second order reflected bsde's and dynkin game under ... · introduction second order...

Introduction Second order RBSDEs Application : Dynkin game under volatility uncertainty

Second order Reflected BSDE’s and Dynkingame under uncertainty

Anis Matoussi

University of Maine (Le Mans, France)and

CMAP, Ecole Polytechnique (Palaiseau, France)

Institut du Risque et de l’Assurance (du Mans)

Centre Henri Lebesgue, Université de Rennes 1May 22-24 2013

Anis Matoussi Stochastic and game under uncertainty 1 / 65


Outline

1 Introduction

2 Second order RBSDEs

3 Application : Dynkin game under volatility uncertainty



Motivations

We study a class of second order Reflected BSDE’s (2RBSDE)which are motivated by the following applications :

1 Optimal Stoping Problem and Dynkin game underuncertainty

2 Pricing American and game options under volatilityuncertainty

A.M, L. Piozin and D. Possamai : Second Order BSDE’s withgeneral reflection and Dynkin games under uncertainty.arXiv :1201 :0746v2.A.M, D. Possamai and C. Zhou : Second Order ReflectedBSDE’s. arXiv :1201.0746 (2012). The Ann. of App. Probab., toappear.



Outline

1 IntroductionReflected BSDE’s and applicationsDoubly Reflected BSDE’s and ApplicationsPricing Game option in the classical financial market

2 Second order RBSDEsMotivation : American option under volatility uncertaintyPDE’s motivationGeneral frameworkQuasi-sure formulation of 2RBSDEs2RBSDE with upper obstacleSecond order doubly RBSDE




Reflected BSDEs : abstract setting

Definition (El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1997))

The solution to a reflected BSDE associated to (ξ, F , L) is a triple ofsquare integrable processes (yt , zt , kt), 0 ≤ t ≤ T such thatP− a.s.,

yt = ξ −∫ T

t

Fs(ys , zs) ds −∫ T

t

zs dWs + kT − kt ,

yt ≥ Lt , ∀t ∈ [0,T ],(kt)t≤T continuous and nondecreasing, k0 = 0 and∫ T

0(yt − Lt)dkt = 0.



Link with Optimal Stoping Problem

Theorem(y , z , k) is solution to the above RBSDE If and only If, for eacht ∈ [0,T ] and

yt = ess supτ∈Tt,T

EPt

[∫ τ

t

Fs(ys , zs)ds + Lτ1τ<T + ξ1τ=T

]where Tt,T is the set of all stopping times valued in [t,T ].

linear case : Snell envelope theorynon-linear case : fixed point theorem



Relations and Applications

Reflected BSDE’s or adapted backward Skorohod formulation isrelated to :

Optimal Stoping Problem

Pricing and hedging American option in complete andincomplete market

Variational inequalities

Obstacle problem for semilinear PDE’s : Viscositéor/and Sobolev solutions



Doubly Reflected BSDE’s : abstract setting

- Cvitanic and Karatzas : Backward SDE’s with reflection and Dynkingames. The Annals of Probability 24, No. 4, 2024-2056 (1996).- Hamadene, Lepeltier, M. : Double barriers reflected backwardSDEs with continuous coefficients, Pitman Research Notes inMathematics, 364, 115-128 (1997).The solution of a Reflected BSDE associated to (ξ, f , L, S) is aquadruple of processes (yt , zt , k+

t , k−t ), 0 ≤ t ≤ T such that

yt = ξ +∫ T

tfs(ys , zs) ds + (k−T − k−t )− (k+

T − k+t )−∫ T

tzs dWs , 0 ≤ t ≤ T ,

Lt ≤ yt ≤ St , ∀t ∈ [0,T ],(k+

t )t≤T and (k−t )t≤T are continuous and increasing processes,k+

0 = k−0 = 0 and∫ T

0 (yt − St)dk−t =

∫ T

0 (St − yt)dk+t = 0.



Game option

Definition of Game option : It’s a contact between a broker(seller) and a trader (buyer) with the specificity that both candecide to exercise before maturity date T .If the trader exercises first at a time t then the broker pays himthe (random) amount Lt .If the broker exercises before the trader at time t, the trader willbe given from him the quantity St ≥ Lt ,The difference St − Lt as to be understood as a penalty imposedon the seller for canceling the contract.In the case where they exercise simultaneously at t, the traderpayoff is Lt and if they both wait untill the maturity of thecontract T , the trader receives the amount ξ.



