phenomenon will be found in the feedback forms of optimal strategies. Proposition 5.4 also

shows that the retirement boundary is expressed by a linear relationship among (x, h, w).

5.2. Numerical illustrations of primal retirement boundary. Proposition 5.4 establishes

the relationship between the retirement boundary in the dual and primal variables, and provides

semi-explicit form of it. As such, we can present some numerical results of the retirement

boundary.

First, we display the retirement boundary in (x, h, w)-axis at time t = 0, 5, 8, which are

illustrated in Figures 5 and 6. Because (x, h, w) has a linear relationship in Proposition 5.4, the

retirement boundaries are all planes for fixed time t. Besides, not all (x, h) are in the allowed

region. The entire side of the plane with the retirement region is contained in Dt × w > 0,thus in the allowed region. The continuation region is composed of the intersection of the other

side of the plane and the allowed region Gt. On the plane w = 0, we know Dt = G0t (see (2.10),

(2.9) and (3.11)), thus the agent does not have initial wealth and will choose to work.

Our figures are the first to describe the retirement boundary with three state variables: wage,

habit and wealth. These figures reveal much more economic implications than previous studies.

We observe from Figures 5 and 6 that the agent with higher habit level will continue to work to

maintain the standard of life. Besides, when the agent has larger wealth or less wage rate, he

will choose to retire to enjoy the leisure gains. As time goes by, the agent is expected to receive

less future labour income. Thus the importance of wage rate on the agent’s retirement decision

decreases. Figure 5 show that the retirement surface evolves closer to the (h,w) plane with time.

The critical level of wage rate that pushes one to retire becomes larger. The above results are

all consistent for 0 < γ < 1 and γ > 1.

(a) t = 0 (b) t = 5 (c) t = 8

Figure 5. Retirement boundary in (x, h, w)-axis, γ = 0.5.

Next, fixing w = 1, we illustrate the movement of the retirement boundary in (x, h)-axis with

time, which is displayed in Figure 7. An elder agent will be more likely to retire and Figure 7


(a) t = 0 (b) t = 5 (c) t = 8

Figure 6. Retirement boundary in (x, h, w)-axis, γ = 1.5.

shows that the retirement region becomes larger with time. For clarity, we show the evolution

(a) γ = 0.5 (b) γ = 1.5

Figure 7. Retirement boundary in (x, h)-axis, w = 1.

of retirement boundaries with time for h and x by fixing x = 80, w = 1 and h = 6, w = 1,

respectively. As we introduce habit formation in our work, we are mostly interested in the

effects of habit parameters on the agent’s retirement decision. The retirement boundaries for

(α, β) = (0.2, 0.4) (the benchmark), (0.1, 0.4) and (0.3, 0.4) are plotted in both cases. The results

are reported in Figures 8 and 9. Figure 8 shows that if the weight α of consumption in habit is

larger, the agent is more likely to retire even with a smaller standard of life. The importance of

habit level on the agent’s retirement decision decreases with time. Therefore, the critical value

of h is a decreasing function of time. We can conclude from the figures that, the introduction

of habit formation dramatically increases the wealth level that pushes the agent to retire, i.e.,

he needs to work longer to satisfy his higher standard of life. Moreover, the greater α, the more

this increase. (2.4) means that with a larger α, the past consumption is more important when


deciding the current standard of life. As such, the agent has to work longer if he is more addicted

to the past level of consumption. This is consistent with the findings of Figure 8, because a larger

α leads to lower critical standard of life and later retirement.

5 10 15 20 25 30

0

1

2

3

4

5

6

7

8

9

10

(a) γ = 0.5

6 8 10 12 14 16 18 20 22 24

0

1

2

3

4

5

6

7

8

9

10

(b) γ = 1.5

Figure 8. Retirement boundary in (h, t)-axis, w = 1, x = 80.

0 10 20 30 40 50 60 70

0

1

2

3

4

5

6

7

8

9

10

(a) γ = 0.5

20 30 40 50 60 70 80

0

1

2

3

4

5

6

7

8

9

10

(b) γ = 1.5

Figure 9. Retirement boundary in (x, t)-axis, w = 1, h = 6.

5.3. Optimal consumption and portfolio. In this subsection, we characterize the optimal

consumption and investment strategies. We expect to express them in feed-back forms in

(t, x, h, w). Because unlike the last subsection, we mainly focus on the behaviors of optimal

strategies in continuation region, V is not enough to derive the optimal strategies. As the finan-

cial market is complete, the strategies can be replicated based on the value of W . However, W

is only known as a semi-explicit form with z∗.


Theorem 5.5. The optimal consumption-investment strategies (c∗, π∗) can be expressed in the

following feed-back forms c∗(t) = C∗(t,X∗(t), h∗(t),W(t)) and π∗(t) = Π∗(t,X∗(t), h∗(t),W(t)),

where X∗ is solved by SDE (2.3), and h∗ is solved by ODE (2.4) after substituting π∗ and c∗ for

π and c. The functions C∗ and Π∗ which are defined on (t, x, h, w) : t ∈ [0, T1), (x, h, w) ∈ Gt,can be expressed by:

C∗(t, x, h, w) =

IU1(yµT (t)) + h, if (x, h) ∈ Cpt,w,IU2(yµT (t)) + h, otherwise,

(5.2)

Π∗(t, x, h, w) =

σ−1(−σwq(t)w − σwwWyw(t, y, w) + κyWyy(t, y, w)), if (x, h) ∈ Cpt,w,(γ + 1)σ−1κ(x− pT (t)h), otherwise,

(5.3)

where y = P−1t,w(x− pT (t)h+ w).


Based on Theorem 5.5, we can show the sensitivity analysis of c∗ and π∗ numerically. x, h

appear as a whole value of de facto wealth x− pT (t)h (see Remark 3) in the strategies. We are

interested in the effects of wage and de facto wealth on the real consumption rate H c∗ = c∗−h.

The strategies at time t = 0 and t = 8 are plotted, respectively.

