representation formulas for solutions of isaacs...
TRANSCRIPT
Representation formulas for solutions of
Isaacs integro-PDE
Shigeaki Koike∗
Mathematical Institute, Tohoku UniversityAoba, Sendai, Miyagi 980-8578, Japan
E-mail: [email protected]
AND
Andrzej Swiech†
School of Mathematics, Georgia Institute of Technology
Atlanta, GA 30332, U.S.A.E-mail: [email protected]
Abstract
We prove sub- and super-optimality inequalities of dynamic programming for vis-cosity solutions of Isaacs integro-PDE associated with two-player, zero-sum stochas-tic differential game driven by a Levy type noise. This implies that the lower andupper value functions of the game satisfy the dynamic programming principle andthey are the unique viscosity solutions of the lower and upper Isaacs integro-PDE.We show how to regularize viscosity sub- and super-solutions of Isaacs equations tosmooth sub- and super-solutions of slightly perturbed equations.
Keywords: viscosity solutions, integro-PDE, Isaacs equation, stochastic differential equa-
tion, Levy process, stochastic differential game.
2010 Mathematics Subject Classification: 35R09, 49L25, 90D15, 90D25.
1 Stochastic differential game
Throughout this paper (Ω,F ,P) will be a complete probability space and Ft, t ≥ 0, will
be a normal filtration (i.e. Ft is right continuous and F0 contains all null sets of P).
∗Supported by Grant-in-Aid for Scientific Research (No. 23340028) of JSPS.†Supported by NSF grant DMS-0856485.
1
Let T > 0, and L be a Levy process on [0, T ] in Rm with cadlag (right continuous
with left limits) sample paths, which is of the form
L(t) = L0(t) + L1(t), (1.1)
where L0, L1 are independent Levy processes,
L0(t) =
∫ t
0
∫
0<‖y‖<1
yπ(ds, dy), L1(t) =
∫ t
0
∫
‖y‖≥1
yπ(ds, dy),
where π is the Poisson random measure of jumps of L and π is the compensated Poisson
random measure of jumps:
π([0, t], B) =∑
0<s≤t
1B(L(s) − L(s−)), B ∈ B(Rm \ 0), L(s−) = limt↑s
L(t),
π(dt, dy) = π(dt, dy)− dt ν(dy).
The measure ν, called the Levy measure of L or the jump intensity measure of L, is a
non-negative measure on (Rm \ 0,B(Rm \ 0)), where B(Rm \ 0) is the Borel σ-field,
for which∫
0<‖y‖<1
‖y‖2ν(dy) +
∫
‖y‖≥1
ν(dy) < +∞. (1.2)
π and π are random measures on B([0, T ])⊗B(Rm \ 0)). We extend ν to a measure on
(Rm,B(Rm)) by setting ν(0) = 0.
We will assume in addition that L is an Ft-Levy process, i.e. that: (i) L is Ft-adapted;
(ii) L(s) − L(t), i = 0, 1 is independent of Ft for all 0 ≤ t < s ≤ T . The process L0 is
then a square integrable Ft-martingale. L1 is a compound Poisson process. According to
the Levy-Ito decomposition (see [4], Theorem 2.4.16), (1.1) is a general form of a Levy
process of pure jump type. Moreover, for every B ∈ B(Rm \ 0), N(t, B) := π([0, t], B)
is a martingale. We will denote
π(dt, dy) =
π(dt, dy) if ‖y‖ < 1π(dt, dy) if ‖y‖ ≥ 1.
The σ-field of predictable sets on [0, T ] × Ω is the smallest σ-field containing all the
sets of the form (s, t] × A, 0 ≤ s < t ≤ T,A ∈ Fs, and 0 × A, A ∈ F0. A stochastic
process with values in a measurable space (E, E) is called predictable if it is measurable
as a map between [0, T ] × Ω and E, where [0, T ] × Ω is equipped with the σ-field of
predictable sets.
We consider the following two-player, zero-sum stochastic differential game. We follow
the Elliot and Kalton [19] definition of the game which was also used by Fleming and
Souganidis [21].
2
Let W,Z be separable metric spaces. They are equipped with the Borel σ-fields
B(W),B(Z). For every t ∈ [0, T ] we define
M(t) := W : [t, T ] × Ω → W, W is predictable,
N(t) := Z : [t, T ] × Ω → Z, Z is predictable.