Contingent claim

we can summurize as following : we consider that the brokerexercises at a stopping time τ ≤ T and the trader at anothertime σ ≤ T then the trader receive from the broker thefollowing payoff :

H(σ, τ) := Sτ1τ<σ + Lσ1σ≤τ + ξ1σ∧τ=T .

In other words, this is an American option which has thespecificity that the seller can also "exercise" early.see Kifer (2006, 2012), Hamadene (2006).



Standard wealth-portfolio

LetM be a standard financial complete market (1 risky asset S anda bond). It is well known that in some constrained cases the couplewealth-portfolio (X , π) satisfies :

X Pt = ξ +

∫ T

t

b(s,X Ps , π

Ps )ds −

∫ T

t

πPs σsdWs , P− a.s

where W is a Brownian motion under the underlying probabilitymeasure P, b is convex and Lipschitz with respect to (x , π).The classical case corresponds to b(t, x , π) = −rtx − π.σtθt , whereθt is the risk premium vector.



Pricing and hedging Game option

TheoremThe fair price of the game option and the corresponding hedgingstrategy are given by the pair (yP, πP) ∈ D2(P)×H2(P) solving thefollowing DRBSDE

yPt = ξ +

∫ T

tb(s, yP

s , πPs )ds −

∫ T

tπPs σsdWs + kP

t − kPt , P− a.s.

Lt ≤ yPt ≤ St , P− a.s.∫ T

0 (yPt− − Lt−)dkP,−

t =∫ T

0 (St− − yPt−)dkP,+

t = 0.

Moreover, the following stopping times are optimal after t for theseller and the buyer respectively

τ ∗,Pt := infs ≥ t, yP

s ≥ Ss

, σ∗,Pt := inf

s ≥ t, yP

s ≤ Ls.



Pricing and hedging Game option

TheoremThe value of the Dynkin game is given by :

yPt = vP

t = vPt

wherev t := ess inf

τ∈Tt,Tess supσ∈Tt,T

E Pt [H(τ, σ)], P− a.s..

andv t := ess sup

σ∈Tt,Tess infτ∈Tt,T

E Pt [H(τ, σ)], P− a.s..



Outline






References

Let us now go to our uncertain volatility framework. The pricing ofEuropean contingent claims has already been treated in thatcontext :

By Avellaneda, Lévy and Paras (1995) (PDE’s framework),By Denis and Martini (2009) with capacity theory,By Nutz, Nutz and Soner (2011, 2012).By Vorbrink (2011) using the G-expectation framework.



The uncertain volatility case

We now consider a financial market with one risky asset S , whosedynamic is given by

dSt

St

= rtdt + σt(ω)dWt , P− a.s.

where σt is a unknown volatility process which may satisfies thefollowing :

σ ≤ σt(ω) ≤ σ

with σ and σ are a given bounds.Avellaneda, Lévy and Paras (1995) (PDE’s framework),Lyons, T. (1998)



The uncertain volatility case : quasi-sure analysis

In our quasi-sure context, we write the dynamic of the risky asset as :

dSt

St

= rtdt + dBt , P− a.s.,∀P ∈ P .

where P is a set of singular probability measures and B is a canonicalprocess defined on the Wiener space which is a local martingaleunder each P.

The volatility is implicitly embedded in the model. Indeed, undereach P ∈ P , we have dBs ≡ a

1/2t dW P

t where W P is a Brownianmotion under P.Therefore, a1/2 plays the role of volatility under each P and thusmakes it possible to model the volatility uncertainty.



Wealth process

We assume as above that our wealth process has the followingdynamic

Xt = ξ +

∫ T

t

b(s,Xs , πs)ds −∫ T

t

πsdBs , P − q.s.

Now the market is no longer complete, we consider thesuper-replication price, which is going to be

Xt =Pess sup

P′∈P(t+,P)X P′t , P− a.s., ∀P ∈ P .

Question : what is the dynamic satisfied by (Xt)t≥0 ? ?