The numerical results are in Figures 10 and 11. When γ > 1, the agent has a consumption

jump-down at retirement, which is consistent with Chen, Hentschel, and Xu (2018) and Dybvig

and Liu (2010). Figure 10 partially explains the retirement consumption puzzle as in Banks,

Blundell, and Tanner (1998). However, γ > 1 in Chen, Hentschel, and Xu (2018), Dybvig and

Liu (2010) and the marginal utility after retirement decreases. As sated in Hurd and Rohwedder

(2003), the agent naturally consumes less due to decrease of the marginal utility. Based on 2001

survey data, Ameriks, Caplin, and Leahy (2007) shows that more than 55% people expect a fall

in consumption, while less than 8% people expect an increase after retirement. Thus there is still

a minority of people consuming more after retirement. We observe from Figure 11 that agent

with risk aversion 0 < γ < 1 has a consumption jump-up at retirement. In the case 0 < γ < 1,

the marginal utility after retirement increases which implies that consumption after retirement

increases. Conversely, we may also deduce that most people have a risk aversion parameter larger

than 1, which is documented in Azar (2006) (γ is between 4.2 and 5.4) and Schechter (2007)

(γ ≈ 1.92).

From Figures 10 and 11, we see that when wage or the de facto wealth increases, the agent

has more wealth and will consume more. Besides, the consumption increases slower when the

“wealth-habit-wage” triplet approaches the retirement boundary. In other words, when the agent

takes retirement into consideration, he will hesitate to consume more, although with more de

facto wealth or wage.


(a) t = 0 (b) t = 8

Figure 10. Optimal consumption, γ = 0.5.

(a) t = 0 (b) t = 8

Figure 11. Optimal consumption, γ = 1.5.

Figures 12 and 13 are results about the optimal proportion of risky investment π∗(t, x, h, w)/x

for γ = 0.5 and γ = 1.5, respectively. For simplicity, we only investigate how this proportion

changes with de facto wealth when fixing h = 1. We observe that the surface of optimal invest-

ment has drastic variations and there are singularities around the retirement boundary. These

singularities arise from the second order derivatives in the expression of Π∗ and the fact that

wzz(t, z) is not continuous at z∗(t) (see Figure 1).

Figures 12 and 13 all suggest a decline of proportion in stock for elder agent, which is ob-

served by the real data in Coile and Milligan (2009)(US) and Spicer, Stavrunova, and Thorp

(2016)(Australia). Interestingly, unlike consumption, the optimal investment strategies always

have a jump down at retirement boundary, no matter γ = 0.5 or γ = 1.5. This means that


people will always withdraw their investment in risky asset upon retirement to hedge the risk of

unemployment, which is the so-called “saving for retirement”. The patterns of the investment be-

havior are different for γ = 0.5 and γ = 1.5. In the continuation region, the evolution of optimal

investment near the retirement boundary is drastic for γ = 1.5 while smooth for γ = 0.5. When

γ > 1 and the agent approaches the retirement boundary, he will suddenly, yet strategically,

raise his risk investment and adjust his consumption simultaneously to prepare for incoming

retirement. However, when γ < 1, the agent behaves more conservatively in consumption while

more aggressive in investment before retirement: he consumes generally less than the case γ > 1,

and adjusts the investment gradually.

(a) t = 0 (b) t = 8

Figure 12. Optimal portfolio proportion, γ = 0.5.

(a) t = 0 (b) t = 8

Figure 13. Optimal portfolio proportion, γ = 1.5.


6. Conclusions

This paper investigates a retirement decision and optimal consumption-investment model with

addictive habit formation. Different from existing literature, three state variables, namely the

wealth x, the habit h and the wage rate w, affect the agent’s decision in this paper. We model this

optimization problem as a StopCP in finite time horizon. The mixture of control and stopping

in this model requires the establishment of dual relation, which becomes more difficult in the

presence of habit formation. Using utility transformation, modified habit reduction method

and dual method, we obtain an equivalent dual pure optimal stopping problem in finite time

horizon. The rather complicated dual relation is then established by investigating the continuity

of optimal stopping time with respect to state variables.

After specifying the preferences, we present and prove the verification theorem that connects

the dual value function with parabolic free boundary problems. Using this, we provide the

retirement boundaries in the dual and primal variables, feedback forms of optimal investment

and consumption strategies. The numerical results have clear economic interpretations. We show

that the enough amount of de facto wealth, when exceeding an critical proportion of wage, will

trigger the retirement. We also find that, people who are more addicted to past consumption

have a larger working region, and will choose to work longer to maintain the standard of life.

“Retirement consumption puzzle” and “saving for retirement” can also be explained to some

extent in our framework.

In this paper, we consider the addictive linear habit formation and a concave utility function for

simplicity. Consideration of drawdown/ratcheting constraints as in Angoshtari, Bayraktar, and

Young (2019) may be interesting. Besides, we can further study non-addictive habit formation,

such as reference to past maximum of consumption in Deng, Li, Pham, and Yu (2020). More

general extensions of the StopCP-based on retirement modeling to the non-concave utility with

habit to characterize loss aversion as in van Bilsen, Laeven, and Nijman (2020) and van Bilsen

and Laeven (2020) are left for future studies. We expect the arguments developed in the first

three sections to be useful when investigating these problems.

Acknowledgements. The authors acknowledge the support from the National Natural Sci-

ence Foundation of China (Grant No.11901574, No.11871036, and No.11471183). The authors

are grateful to Lin He and Yang Liu for their comments and suggestions. The authors also thank

the members of the group of Insurance Economics and Mathematical Finance at the Department

of Mathematical Sciences, Tsinghua University for their feedbacks and useful conversations.


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Appendix A. Proofs of Lemmas 3.3 and 3.4

Proof of Lemma 3.3. The proof is similar with various literature with habit formation by modi-

fying the terminal fixed time to a stopping time, see Proposition 2.3.3 in Yu (2012) for example.

However, as some details of calculations change, we provide the proof for completeness. We first

see from (2.7) that for s ≤ u ≤ T1:

h(u) = h(s)m(u)/m(s) + αm(u)

∫ u

sc(v)/m(v)dv. (A.1)




A combination of c = c+ h and Fubini’s theorem shows for any fixed τ ∈ ST1t ,∫ τ

sc(u)ξt(u)du =

∫ τ

sc(u)ξt(u)du+

∫ τ

sh(u)ξt(u)du

=

∫ τ

sc(u)ξt(u)du+

∫ τ

sh(s)m(u)/m(s)ξt(u)du

+α

∫ T1

s

∫ u

s

m(u)

m(v)ξt(u)c(v)Iu≤τdvdu

=

∫ τ

sc(u)ξt(u)du+

∫ τ

sh(s)m(u)/m(s)ξt(u)du

+α

∫ T1

sc(u)

(∫ T1

u

m(v)

m(u)ξt(v)Iv≤τdv

)du

=

∫ τ

sc(u)

(ξt(u) + α

∫ τ

u

m(v)

m(u)ξt(v)dv

)du

+h(s)ξt(s)

∫ τ

s

m(u)

m(s)

ξt(u)

ξt(s)du.