We will call M(t) the set of controls for player I, and N(t) the set of controls for player
II. We identify two controls Z1, Z2 ∈ N(t) on [t, s] if Z1 = Z2, dt ⊗ dP a.e. on [t, s] × Ω.
We will then write Z1 = Z2 on [t, s]. The same convention applies to controls in N(t).
The admissible strategies for player I are defined by
Γ(t) := α : N(t) →M(t), non-anticipating,
and the admissible strategies for player II by
∆(t) := β : M(t) → N(t), non-anticipating.
Strategy α (resp., β) is non-anticipating if whenever Z1 = Z2 (resp., W1 = W2) on [t, s]
then α[Z1] = α[Z2] (resp., β[W1] = β[W2]) on [t, s] for every s ∈ [t, T ].
Remark 1.1. Various kinds of controls are considered in the optimal control literature.
Perhaps the more typical choice is to assume that controls are cadlag and Fs-adapted.
Such controls are considered in [24, 32]. Predictable controls give a broader class and
may seem more natural. Predictable controls are used in [11, 13]. We remark that if a
process M(s) is cadlag and Fs-adapted then the process M(s−) is predictable. In general
any measurable, stochastically continuous and adapted process with values in a separable
Banach space has a predictable modification (see for instance [33], Proposition 3.21).
For an initial time t ∈ [0, T ] and x ∈ R, the dynamics of the game are given by a
stochastic differential equation (SDE)
dX(s) = b(s,X(s),W (s), Z(s))ds+∫
Rm\0γ(s,X(s),W (s), Z(s), y)π(ds, dy)
X(t) = x,(1.3)
where b : [0, T ] × Rn ×W ×Z → R
n, γ : [0, T ] × Rn ×W ×Z × R
m → Rn.
The pay-off functional is given by
J(t, x;W (·), Z(·)) = IE
∫ T
t
l(s,X(s),W (s), Z(s))ds+ h(X(T ))
(1.4)
for some functions l : [0, T ] × Rn ×W ×Z → R, h : R
n → R.
The game is played in continuous time. Player I controls W and wants to maximize J
over all choices of Z whereas Player II controls Z and wants to minimize J over all choices
3
of W . The game thus depends on which player moves first and we are forced to consider
two versions of the game, the so called lower and upper games. In the lower game, Player
II chooses Z(s) knowing W (s), and in the upper game, Player I chooses W (s) knowing
Z(s).
The value functions of the lower and upper games, called respectively the lower and
upper values are defined as follows:
V (t, x) = infβ∈∆(t)
supW∈M(t)
J(t, x;W (·), β[W ](·)) (lower value of the game), (1.5)
U(t, x) = supα∈Γ(t)
infZ∈N(t)
J(t, x;α[Z](·), Z(·)) (upper value of the game). (1.6)
We remark that in this formulation of the game the reference probability space (Ω,F ,Fs,P,
L) is fixed and does not change when we define M(t) and N(t). If the filtration Fs was
generated by the Levy process L this would mean that the controls in M(t) and N(t) may
depend on the past of L before time t, i.e. on Ft. However the techniques of this paper
work as well for other formulations of the game, see comments after the proof of Theorem
7.1.
The lower value function V should satisfy the lower Isaacs integro-PDE
Vt + F−(t, x,DV, V (·)) = 0V (T, x) = h(x),
(1.7)
where the lower value Hamiltonian F− : [0, T ] × Rn × R
n × C2b(Rn) → R is defined by
F−(t, x, p, ϕ) = supw∈W
infz∈Z
〈b(t, x, w, z), p〉 + l(t, x, w, z) (1.8)
+
∫
Rm
(
ϕ(x+ γ(t, x, w, z, y))− ϕ(x) − 1‖y‖<1〈γ(t, x, w, z, y), Dϕ(x)〉)
ν(dy)
.
The upper value function U should satisfy the upper Isaacs integro-PDE
Ut + F+(t, x,DU, U(·)) = 0U(T, x) = h(x),
(1.9)
where the upper value Hamiltonian F+ : [0, T ] × Rn × R
n × C2b(R
n) → R is defined by
F+(t, x, p, ϕ) = infz∈Z
supw∈W
〈b(t, x, w, z), p〉 + l(t, x, w, z) (1.10)
+
∫
Rm
(
ϕ(x+ γ(t, x, w, z, y))− ϕ(x) − 1‖y‖<1〈γ(t, x, w, z, y), Dϕ(x)〉)
ν(dy)
.