BSDEs and semilinear PDEs

The Markov case corresponds to

Ft(ω, y , z) = f (t,Xt(ω), y , z) and ξ(ω) = g(XT (ω)

)where Xt = X0 +

∫ t

0 b(s,Xs)ds +∫ t

0 σ(s,Xs)dWs

In this context, under some conditions, we have

Yt = V (t,Xt), Zt = σTDV (t,Xt)

And V is a classical solution (V ∈ C 1,2) of the semilinear PDE

∂tV + b · DV +12Tr[σσTD2V

]= f

(.,V , σTDV

)Anis Matoussi Stochastic and game under uncertainty 19 / 65


A fully nonlinear PDE : Black -Scholes andBarenblatt PDE’s

G (γ) := 12 supa≤a≤a(aγ) = 1

2 (aγ+ − aγ−)

Suppose that the PDE

∂tu + G (D2u) = 0, and u(T , .) = g

has a smooth solution. Then, with Xt = X0 +∫ t

0 α1/2s dWs ,

(Yt := u(t,Xt), Zt := Du(t,Xt)) verifies the following equation

Yt = g(XT )−∫ T

t

ZsdXs + KT − Kt ,

where Kt :=∫ t

0

(G (D2u)− 1

2αsD2u)

(s,Xs)ds.



Semilinear to fully nonlinear PDE’s

Let u be a solution of

∂tu + H(., u,Du,D2u) = 0, u(T , .) = g

with H(x , r , p, γ) = supa≥0

12aγ − f (x , r , p, a)

.

Then u = supa ua where ua is a solution of

∂tua +

12aD2ua − f (., ua,Dua, a) = 0, ua(T , .) = g

a semilinear PDE which corresponds to a BSDE...



Link with the Quasi-sure stochastic analysis

This suggests to introduce

”Yt = supαYαt ”

Yαt = g(XαT )−

∫ T

t

f (.,Xαs ,Yαs ,Zαs , αs)ds −

∫ T

t

Zαs dXαs ,

where dXαs = α

1/2s dWs .

This is similar to stochastic control theory, since we end up with afamily of processes Yα. Then, changing α amounts to changingthe underlying probability measure.



General framework

Soner, Touzi and Zhang (2012)

Ω :=ω ∈ C ([0,T ],Rd), ω0 = 0

, ‖ω‖∞ := sup0≤t≤T |ωt |, B :

the canonical process (Bt(ω) = ωt), P0 : Wiener measure.F := Ft0≤t≤T : filtration generated by B , F+ := F+

t 0≤t≤T :right limit of F.

Karandikar(1995) :∫ t

0 BsdBTs , defined ω−wise, coincides with

Itô’s integral, P−a.s. for all local martingale measure P. Then〈B〉t := BtB

Tt − 2

∫ t

0 BsdBTs and at := lim

ε↓01ε

(〈B〉t − 〈B〉t−ε

)are defined ω−wise.



Non-dominated family of measures

For every F−prog. meas. α valued in S>0d with

∫ T

0 |αt |dt <∞,P0−a.s. Define

Pα := P0 (Xα)−1 where Xαt :=

∫ t

0α1/2s dBs , P0 − a.s.

(P0,B , α,Xα) ≡ (Pα,W Pα , a,B)

Then every Pα satisfiesthe Blumenthal zero-one lawthe martingale representation property

PS : collection of all such Pα



Generator Ht(ω, y , z , γ) : [0,T ]× Ω× R× Rd × DH → R

Convex conjugate :

Ft(ω, y , z , a) := supγ∈DH

12Tr[aγ]− Ht(ω, y , z , γ)

, a ∈ S>0

d

Ft(y , z) := Ft(y , z , at) and F 0t := Ft(0, 0)

PH consists of all P ∈ PS such that

aP ≤ a ≤ aP, dt × dP− a.s., aP, aP ∈ S>0d ,

and EP[(∫ T

0

∣∣∣F 0t

∣∣∣2 dt)] < +∞

Definition PH−quasi-surely (PH−q.s.) means P−a.s. for allP ∈ PH



Spaces and norms

L2H :=

ξ,FT −meas. : ‖ξ‖L2

H<∞

, ‖ξ‖2L2

H:= sup

P∈PH

E P[|ξ|2]

D2H :=

Y , F+−prog. in R càdlàg PH − q.s. ‖Y ‖D2

H<∞

‖Y ‖2D2

H:= supP∈PH

E P[sup0≤t≤1 |Yt |2

]H2

H :=Z , F+−prog. meas. in Rd : ‖Z‖2H2

H<∞

‖Z‖2H2

H:= supP∈PH

E P[ ∫ 1

0 |a1/2t Zt |2dt

]Anis Matoussi Stochastic and game under uncertainty 26 / 65


2BSDEs : definition (Soner, Touzi and Zhang,2010-2012)For FT−meas. ξ, we consider the 2BSDE :

Yt = ξ −∫ T

tFs(Ys ,Zs)ds −

∫ T

tZsdBs + KT − Kt , PH − q.s.