Taking Es on both sides of the last equation derives the desired result.

Proof of Lemma 3.4. Let M(s) = Es[ξt(τ)pT (τ)m(τ)]. Obviously, M(s ∧ τ) : t ≤ s ≤ T1is a square-integrable and uniformly integrable continuous martingale. Based on definitions in

Lemma 3.3, we have that for s ∈ [t, τ ]pτ (s) = Es

[∫ Ts

m(u)m(s)

ξt(u)ξt(s) du

]− Es

[∫ Tτ

m(u)m(s)

ξt(u)ξt(s) du

]= pT (s)− EsEτ

[∫ Tτ

m(u)m(τ)

ξt(u)ξt(τ)du

]m(τ)ξt(τ)(m(s)ξt(s))−1

= pT (s)− (ξt(s)m(s))−1M(s),µτ (s) = µT (s)− α(ξt(s)m(s))−1M(s).

(A.2)

As such,

Es[∫ τ

sc(u)ξt(u)µT (u)du

]−Es

[∫ τ

sc(u)ξt(u)µτ (u)du

]=αEs

[∫ τ

sc(u)m(u)−1M(u)du

]. (A.3)

On the other hand, we see from (A.1) that

d(h(u)/m(u)) = αc(u)/m(u)du.

Define τn = inft ≤ t′ ≤ T :∫ t′t |m(u)−1h(u)|2d < M >u≥ n ∨ s. Applying Ito’s formula to

h(u)M(u)m(u)−1, we obtain

h(τ ∧ τn)M(τ ∧ τn)m(τ ∧ τn)− h(s)m(s)−1M(t) = α

∫ τ∧τn

sM(u)m(u)−1c(u)du

+

∫ τ∧τn

sm(u)−1h(u)dM(u). (A.4)

Taking Es on both sides of (A.4),

Esh(τ ∧ τn)M(τ ∧ τn)m(τ ∧ τn)− h(s)m(s)−1M(s) = αEs∫ τ∧τn

sM(u)m(u)−1c(u)du.


Letting n→∞ and using monotonous convergence theorem and dominated convergence theorem,

we have

Esh(τ)M(τ)m(τ)− h(s)m(s)−1M(s) = αEs∫ τ

sM(u)m(u)−1c(u)du. (A.5)

Combining (A.3) and (A.5) gives

Esh(τ)pT (τ)ξt(τ)−h(s)m(s)−1M(s) = Es[∫ τ

sc(u)ξt(u)µT (u)du

]−Es

[∫ τ

sc(u)ξt(u)µτ (u)du

].

Thus, plugging (A.2) to replace the term h(s)m(s)−1M(s) in the last equation completes the

proof.

Appendix B. Proof of Theorem 3.5

To verify the dual relation, we quote some classical arguments in the proof of main results

in Karatzas and Wang (2000). Because our financial model with habit formation is different

from Karatzas and Wang (2000), the procedure to prove the theorem is similar while much more

complex. Recall that we have already proved the inequality (3.17), i.e., we have the one side

inequality of (3.18):

V (t, x, h, w) ≤ infy>0W (t, y, w) + y(x− pT (t)h) + yq(t)w.

Then we want to verify the converse inequality of (3.18). It is sufficient to show that (3.18) holds

for some particular y. First, we present the following lemmas:

Lemma B.1. We have the following asymptotic results for function U :

limy→0

inf0≤t≤T1

IU (t, y) =∞, (B.1)

limy→∞

sup0≤t≤T1

IU (t, y) = 0, (B.2)

sup0≤t≤T1

IU (t, y) ≤ C(1 + y−K), (B.3)

where C and K > 0 are constants.

Proof. By similar arguments in the proof of Lemma B.2 (using convexity of V ), we conclude that

IU (t, y) = −∂yV (t, y) = E[∫ T

tξt(u)IU2(u, yY t,1(u))du

].

(B.3) follows from bounded assumption of IU2 and (B.2) follows from the asymptotic properties of

IU2 and monotone convergence theorem. In fact, (B.1) can be obtained by the same arguments,


as we have the following estimation:

IU (t, y) ≥ E[∫ T1

Tξt(u)IU2(u, yY t,1(u))du

]≥ E

[(∫ T1

Tξ(u)IU2(u, e2ρT ξ(u)/ inf

0≤t≤T1ξ(u))du

)/ sup

0≤t≤T1ξ(t)

].

Lemma B.2. For τ ∈ ST1t and fixed t ∈ [0, T1], define

Xt,τ (y) = E[∫ τ

tξt(u)µT (u)IU1(Y t,y(u)µT (u)) + IU (t, Y t,y(τ))ξt(τ)

].

Then, ∀x > 0, there exists y > 0 such that

Xt,τy(y) = x.

Furthermore, W is differentiable in y in (0,∞) and

Wy(t, y, w) = −Xt,τy(y)− wEe−ρ(τy−t)Y t,1(τy)Wt,1(τy)q(τy). (B.4)

Proof. To prove (B.4), first we have

limy→0Xt,τy(y) =∞, lim

y→∞Xt,τy(y) = 0,

which can be easily verified by Lemma B.1. We claim that Xτy(y) is continuous in y. In fact,

the estimation

|Xt,τy(y)| ≤ C(2T1M + 1)( supt≤s≤T1

ξt(s))(1 + δ−K supt≤s≤T1

(Y t,1(s))−K)

holds uniformly for y ∈ [δ,∞), where M = supt∈[0,T ] µT (t). The right side of the inequality

is integrable by Burkholder-Davies-Gundy’s inequality. As such, using dominated convergence

theorem, we conclude that Xt,τy(y) is continuous in y on [δ,∞). As δ > 0 is arbitrary , Xt,τy(y)

is continuous in (0,∞). Therefore, the mapping (0,∞)→ (0,∞) : y 7→ Xτy(y) is surjective. To

prove (B.4), we begin with some estimation of the difference of W based on convexity. For η ∈ R


near 0, we have

W (t, y + η, w)−W (t, y, w)