Equations (1.7) and (1.9) will be understood in the viscosity sense. Since supw∈W infz∈Z ≤infz∈Z supw∈W , it is always true that F− ≤ F+. Thus, if V is a viscosity solution of
4
(1.7), it is a viscosity subsolution of (1.9). If comparison holds for (1.9), i.e. if a viscosity
subsolution is less than or equal to a viscosity supersolution, we obtain V ≤ U . If the
Isaacs condition F− = F+ is satisfied and viscosity solutions of (1.7) and (1.9) are unique,
it follows that V = U and we then say that the game has value.
The main purpose of this paper is to show how PDE techniques can be used to prove
sub- and super-optimality inequalities of dynamic programming contained in Theorem
7.1. This result in particular implies that the dynamic programming principle is satisfied
and that the unique solutions of the lower and upper Isaacs integro-PDE (1.7) and (1.9)
are satisfied by the lower and upper value functions (1.5) and (1.6). We adapt to the
jump diffusion/integro-PDE case a method used in [40]. It is based on regularization of
viscosity sub- and super-solutions of Isaacs equations and approximate optimal synthesis.
The method is constructive and provides a fairly explicit way to produce almost optimal
controls and strategies. We show how to regularize viscosity sub- and super-solutions of
Isaacs equations (1.7) and (1.9) to smooth sub- and super-solutions of slightly perturbed
equations. Our method however has some drawbacks. We need to introduce an elliptic
regularization by a small Laplacian term which corresponds to the introduction of an
independent Wiener process on the level of the stochastic state equation of the game.
Therefore we have to assume that the probability space can support a standard Wiener
process independent of L. Another limitation is that we have to restrict the dynamics of
the game to the noise of pure jump type. If the state equation (1.3) had a continuous part
of the noise (i.e. a Wiener process term), the Isaacs equations would have second-order
PDE terms. While some results (like Lemma 6.1) would still hold in this case, we do not
know if our methods could be modified to show Theorem 7.1 in this case, see Remarks
6.2, 6.5, and 6.7.
The first proof of the dynamic programming principle for two-player, zero-sum stochas-
tic differential game driven by continuous noise in the above formulation appeared in the
fundamental paper [21]. It used some ideas from earlier papers on the subject of deter-
ministic and stochastic games [19, 20, 31]. The proof and the methods of [21] have been
widely used in many subsequent works on the subject, for instance to obtain represen-
tation formulas for second order parabolic equations [27]. Recently stochastic differential
games and the dynamic programming principle have been approached using backwards
stochastic differential equations [12]. We refer to this paper and to an excellent research
expository article [14] for more on this topic, references, and general overview of the sub-
ject of stochastic differential games, methods, techniques, current trends and challenges.
The subject of two-player, zero-sum stochastic differential games driven by jump diffusions
is studied in [11, 13]. Both papers provide fairly general results. [11] adapts the methods
of [21] to the jump diffusion case and [13] is based on the use of backwards stochastic
5
differential equations. Our results apply to a less general case than these considered in
[11, 13], however they are somehow complementary to the results in these papers.
We refer the readers to [1, 2, 3, 5, 6, 7, 8, 9, 15, 16, 23, 25, 26, 37, 39] for more
on basic theory of viscosity solutions of integro-PDE and to [11, 13, 24, 32, 34, 38] and
the references therein for applications to stochastic control and stochastic differential
games for jump diffusions. Books [4, 10, 33, 36] are good references on the theory of Levy
processes and stochastic differential equations with Levy noise.
2 Notation
For a real valued, cadlag, square integrable martingale K we will denote by [K,K]t its
quadratic variation process (see [35], p. 57, or [30], p. 150). By the definition of the
quadratic variation we have
IE[K,K]T = IEK2(T ). (2.11)
For an interval I ⊂ R, we will be using the following function spaces.
B(Rn) = u : Rn → R : u is Borel measurable and bounded,
C2(Rn) = u : Rn → R : u,Du,D2u are continuous,
C2b(R
n) = u ∈ C2(Rn) : u is bounded,
C1,2(I × Rn) = u : I × R
n → R : u, ut, Du,D2u are continuous,
C1,2b (I × R
n) = u ∈ C1,2(I × Rn) : u is bounded,
For a metric space Z we will denote by B(Z) its Borel σ-field and by dZ its metric. We
will write Br(x) for the open ball of radius r centered at x. A modulus is a continuous,
nondecreasing and subadditive function σ : [0,+∞) → [0,+∞) such that σ(0) = 0.