We say (Y ,Z ) is a solution to the 2BSDE ifYT = ξ, PH−q.s.For each P ∈ PH , KP has nondecreasing paths, P−a.s. :

KPt :=Y0−Yt+

∫ t

0Fs(Ys ,Zs)ds+

∫ t

0ZsdBs , t ∈ [0,T ], P− a.s.

The family of processes KP,P ∈ PH satisfies :KP

t = ess infP′∈PH(t+,P)

P E P′t [KP′

T ], P− a.s. ∀P ∈ PH

where PH(t+,P) :=P′ ∈ PH : P′ = P on F+

t

.



2BSDEs : assumptions

• PH is not empty and the domain of a 7−→ Ft(y , z , a) isindependent of (ω, y , z)

• F is F−progressively measurable and uniformly continuous in ω forthe ‖·‖∞-norm• Uniformly Lipschitz in (y , z) :∣∣∣Ft(y , z)− Ft(y

′, z ′)∣∣∣ ≤ C (|y − y ′|+ |a1/2(z − z ′)|), PH − q.s.



Definition : Second order RBSDE

For FT−meas. ξ, consider the 2RBSDE with lower obstacle L :

dYt = Ft(Yt ,Zt)dt + ZtdBt − dKPt .

We say (Y ,Z ) ∈ D2H ×H2

H is a solution to the 2RBSDE if

YT = ξ, and Yt ≥ Lt , PH−q.s.For each P ∈ PH , KP has nondecreasing paths, P−a.s. :

KPt :=Y0−Yt +

∫ t

0 Fs(Ys ,Zs)ds+∫ t

0 ZsdBs , t ∈ [0,T ], P− a.s.

The family of processes KP,P ∈ PH satisfies :

KPt − kP

t =P

ess infP′∈PH(t+,P)

EP′t

[KP′

T − kP′T

], P− as, ∀P ∈ PH .



Definition

We have denoted by (yP, zP, kP) a solution of standard RBSDEassociated to (ξ, F , S) , namely :

yPt = ξ +

∫ T

t

Fs(yPs , z

Ps ) ds + kP

T − kPt −

∫ T

t

zs dWs ,

yPt ≥ Lt , ∀t ∈ [0,T ],

(kPt )t≤T is continuous and increasing process, k0 = 0 and∫ T

0(yP

t− − Lt−)dkPt = 0.



Remark

Actually, using recent result of M. Nutz (2012) on the pathwiseconstruction of stochastic integrals , we can deduce that thefamily

(KP)P∈PH

can be aggregated into a universal process K ,

Therefore, one may denote by (Y ,Z ,K ) the solution of the2RBSDE’s.



Back to standard RBSDE’s

If H is linear in γ, that is to say

Ht(y , z , γ) :=12Tra0

t γ − ft(y , z),

where a0 : [0,T ]× Ω→ S>0d is F-progressively measurable and

has uniform upper and lower bounds.Besides, the domain of F is restricted to a0 and we have

Ft(y , z) = ft(y , z).

Assume that there exists some P ∈ PS such that a and a0

coincide P− a.s. and EP[∫ T

0 |ft(0, 0)|2dt]< +∞, then

PH = P.



Back to standard RBSDE’s

However, we know that the process KP − kP is a martingale withfinite variation. Since P satisfy the martingale representationproperty, this martingale is also continuous, and therefore it isnull. Thus we have

kP = KP, P− a.s.

Therefore the 2RBSDE is equivalent to a standard RBSDE.In particular, the part of KP which increases only whenYt− > Lt− is null, which means that KP satisfies the Skorohodcondition, and then (Y ,Z ,K ) is a solution of a standardRBSDE’s.