≤E[∫ τy+η

te−ρ(u−t)[V1((y + η)µT (u)Y t,1(u))− V1(yµT (u)Y t,1(u))]du

]+ Ee−ρ(τy+η−t)(V (t, (y + η)Y t,1(τy+η))− V (t, yY t,1(τy+η)))

− ηwEe−ρ(τy+η−t)Y t,1(τy+η)Wt,1(τy+η)q(τy+η)

≤E[∫ τy+η

tηe−ρ(u−t)µT (u)Y t,1(u)V ′1(µT (u)Y t,y+η(u))du

]+ Ehe−ρ(τy+η−t)Y t,1(τy+η)V

′y(t, Y t,y+η(τy+η))

− ηwEe−ρ(τy+h−t)Y t,1(τy+h)Wt,1(τy+h)q(τy+h)

=− ηE[∫ τy+η

tξt(u)IV1(µT (u)Y t,y+η(u))du+ ξt(τy+η)IU (t, Y t,y+η(τy+η))

]+ ηwEe−ρ(τy+η−t)Y t,1(τy+η)Wt,1(τy+η)

=− ηXt,τy+η(y + η)− ηwEe−ρ(τy+η−t)Y t,1(τy+η)Wt,1(τy+η)q(τy+η).

Dividing by η on both sides of the last inequalities and letting η → 0+, we have

Wy+(t, y, w) ≤ −Xτy(y)− wEe−ρ(τy−t)Y t,1(τy)Wt,1(τy)q(τy).

Conversely, letting η → 0−, we get

Wy−(t, y, w) ≥ −Xτy(y)− wEe−ρ(τy−t)Y t,1(τy)Wt,1(τy)q(τy).

Combining the last two inequalities, (B.4) holds.

Proof of Theorem 3.5. First, Eb(τy)ξt(τy) is continuous in y, which can be proved similarly

as the continuity of Xτy(y). As Xt,τy(y) maps (0,∞) onto (0,∞), Xτy(y) + Eb(τy)ξt(τy)) maps

(0,∞) onto (q(t)w,∞). Now, we choose y∗ > 0 such that

Xt,τy∗ (y∗) + Eb(τy∗)ξt(τy∗) = x− pT (t)h+ q(t)w.

Based on Lemma 3.4, the last equation is

E[∫ τy∗

tξt(u)µτy∗ (u)c∗(u)du+B∗ξt(τy∗)

]+ hpτy∗ (t) = x+ b(t),

with c∗(u) = IU1(u, Y t,y∗(u)µT (u)) and B∗ = IU (τy∗ , Yt,y∗(τy∗)) + h(τy∗)p

T (τy∗) + b(τy∗). This

implies that (c∗, τy∗) ∈ B= and all inequalities in (3.13) are in fact equalities. Thus, the proof is

completed.


Appendix C. Proofs of the results in Section 4

In this section we prove the properties of w satisfying (4.6) and verify Theorem 4.2. All

properties will be satisfied by w. Let F = w − V + q. We need the following results.

Lemma C.1. Let Fα(t, z) = F (t, α(t)z), then

(∂t + Lα)Fα∣∣(t,z)

= (∂t + LZ)F∣∣(t,α(t)z)

,

where

Lα =σ2z

2z2∂zz + (µz − α′(t)/α(t))z∂z + ϑ.

Proof. This lemma is easily verified by directly calculating the derivatives of Fα as follows:

∂tFα = ∂tF + α′(t)z∂zF∣∣(t,α(t)z)

,

∂zFα = α(t)∂zF∣∣(t,α(t)z)

,

∂zzFα = α(t)2∂zzF∣∣(t,α(t)z)

.

The left sides of the last three equations are all calculated at point (t, z).

We now turn to showing the monotonicity of F , which is the basis of the analysis of z∗.

Lemma C.2. (1) If 0 < γ < 1, F increases with respect to z.

(2) If γ > 1, F decreases with respect to z.

Proof. We first prove the case γ > 1. ∀δ′ > 1, consider Fδ′(t, z) = F (t, δ′z) . Based on Lemma

C.1 and (4.8), we conclude that Fδ′ satisfies(−∂t −LZ)Fδ′ = 4(t)(µT (t)δ′z)−γ + 1, if Fδ′(t, z) > 0,

(−∂t −LZ)Fδ′ ≥ 4(t)(µT (t)δ′z)−γ + 1, if Fδ′(t, z) = 0,

Fδ′(T1, ·) = 0.

It follows that

4(t)(µT (t)δ′z)−γ < 4(t)(µT (t)z)−γ .

As such, if the comparison principles of variational inequalities (cf. Lemma A.6 in Yan, Liang,

and Yang (2015)) can be applied, we can get the desired conclusion. Note that in order to

apply Lemma A.6 in Yan, Liang, and Yang (2015), we only need to verify that w and V are of

polynomial growth. This is true because

(−∂t −LZ)F ≤ −δ2z−γ + 1, in C.

Hence, we choose

F0(t, z) = −1/ϑ−Az−γ +B′zλ+


such that

(−∂t −LZ)F0 = −δ2z−γ + 1, in C,

where the constants A and λ+ are explained in the proof of Theorem 4.5 (see Figure 14 for the

relation of F0, F and F ). B′ is a constant which is large enough. Thus, we have F0 ≥ F ≥ 0, and

in particular, F0(T1, ·) ≥ 0, F0 ≥ 0 = F on ∂C. Then we apply strong maximum principle for

elliptic equations to conclude that F has growth rate no more than z−γ+λ+ , hence has polynomial

growth. The fact that −γ > 0 when γ > 1 is crucial in this case. Note that the above argument

also guarantees the boundness of F when z → 0+. As such, the comparison on z = 0 is always

valid. The monotonicity when γ > 1 has been proved. To prove the case 0 < γ < 1, define

F (t, z) = F (t, 1/z), we have(−∂t −Linv)F = 4(t)(µT (t))−γzγ + 1, if F (t, z) > 0,

(−∂t −Linv)F ≥ 4(t)(µT (t))−γzγ + 1, if F (t, z) = 0,

F (T1, ·) = 0,

where

Linv =σ2z

2z2∂zz + (σ2

z − µz)z∂z + ϑ.

Noting that in this case F has polynomial growth, comparison principles are valid. As such,

repeating the argument for the case γ > 1, we conclude that F (t, 1/z) is decreasing in z, and F

is increasing in z.

Now we present the characterization of the stopping boundary in Problem (4.3).

Proposition C.3. If 0 < γ < 1, C = (t, z) : 0 ≤ t < T1, z > z∗(t). If γ > 1, C = (t, z) : 0 ≤t < T1, 0 < z < z∗(t). Here, z∗(·) is a [0,∞]-valued function, and is Lipschitz continuous on

t : z∗(t) <∞.

Proof. We only prove the case γ > 1 as the proof when 0 < γ < 1 is similar. Based on Lemma

C.2, we know that F is decreasing in z. For any t ∈ [0, T1), let z∗(t) , infz > 0 : F = 0. Then

it is easy to verify that C = (t, z) : 0 < z < z∗(t).

Proof of Theorem 4.5. All things have been proved except the boundness of z∗ (above and be-

low). We first prove the case γ > 1. Assume that µz < 0 and −µz is large enough (to be de-

termined later). Based on Proposition 4.4 and Assumption 3, we know that if z < M1/γ/µT (0),

then ∀t ∈ [0, T1), z < M γ/1/µT (t) ≤ (−4(t))γ/1/µT (t). Thus 4(t) > −(µT (t)z)γ and (t, z) ∈ C.This gives a lower bound: z∗(t) ≥ M1/γ/µT (0) > 0. For the upper bound, we consider the

following auxiliary function:

F (t, z) =

0, z ≥ z∞,−1/ϑ−Az−γ +Bzλ+ , 0 < z < z∞.

(C.1)


The constants A, B, z∞ in (C.1) will be explained successively. Here, we consider the quadratic

function:

Q(λ) =σ2z

2λ2 + (µz −

σ2z

2)λ+ ϑ.

Now in (C.1), λ+ is the positive root of Q(λ) = 0. It is easy to check that λ+ > 1 (as µz < 0).

Moreover, we set the constants as

A = −δ/2Q(−γ),

z∞ =

(λ+

−ϑA(λ+ + γ)

)−1/γ

,

B =−γAλ+

z−γ−λ+∞ .

By the fact 0 < −γ < 1 < λ+, obviously, Q(−γ) < 0 and A, B, z∞ > 0. We see that F is

independent of t and is in C1(R+) ∩ C2(R+\z∞). Moreover, ∂F∂z < 0 for z ∈ (0, z∞). As such, it

follows that F is decreasing on (0, z∞) with F (z∞) = 0, and satisfies

(−∂ −LZ)F = −δ2z−γ + 1, if 0 < z < z∞.

Besides, ∀z > z∞, we have

−δ2z−γ + 1 < −δ

2z−γ∞ + 1

= −δ2

λ+

−ϑA(λ+ + γ)+ 1

= 1−Q(−γ)λ+

ϑ(λ+ + γ).

Direct calculations show that the right side of the last equation is not greater than 0 if and only

if

λ+γ +2

σ2z

ϑ ≤ ϑ.

Noting that

λ+ =

σ2z2 − µz +

√(σ

2z2 − µz)2 − 2σ2

zϑ

σ2z

,

we have λ+ → +∞ if µz → −∞. Choosing µz small enough, we can prove that

λ+γ +2

σ2z

ϑ ≤ ϑ.

In this case we conclude that F satisfies(−∂t −LZ)F = −δ

2z−γ + 1, if F (t, z) > 0,

(−∂t −LZ)F ≥ −δ2z−γ + 1, if F (t, z) = 0,

F (T1, ·) ≥ 0.


Applying the comparison principles, we obtain that F ≤ F . In particular, (t, z) : 0 ≤ t <

T1, z > z∞ ⊂ S. This implies that z∗(t) ≤ z∞,∀t ∈ [0, T1), which is what we need.

Next we reduce the condition that µz < 0 is small enough. Choosing α(t) = eKt with K > 0

large enough in Lemma C.1, we find that Fα(t, z) = F (t, eKtz) satisfies the variational inequalities

with first order term µz−K in the differential operator. Then, by the arguments above, we know

that e−Ktz∗(t) is bounded above, thus so is z∗. Finally, an application of transformation z 7→ 1/z

just like the proof of Proposition C.2 completes the proof in the case 0 < γ < 1.

0z

−1/ϑ

z∞z∗(t)

Figure 14. The images of F (t, ·) for some fixed t ∈ [0, T1). Dashed line: F .Dotted line: F0.

Remark 9. It is easily seen from the proof that (−4(t))1/γµT (t)−1 serves as a(n) lower/upper

bound for t ∈ [0, T1) when γ > 1 and 0 < γ < 1, respectively.

Finally, we will need some properties of w that are useful in the proof of Theorem 4.2.

Lemma C.4. z 7→ w(t, z) is convex on (0,∞).

Proof. First, we consider γ > 1. For fixed λ ∈ [0, 1], denote G1λ(t, z) = w(t, λz1 + (1 − λ)z2),

G2λ(t, z) = λw(t, z1) + (1 − λ)w(t, z2), H1

λ(t, z) = V (t, λz1 + (1 − λ)z2) − q(t) and H2λ(t, z) =

λV (t, z1) + (1− λ)V (t, z2)− q(t), where zT = (z1, z2) ∈ R2+. Define

L =σ2z

2zT∂zzz + µzz

T∂z + ϑ,


where ∂zz and ∂z are interpreted as Hessian and gradient operators. We can easily check(−∂t − LZ)G1

λ = (λz1 + (1− λ)z2)−γµT (t)−γ/γ, if G1λ(t, z) > H1

λ(t, z),

(−∂t − LZ)G1λ ≥ (λz1 + (1− λ)z2)−γµT (t)−γ/γ, if G1

λ(t, z) = H1λ(t, z),

G1λ(T1, ·) = H1

λ(T1, ·),

and (−∂t − LZ)G2

λ = (λz−γ1 + (1− λ)z−γ2 )µT (t)−γ/γ, if G2λ(t, z) > H2

λ(t, z),

(−∂t − LZ)G2λ ≥ (λz−γ1 + (1− λ)z−γ2 )µT (t)−γ/γ, if G2

λ(t, z) = H2λ(t, z),

G2λ(T1, ·) = H2

λ(T1, ·).