Representation and uniqueness

Theorem (M., Possamai, Zhou (2012))

Assume ξ ∈ L2H and that (Y ,Z ) ∈ D2

H ×H2H is a solution to

2RBSDE. Then, for any P ∈ PH and 0 ≤ t1 < t2 ≤ T ,

Yt1 =Pess sup

P′∈PH(t+1 ,P)

yP′t1

(t2,Yt2), P− a.s.

Consequently, the 2RBSDE has at most one solution in D2H ×H2

H .



Existence result


Let ξ ∈ L2H . Then

1) There exists a unique solution (Y ,Z ) ∈ D2,κH ×H2,κ

H of the2RBSDE.2) Moreover, if in addition we choose to work under the followingmodel of set theory :(i) Zermelo-Fraenkel set theory with axiom of choice (ZFC) plus the

Continuum Hypothesis (CH).Then there exists a unique solution (Y ,Z ,K ) ∈ D2,κ

H ×H2,κH ×A2,κ

H ofthe 2RBSDE.



More information about the non-decreasingprocesses KP

We now show that we can obtain more information about thenon-decreasing processes KP.

Proposition (M., Possamai, Zhou (2012))

Let (Y ,Z ) ∈ D2H ×H2

H is a solution to the 2RBSDE. Let(yP, zP, kP)

P∈PH

be the solutions of the corresponding BSDEs.Then, for all t ∈ [0,T ], we have :∫ t

01Ys−=Ls−dK

Ps =

∫ t

01Ys−=Ls−dk

Ps , P− a.s.



Remarks

Recall that the role of the non-decreasing processes KP is on theone hand to keep the solution of the 2RBSDE above the obstacleL and on the other hand to keep it above the correspondingRBSDE solutions yP, as confirmed by the representation formula.What the above result tells us is that if Y becomes equal to theobstacle, then it suffices to push it exactly as in the standardRBSDE case. This is conform to the intuition. Indeed, when Yreaches L, then all the yP are also on the obstacle, therefore,there is no need to counter-balance the second order effects.



More remarks..

The above result leads us naturally to think that one coulddecompose the non-decreasing process KP into twonon-decreasing processes AP and V P such that AP satisfies theusual Skorohod condition and V P satisfies

V Pt =

Pess inf

P′∈PH(t+,P)EP′t

[V P′T

], 0 ≤ t ≤ T , P− a.s., ∀P ∈ PH .

Such a decomposition would isolate the effects due to theobstacle and the ones due to the second-order. Of course, thechoice AP := kP would be natural, given the minimum condition.However the situation is not that simple. Indeed, we know that∫ t

01Ys−=Ls−dK

Ps =

∫ t

01Ys−=Ls−dk

Ps .



More remarks..

But kP can increase when Y is strictly above the obstacle, sincewe can have Yt− > yP

t− = Lt− . We can thus only write

KPt =

∫ t

01Ys−=Ls−k

Ps + V P

t .

Then V P satisfies the minimum condition when Yt− = Lt− andwhen yP

t− > Lt− .However, we cannot say anything when Yt− > yP

t− = Lt− . Theexistence of such a decomposition, which is also related to thedifficult problem of the Doob-Meyer decomposition for theG -submartingales of Peng, is therefore still an open problem.



The case of regular obstacle

PropositionAssume that L is a semi-martingale of the form

Lt = L0 +

∫ t

0Usds +

∫ t

0VsdBs + Ct , PH − q.s.

where C is càdlàg process of integrable variation such that themeasure dCt is singular with respect to the Lebesgue measure dt andCt = C+

t − C−t , where C+ and C− are non-decreasing processes.Then, on the set

Yt− = Lt−

, we have

Zt = Vt , dt × PH − q.s.,0 ≤ dKP

t 1Yt−=Lt− ≤ 1Yt−=Lt−([Ft(St ,Vt) + Ut

]−dt + dC−t

).



2RBSDE solution as value of an optimal stoppingproblem


Let (Y ,Z ) be the solution to the above 2RBSDE. Then for eacht ∈ [0,T ] and for all P ∈ PH

Yt =

ess supP

P′∈PH(t+,P)ess supτ∈Tt,T

EP′t

[∫ τtFs(y

P′s , z

P′s )ds + Lτ1τ<T + ξ1τ=T

]Yt =

ess supτ∈Tt,T

EPt

[∫ τtFs(Ys ,Zs)ds + AP

τ − APt + Lτ1τ<T + ξ1τ=T

],

where Tt,T is the set of all stopping times valued in [t,T ] and whereAP

t :=∫ t

0 1Ys−>Ls−dKP

s is the part of KP which only increases whenYs− > Ls− .Anis Matoussi Stochastic and game under uncertainty 41 / 65


Lower or upper obstacle ?