Note that comparison principles are also valid for higher dimensional variational inequalities.

Thus convexity of w follows from convexity of z 7→ z−γ/γ as 0 < −γ < 1.

In the case 0 < γ < 1, we consider w(t, z) = w(1, 1/z). Then growth estimation for γ > 1

derives |F | ≤ C(1 + zK), i.e., |w| ≤ C(1 + z−K). Based on Theorem 4.5, when z is small enough,

w = V − q is convex. Then we restrict our domain to z1 ≥ z01 and z2 ≥ z0

2 such that G1λ ≤ G2

λ

holds true on the part of parabolic boundary [0, T1) × z01 × [z0

2 ,∞) ∪ [0, T1) × [z01 ,∞) × z0

2.Because G1

λ and G2λ are of polynomial growth on the restricted domain, comparison principles

are also valid. Thus, we complete the proof because z 7→ z−γ/γ is still convex.

Lemma C.5. w satisfies (4.5).

Proof. We only need to prove Lemma C.5 when γ > 1. Using the bounds of F and V , we have

|w| ≤ C(1 + zK). Based on the fact that V is decreasing in z (which can be checked directly,

noticing V2 < 0), and F = w− V +q is decreasing in z, we know that w is decreasing in z. Noting

the convexity of w, we have that ∂zw(t, ·) ≤ 0 is increasing in z. Furthermore, for 0 < z < z0,

we have

w(t, z)− w(t, z0) ≥ ∂zw(t, z0)(z − z0),

which implies ∂zw(t, z0) ≥ −Cz−10 because w is bounded when z → 0. Therefore when z → 0,

∂zw ≥ −C(1 + z−K). Thus, |∂zw| ≤ C(1 + z−K).

Based on Lemmas C.4 and C.5, we now give the proof of Theorem 4.2.

Proof of Theorem 4.2. Step1: w ≤ w. Fix any (t, z) ∈ NT1 and τ ∈ ST1t . We know from the

proofs above that ∂C ∩NT1 can be expressed as a curve (t, z) : 0 ≤ t ≤ T1, z = z∗(t), where z∗

is Lipschitz continuous, and bounded above and below from 0 on [0, T1]. As such, the local time

formula holds (see Section II.3.5 in Peskir and Shiryaev (2006)). The conditions (3.5.10)-(3.5.13)

in Peskir and Shiryaev (2006) should be satisfied, i.e., the convexity of w in Lemma C.4 is needed.


We conclude that, Pt,z-almost surely,

eϑτ w(τ, Z(τ))− eϑtw(t, z) =

∫ τ

teϑu(∂t + LZ)w(u, Z(u))IZ(u) 6=z∗(u)du

+

∫ τ

tσze

ϑuZ(u)∂zw(u, Z(u))IZ(u)6=z∗(u)dB(u)

≤ −∫ τ

teϑuV1(u, Z(u))du

+

∫ τ

tσze

ϑuZ(u)∂zw(u, Z(u))IZ(u)6=z∗(u)dB(u).

The indicator in the first term of right side can be estimated by property (4.4). Then taking

expectation Et,z on both sides of the last inequality, we conclude from the growth condition of

∂zw that the expectation of the Ito’s integral is zero. Noting the fact w ≥ V − q, we have

Et,z[eϑτ (V (τ, Z(τ))− q(τ)) +

∫ τ

teϑuV1(u, Z(u))du

]≤ eϑtw(t, z),

which yields w ≤ w.

Step2: w ≥ w. Let (t, z) ∈ C and define τ∗ = infs ≥ t : (u, Z(u)) /∈ C. Now let τ = τε ,

(τ∗ − ε) ∧ t in the calculation of Step 1. Then we get

Et,zeϑτεw(τε, Z(τε)) = eϑtw(t, z)− E[∫ τε

teϑuV1(u, Z(u))du

],

as (u, Z(u)) ∈ C on [t, τε]. Letting ε→ 0 on both sides of the last equation, because of the growth

conditions of w and V1, we have

Et,z[eϑτ

∗(V (τ, Z(τ∗))− q(τ∗)) +

∫ τ∗

teϑuV1(u, Z(u))du

]= eϑtw(t, z),

which implies w ≥ w in C. Moreover, in NT1\C we have w = V − q ≤ w. The proof is thus

completed.

Proof of Theorem 4.6. Applying Ito’s formula involving local time to eϑT1F (T1, Z(T1))−eϑtF (t, z)

and setting z = z∗(t), we obtain (4.9) and (4.10), which is valid as F is the linear combination

of two convex functions. The fact that F = 0 when z = z∗(t) is also used.

Next we prove the terminal condition in Theorem 4.6. Again we only need to prove it when

γ > 1. By transformation z 7→ 1/z, the result remains true when 0 < γ < 1.

The boundness and Lipschitz continuous properties guarantee the existence of limit z∗(T1−) =

limt→T1 z∗(t). Proposition 4.4 derives z∗(T1−) ≥ α(T1) with the definition α(t) , (−4(t))1/γµT (T1)−1.

If z∗(T1−) > α(T1), we find an interval α(T1) < z1 < z2 < z∗(T1−) and sufficiently small

ε > 0 such that [T1 − ε, T1) × [z1, z2] ⊂ C. Define FK(t, z) = F (t, eKtz). Based on Lemma C.1,


we know that FK satisfies(−∂t −LK)FK = −(α(t)−1eKtz)−γ + 1, if FK(t, z) > 0,

(−∂t −LK)FK ≥ −(α(t)−1eKtz)−γ + 1, if FK(t, z) = 0,

FK(T1, ·) = 0.

Now choose K > 0 large enough such that −(α(t)−1eKtz)−γ decreases with t, which is possible

because |4(t)′| and |µT (t)′| are both bounded from above. Then, ∀t ∈ [t0, T1), comparing

FK(t− t0, z) and FK(t, z) on [t0, T1)× R+, and noting the fact FK(T1 − t0, z) ≥ 0 = FK(T1, z),

we conclude that FK decreases with t. As such, ∂tFK ≤ 0 in NT1 . If we let ε < 1K log(z2/z1),

then for any z ∈ I , (z1e−K(T1−ε), z2e

−KT1), we have z1e−Kt < z < z2e

−Kt, ∀t ∈ [T1− ε, T1]. As

such, (t, eKtz) ∈ C, (FK(t, z) > 0 and

−∂tFK(t, z)−LKFK(t, z) = −(α(t)−1eKtz)−γ + 1.

Letting t→ T1 on both sides of the last equation, as z > α(T1)e−KT1 , we have

lim supt→T1

(−(α(t)−1eKtz)−γ + 1) ≤ −(α(T1)−1eKT1z)−γ + 1 < 0.

As FK(T1, z) = 0, if we can prove LKFK(t, z) → 0 for a.s. z on I, as t → T1−, then

∂tFK(T1−, z) > 0 a.s. on I, which is a contradiction. In fact, ∀z3 > z4 > 0,

FK(t, z3)− FK(t, z4) =

∫ z4

z3

∂zFK(t, z′)dz′.

Dominated convergence theorem shows

0 =

∫ z4

z3

∂zFK(T1−, z′)dz′.

However, FK is decreasing in z. Thus we have FK(T1−, z′) = 0 on [z3, z4]. Moreover

∂zFK(t, z3)− ∂zFK(t, z4) =

∫ z4

z3

∂zzFK(t, z′)dz′.

Applying Fatou’s lemma, we have∫ z4

z3

∂zzFK(T1−, z′)dz′ ≤ 0.

Then convexity of FK(t, ·) implies ∂zzFK(T1−, z′) ≥ 0, which in turn implies ∂zzFK(T1−, z′) = 0,

a.s. on [z3, z4]. Thus, by arbitrariness of z3 and z4, it is also true a.s. on I. As such, LKFK → 0,

which is a contradiction. Therefore, the assumption that z∗(T1−) > α(T1) is not true, thus, we

have z∗(T1−) = α(T1).


Appendix D. Proof of results in Section 5

Proof of Lemma 5.1. We only prove Lemma 5.1 when 0 < γ < 1, because the proof of the case

γ > 1 is similar. Denote zy = y1/(1−γ)wγ/(1−γ). We first note that

Law(τy|P) = Law(τy|P) (D.1)

with τy = inft ≤ s ≤ T1 : Zt,zy(s) ≤ z∗(s). Based on optimal stopping theory (cf. Chapter I,

Theorem 2.4 in Peskir and Shiryaev (2006)), τy is optimal for Problem (4.3). Based on Lemma

4.1, τy is also optimal for Problem (3.16). Then we need to prove the continuity. By (D.1), it is

sufficient to prove the continuity of τy in the sense of P-almost surely.

If zy < z∗(t), then for sufficiently small ε, τy+ε ≡ t. As such, we only need to consider

zy ≥ z∗(t). We prove it in the following two steps:

Step 1: Show that τy+ε → τy, as ε 0, for zy ≥ z∗(t).Step 2: Show that τy′ → τy, as y′ y, for zy > z∗(t).

Proof of Step 1: We rewrite τy as τy = inft ≤ s ≤ T1 : Zt,1(s) ≤ z∗(s)/zy and similarly

for τy+ε (see Figure 15-(a) for graphical illustration). Using the fact that zy increases with y,

we have τy+ε τ ′ ≥ τy. Define Bk = ω : τ ′ − τy > 1/k and B =⋃∞k=1Bk, we aim to prove

P(B) = 0. It is sufficient to prove P(Bk) = 0. Define Aεk , ω : τy+ε − τy ≥ 1/k for any k.

Noting the following observations: under the conditional law of P(·|τy = t′),

τy+ε − τy = τy+ε − t′

= inft′ ≤ s ≤ T1 : Zt′,z∗(t′)/zy(s) ≤ z∗(s)/zy+ε − t′

= inf0 ≤ s ≤ T − t′ : Zt′,z∗(t′)zy+ε/zy(s+ t′) ≤ z∗(s+ t′),

we have (see Figure 15-(b))

Law(τy+ε − τy|P(·|τy = t′)) = Law(τ∗|Pt′,z∗(t′)zy+ε/zy),

where

τ∗ = inf0 ≤ s ≤ T1 − t′ : Z(s+ t′) ≤ z∗(s+ t′).

The regularities of the boundary z = z∗(s) and the process (·+ t′, Z(·+ t′)) derive

limε0

Pt′,z∗(t′)zy+ε/zy(τ∗ ≥ 1/k) = 0,

which in turn implies

limε0

P(Aεk|τy = t′) = 0.

Then dominated convergence theorem gives

limε0

P(Aεk) = limε0

∫ T1

tP(Ak|τy = t′)P(τy ∈ dt)

= 0.


As such,

P(limε0

τy+ε − τy ≥ 1/k) = limε0

P(τy+ε − τy ≥ 1/k)

= limε0

P(Aεk)

= 0.

Thus the proof of Step 1 is completed.

(t, 1)

z∗(·)/zyz∗(·)/zy+ε

t′

t

z∗(·)

(t‘, z∗(t′)zy+ε/zy)

t′

z∗(·)

t′ + τ∗

(a) (b)

Figure 15. Continuity of τy.

Proof of Step 2: Using monotonicity of zy′ in y′, we know τy′ τ2 ≤ τy. Based on regularity

of process Z, we have

zy′Zt,1(τy′) ≤ z∗(τy′).

Letting y′ → y,

zyZt,1(τ2) ≤ z∗(τ2),

which in turn shows τ2 ≥ τy. Thus, τ2 = τy.

Proof of Lemma 5.2. Based on Remark 5, we need to prove limy→∞ E[b(τy)ξt(τy)] = 0. Again we

only prove it when 0 < γ < 1. To this end, we claim that τy → T1 as y →∞. The claim is then

equivalent to τz → T1 when z →∞ with the notation τz = inft ≤ s ≤ T1 : Zt,z(s) ≤ z∗(s). We

expand z∗ to z∗∗ as follows

z∗∗(s) =

z∗(s), s ∈ [0, T1),z∗(T1−), s ∈ [T1,∞).

Define τ ′z = infs ≥ t : Zt,z(s) ≤ z∗∗(s). Then it is sufficient to prove τ ′z →∞ as z →∞. As τ ′z

increases with z, we only need to prove for any fixed K > 0, limz→∞ P(τ ′z ≤ K) = 0 (using the

same argument as in the proof of Lemma 5.1).