We want to highlight here that unlike with classical RBSDEs,considering an lower obstacle in our context is fundamentallydifferent from considering a upper obstacle.Indeed, having an lower obstacle corresponds, at least formally,to add an increasing process in the definition of a 2BSDE.Since there is already an increasing process in that definition, westill end up with an increasing process. However, in the case of aupper obstacle, we would have to add a decreasing process inthe definition, therefore ending up with a finite variation process.Furthermore, in that case the above representation would holdwith a sup-inf instead of a sup-sup, indicating that this situationshould be closer to stochastic games than to stochastic control.



Definition : 2RBSDE with upper obstacle

For FT−meas. ξ, consider the 2RBSDE with lower obstacle L :

dYt = Ft(Yt ,Zt)dt + ZtdBt − dV Pt .



YT = ξ, and Yt ≤ St , PH−q.s.For each P ∈ PH , V P has paths of bounded variation, P−a.s. :V Pt :=Y0−Yt +

∫ t


0 ZsdBs , t ∈ [0,T ], P− a.s.

The family of processes V P,P ∈ PH satisfies the minimalcondition :

V Pt + kP

t =P

ess infP′∈PH(t+,P)

EP′t

[V P′T + kP′

T

], P− as, ∀P ∈ PH .



Definition

We have denoted by (yP, zP, kP) a solution of standard RBSDEassociated to (ξ, F , S) , namely :

yPt = ξ +

∫ T

t

Fs(yPs , z

Ps ) ds − kP

T + kPt −

∫ T

t

zs dWs ,

yPt ≤ St , ∀t ∈ [0,T ],


0(St− − yP

t−)dkPt = 0.



Assumptions

In addition to the Assumptions in the 2BDSE case, we assumeThe upper obstacle S is a semimartingale for every P ∈ PH , withthe decomposition

St = S0 +

∫ t

0PsdBs + AP

t , P− a.s., for all P ∈ PH ,

where the AP are bounded variation process with Jordandecomposition AP,+ − AP,− with some integrability conditions.



About the bounded variation process

Proposition (M., Piozin, Possamai (2012))

Let (Y ,Z ,V ) ∈ D2H ×H2

H × A2H is a solution to the 2RBSDE. Let

(yP, zP, kP)P∈PH

be the solutions of the corresponding BSDEs.Then we have the following results

(i) For all t ∈ [0, 1], VPt :=

∫ t

0 1yPs−<Ss−

dV Ps is a non-decreasing

process, P− a.s.

(ii) For all t ∈ [0, 1], V Pt :=

∫ t

0 1yPs−

=Ss−dV P

s = −kPt , P− a.s., and is

therefore a non-increasing process.



Remark

The above Proposition is crucial for us. Indeed, we have actuallyshown that

V Pt = V

Pt − kP

t , P− a.s.,

whereV

Pand kP are two non-decreasing processes which never act at

the same time.This decomposition allow us to obtain the desired a prioriestimates :



A priori estimates

Theorem (M., Piozin, Possamai (2012))

Let (Y ,Z ,V ) ∈ D2H ×H2

H be a solution to the 2RBSDE (upperobstacle) . Let

(yP, zP, kP)

P∈PκH

be the solutions of thecorresponding RBSDEs. Then, there exists a constant C dependingonly on T and the Lipschitz constants of F such that‖Y ‖2D2

H+ ‖Z‖2H2

H+ sup

P∈PκHEP[Var0,T

(VP)2] ≤ C

(‖ξ‖2L2

H+ φ2

H + ψ2H

).

supP∈PH

∥∥yP∥∥2D2(P) +

∥∥zP∥∥2H2(P) + EP

[(kPT

)2] ≤C(‖ξ‖2L2

H+ φ2

H + ψ2H

).



Optimal stopping problem : sup inf

We can show, as in the classical framework, that the solution Y ofthe 2RBSDE is linked to an optimal stopping problem.


Let (Y ,Z ) be the solution to the above 2RBSDE. Then for eacht ∈ [0,T ] and for all P ∈ PH

Yt = ess supP

P′∈PH(t+,P)essinfτ∈Tt,T

EP′t

[∫ τtFs(y

P′s , z

P′s )ds + Sτ1τ<T + ξ1τ=T

]Yt =

essinfτ∈Tt,T

EPt

[∫ τtFs(Ys ,Zs)ds + V P

τ − V Pt + Sτ1τ<T + ξ1τ=T

],

where Tt,T is the set of all stopping times valued in [t,T ].



Definition : Second order double reflected BSDE’s

For FT−meas. ξ, consider the Doubly 2RBSDE with obstacles L andS :

dYt = Ft(Yt ,Zt)dt + ZtdBt − dV Pt .



YT = ξ, and Lt ≤ Yt ≤ St , PH−q.s.For each P ∈ PH , V P has paths of bounded variation, P−a.s. :V Pt :=Y0−Yt +

∫ t


0 ZsdBs , t ∈ [0,T ], P− a.s.

The family of processes V P,P ∈ PH satisfies the minimalcondition :

V Pt + kP,+

t − kP,−t =

Pess inf

P′∈PH(t+,P)EP′

t

[V P′T + kP′ ,+

T − kP′ ,−T

], P− as, ∀P ∈ PH .



Definition

We have denoted by (yP, zP, kP,+, kP,−) a solution of standarddoubly RBSDE associated to (ξ, F , L, S) , namely :

yPt = ξ+

∫ T

t

Fs(yPs , z

Ps ) ds+kP,−

T −kP,−t −kP,+

T −kP,+t −

∫ T

t

zs dWs ,

Ly ≤ yPt ≤ St , ∀t ∈ [0,T ],


0(St− − yP

t−)dkP,+t = 0 and

∫ T

0(yP

t− − Lt−)dkP,−t = 0.



Assumption : Regular lower obstacle

The setting is the same as for upper reflected 2BSDEs :

S is a F-progressively measurable càdlàg semimartingale forevery P ∈ PH , with the decomposition

St = S0 +

∫ t

0PsdBs + AP

t , P− a.s., for all P ∈ PH ,

where the AP are bounded variation process with Jordandecomposition AP,+ − AP,− with some integrability conditions.For all t ∈ [0,T ], we have

Lt < St and Lt− < St− , PH − q.s.

integrability conditions.



About the bounded variation process

PropositionLet (Y ,Z ,V ) ∈ D2

H ×H2H × I2H is a solution to the Doubly 2RBSDE.

Let

(yP, zP, kP,+, kP,−)P∈PH

be the solutions of the correspondingdoubly reflected BSDEs. Then we have the following results

(i) For all t ∈ [0, 1], VPt :=

∫ t

0 1yPs−<Ss−

dV Ps is a non-decreasing

process, P− a.s.

(ii) For all t ∈ [0, 1], V Pt :=

∫ t

0 1yPs−

=Ss−dV P

s = −kP,+t , P− a.s., and

is therefore a non-increasing process.



Remark

We can go further into the structure of the bounded variationprocesses V P. Indeed, we could also show that

1Yt−=Lt−dV P

t = 1Yt−=Lt−dkP,−

t .

Notice however that we, a priori, cannot say anything about V P whenLt− = yP

t− < Yt− , even though we showed that it could be knownexplicitely when St− = yP

t− . This emphasizes once more the fact thatthe upper and the lower obstacle in our context do not play asymmetric role.



Representation and uniqueness


Assume ξ ∈ L2H and that (Y ,Z ,V ) ∈ D2

H ×H2H is a solution to

2DRBSDE. Then, for any P ∈ PH and 0 ≤ t1 < t2 ≤ T ,

Yt1 =Pess sup

P′∈PH(t+1 ,P)

yP′t1

(t2,Yt2), P− a.s.

Consequently, this gives uniqueness of the solution.



Existence result


Let ξ ∈ L2H . Then

1) There exists a unique solution (Y ,Z ) ∈ D2,κH ×H2,κ

H of the2DRBSDE.2) Moreover, if in addition we choose to work under the followingmodel of set theory :(i) Zermelo-Fraenkel set theory with axiom of choice (ZFC) plus the

Continuum Hypothesis (CH).Then there exists a unique solution (Y ,Z ,V ) ∈ D2,κ

H ×H2,κH ×V2,κ

H ofthe 2DRBSDE.