Denote z∞ = sups≥0

z∗∗(s) > 0. We have that τ ′z ≥ t + τ ′′z , s ≥ 0 : σzW (s) + (µz −

σ2z2 )s ≤ log(z∞/z). Using measure change, we conclude that for some equivalent measure P,

and standard Brownian Motion W under P, τ ′′z is the first hitting time of W to log(z∞/z)/σz.

Thus, we may estimate

P(τ ′z ≤ K) ≤ P(τ ′′z ≤ K − t) =2√2π

∫ ∞log(z∞/z)/(−

√K−t|σz |)

e−x2/2dx→ 0,

as z →∞. The validity of (3.18) on Gt follows naturally from the proof of Theorem 3.5 as −Wy

maps R+ onto R+. The proof is completed.

Proof of Lemma 5.3. Based on Theorem 3.5 and convexity of W , we have

V (t, x, h, w) = (x− pT (t)h+ q(t)w)y +W (t, y, w), y = P−1t,w(x− pT (t)h+ q(t)w).

If y ∈ Cdt,w, then (see Lemma 3.19):

(x− pT (t)h+ q(t)w)y∗ +W (t, y∗, w) > (x− pT (t)h+ q(t)w)y∗ + V (t, y∗)− q(t)y∗w

= y∗(x− pT (t)h) + V (t, y∗).

≥ U(t, x− pT (t)h) = U(t, x, h),

which shows (x, h) ∈ Cpt,w. As such, if (x, h) ∈ Cpt,w, it is clear that P−1t,w(x−pT (t)h+q(t)w) ∈ Cdt,w.

Thus, (x, h) ∈ Cpt,w and Cpt,w ⊂ Cpt,w.

On the other hand, if (x, h) ∈ Cpt,w, then for y∗ = P−1t,w(x− pT (t)h+ q(t)w), based on the last

part of proof in Lemma B.2, we know that the optimal stopping time τy∗ for W is also optimal for

V . As such, τy∗ > t, which implies y∗ ∈ Cdt,w. Therefore x− pT (t)h+ q(t)w = Pt,wy∗ ∈Pt,wCd

and the proof is completed.

Proof of Proposition 5.4. Based on Lemma B.2 and the proof of Lemma 5.2, we know that

Pt,wR+ = (0,∞). As such, if (x, h, w) ∈ Gt, then x − pT (t)h + q(t)w /∈ Pt,wCdt,w if and only if

x− pT (t)h+ q(t)w ∈Pt,w(R+\Cdt,w).

Proof of Theorem 5.5. The expressions of c∗ and π∗ outside Cpt,w can be easily obtained by clas-

sical methods. The readers are referred to, for example, (6.24) and (6.25) in Englezos and

Karatzas (2009). We only need to prove the expressions for (x, h) ∈ Cpt,w. Let τt,y,w be as in

Lemma 5.1. Based on the proof of Theorem 3.5, defining c∗(u) = IU1(u, Y t,y(u)µT (u)) and

B∗ = IU (τt,y.w, Yt,y(τt,y,w)) + h(τt,y,w)pT (τt,y,w) + b(τt,y,w), we know that (τt,y,w, c

∗) ∈ B= and

all inequalities in (3.13) become equalities. As such, H −1(τt,y,w, c∗) ∈ B= attains maxima in

Problem (3.3). Thus, the expression (5.2) is verified.

Now we prove (5.3). Let s ≥ t, for (s, y′, w′) such that y′ ∈ Cds,w′ , conditioned on s <

τt,y,w, Yt,y(s) = y′,Wt,w(s) = w′, τt,y,w = τs,y′,w′ holds in the sense of distribution under P.


Moreover, using Markovian property of Y , we have

E[e−ρ(τt,y,w−t)Y t,y(τt,y,w)IU (τt,y,w, Y

t,y(τt,y,w)) + e−ρ(τt,y,w−t)Y t,y(τt,y,w)q(τt,y,w)Wt,w(τt,y,w)

+

∫ τt,y,w

se−ρ(τt,y,w−t)Y t,y(u)µT (u)IU1(u, Y t,y(u)µT (u))du

∣∣Y t,y(s) = y′,Wt,w(s) = w′]

= E[e−ρ(τs,y′,w′−t)Y s,y′(τs,y′,w′)IU (τs,y′,w′ , Y

s,y′(τs,y′,w′))

+[e−ρ(τs,y′,w′−t)Y s,y′(τs,y′,w′)q(τs,y′,w′)Ws,w′(τs,y′,w′)

+

∫ τs,y′,w′

se−ρ(τs,y′,w′−t)Y s,y′(u)µT (u)IU1(u, Y s,y′(u)µT (u))du

]= −Wy(s, y

′, w′)y′.

Integrating with respect to the distribution of Y t,y(s) and Wt,w(s), for s ∈ [t, τt,y,w), we have

Es[e−ρ(τt,y,w−t)Y t,y(τt,y,w)IU (τt,y,w, Y

t,y(τt,y,w)) + e−ρ(τt,y,w−t)Y t,y(τt,y,w)q(τt,y,w)Wt,w(τt,y,w)

+

∫ τt,y,w

se−ρ(τt,y,w−t)Y t,y(u)µT (u)IU1(u, Y t,y(u)µT (u))du

]= −Wy(s, Y

t,y(s),Wt,w(s))Y t,y(s).

Combining the last equation with (3.6), (3.7) and (3.10), we have

ξt(s)(X(s) + q(s)Wt,w(s)) = −Wy(s, Yt,y(s),Wt,w(s))eρ(s−t)ξt(s) + ξt(s)pT (s)h(s).

Then dividing both sides of the last equations by ξt, taking differential and applying Ito’s for-

mula, we obtain the stochastic differential equation of ξt(s)(X(s) + q(s)Wt,w). Comparing the

coefficients of diffusion terms in this SDE with that obtained from (2.3), we have

π∗(s) = σ−1(−σwq(s)Wt,w(s)− σwWyw(s, Y t,y(s),Wt,w(s))Wt,w(s)eρ(s−t)

+ κWyy(s, Yt,y(s),Wt,w(s))Y t,y(s)eρ(s−t)),

which shows that the feed-back form of π∗ in (y, w) is

Π∗(t, y, w) = σ−1(−σwq(t)w − σwwWyw(t, y, w) + κyWyy(t, y, w)).

At last, applying dual-primal relation x− pT (t)h+ q(t)w = Pt,wy proves (5.3).

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