Outline






The zero-sum differential game under uncertainty

Let us now describe what we mean precisely by a Dynkin game withuncertainty :

Two players P1 and P2 are facing each other in a game. Astrategy of a player consists in picking a stopping time.The player P1 chooses τ ∈ T0,T and P2 chooses σ ∈ T0,T .Then the game stipulates that P1 will pay to P2 the followingrandom payoff

Rt(τ, σ) :=

∫ τ∧σ

t

gsds + Sτ1τ<σ + Lσ1σ≤τ,σ<T + ξ1τ∧σ=T ,

where g , S and L are F-progressively measurable processes



The zero-sum differential game under uncertainty

P1 will try to minimize the expected amount that he will have topay, but taking into account the fact that both P2 and the"Nature" (which we interpret as a third player, represented bythe uncertainty that our player have with respect to theunderlying probability measure) can play against him.Symmetrically, P2 will try to maximize his expected returns,considering that both P1 and the Nature are antagonist players.This leads us to introduce the following upper and lower valuesof our robust Dynkin game

V t := ess infτ∈Tt,T

ess supσ∈Tt,T

Pess supP′∈PκH (t+,P)

E P′t [Rt(τ, σ)], P− a.s.

V t := ess supσ∈Tt,T

ess infτ∈Tt,T

Pess inf

P′∈PκH (t+,P)E P′

t [Rt(τ, σ)], P− a.s.



Remark

V is the maximal amount that P1 will agree to pay in order totake part in the game.Symmetrically, V is the minimal amount that P2 must receive inorder to accept to take part to the game.Unlike in the classical setting without uncertainty, for whichthere is only one value on which the 2 players can agree, in ourcontext there is generally a whole interval of admissible valuesfor the game. Indeed, we have the following easy result :

Lemma

∀t ∈ [0,T ], ,V t ≥ V t , PκH − q.s.

Therefore the admissible values for our game are the interval [V t ,V t ].



Definition

DefinitionFor ξ ∈ L2

H , we consider the following type of equations satisfied by apair of progressively-measurable processes (Y ,Z )

• YT = ξ, PH − q.s.• ∀P ∈ PκH , the process V P defined below has paths of bounded

variation P− a.s.

V Pt := Y0−Yt−

∫ t

0Fs(Ys ,Zs)ds+

∫ t

0ZsdBs , 0 ≤ t ≤ T , P−a.s.

• We have the following maximum condition for 0 ≤ t ≤ T

V Pt +kP,+

t −kP,−t =

Pess supP′∈PH(t+,P)

EP′t

[V P′T + kP′ ,+

T − kP′ ,−T

], ∀P ∈ PH .

• Lt ≤ Yt ≤ St , PH − q.s.Anis Matoussi Stochastic and game under uncertainty 61 / 65


Remark

This Definition is symmetric in the sense that if (Y ,Z ) solves anequation as in the last Definition, then (−Y ,−Z ) solves a2DRBSDE (in the sense of first Definition) with terminalcondition −ξ, generator g(y , z) := −g(−y ,−z), lower obstacle−S and upper obstacle −L.With this remark, it is clear that we can deduce a wellposednesstheory for the above equations. In particular, we have thefollowing representation

Yt = ess infP′∈PκH (t+,P)

yP′t , P− a.s., for any P ∈ PκH .



Dynkin game value under uncertainty


Assume "min-max" condition. Let (Y ,Z ) (resp. (Y , Z )) be asolution to the 2DRBSDE (resp. in the sense of Definition givenabove) with terminal condition ξ, generator g , lower obstacle L andupper obstacle S . Then we have for any t ∈ [0,T ]

V t = Yt , V t = Yt , PH − q.s.

Moreover, unless PH is reduced to a singleton, we haveV > V , PH − q.s.



Remark

We assumed a min-max property which is closely related to theusual Isaacs condition for the classical Dynkin game, in order tolink the solution of the Dynkin game under uncertainty to2DRBSDEs.Nutz and Zhang [NZ12] also showed such a result (at least attime 0) when there is only one player. We intend to prove similarresult in our context . ? ?



Questions ?

Thank you for your attention !


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