A local modulus is a continuous function σ : [0,+∞) × [0,+∞) → [0,+∞) which is
nondecreasing in both arguments, subadditive in the first argument, and such that for
every s ≥ 0, σ(0, s) = 0.
3 Assumptions
We impose the following assumptions throughout the paper. Many of them, in particular
these related to the boundedness of cost functions and coefficients, can be relaxed however
we do not do this in order not to obscure the main ideas of the paper. Recall that ν is
the Levy measure satisfying (1.2).
6
(A1) There exists a Borel measurable, locally bounded function ρ on Rm, such that
inf‖y‖>r ρ(y) > 0 for every r > 0, and
∫
Rm
(ρ(y))2ν(dy) < +∞. (3.1)
(A2) b : [0, T ] × Rn ×W ×Z → R
n is bounded, uniformly continuous, and such that
‖b(s1, x1, w, z) − b(s2, x2, w, z)‖ ≤ C‖x1 − x2‖ + σ(|s1 − s2|), (3.2)
γ : [0, T ]×Rn×W×Z×R
m → Rn is such that γ(·, ·, ·, ·, y) is uniformly continuous
on [0, T ] × Rn ×W ×Z for every y, γ is Borel measurable with respect to y, and
‖γ(s1, x1, w, z, y)− γ(s2, x2, w, z, y)‖ ≤ Cρ(y) (‖x1 − x2‖ + σ(|s1 − s2|)) (3.3)
for all x1, x2 ∈ Rn, s1, s2 ∈ [0, T ], w ∈ W, z ∈ Z, y ∈ R
m for some modulus σ.
(A3) l : [0, T ] × Rn ×W × Z → R, h : R
n → R are bounded, uniformly continuous and
such that
|l(s1, x1, w, z) − l(s2, x2, w, z)| ≤ ω(‖x1 − x2‖ + |s1 − s2|), (3.4)
|h(x1) − h(x2)| ≤ ω(‖x1 − x2‖) (3.5)
for all x1, x2 ∈ Rn, s1, s2 ∈ [0, T ], w ∈ W, z ∈ Z, for some modulus ω.
(A4)
‖γ(s, x, w, z, y)‖ ≤ Cρ(y) (3.6)
for all s ∈ [0, T ], x ∈ Rn, w ∈ W, z ∈ Z, y ∈ R
m.
(A5) The filtered probability space (Ω,F ,Ft,P) is such that there exists an Ft-standard
Wiener process B in Rn defined on this space which is independent of L.
Under assumptions (A1) − (A4) the Hamiltonians F± are well defined.
Condition (A5) can always be achieved in the following way. Let L be initially de-
fined on a probability space (Ω1,F1,F1t ,P
1). We take a complete filtered probability space
(Ω2,F2,F2t ,P
2) and an F2t -standard Wiener process B in R
n. We define a product space
(Ω1 × Ω2,F1 ⊗F2,F1t ⊗ F2
t ,P1 ⊗ P
2), where we augmented the sigma field and the fil-
tration by the P1 ⊗ P
2 null sets. Finally to ensure that the filtration is right continuous
we can take Ft = ∩s>t(F1s ⊗F2
s ). The processes L,B are defined on this new probability
space in a natural way and are independent.
7
4 Viscosity solutions
In this section we recall the definition of viscosity solution for terminal value problems for
integro-PDE of the form
ut + F (t, x,Du, u(t, ·)) = 0, in (0, T ) × Rn,
u(T, x) = g(x),(4.1)
where F is defined by (1.8) or (1.10). To avoid too many purely technical complications,
in this paper we will only deal with bounded viscosity solutions. However we remark that
under assumption (3.1) we could consider solutions growing quadratically at infinity.
Definition 4.1. A bounded upper semicontinuous function u : (0, T ] × Rn → R is a
viscosity subsolution of (4.1) if u(T, x) ≤ g(x) on Rn and whenever u − ϕ has a global
maximum at a point (t, x) for a test function ϕ ∈ C1,2b ((0, T ) × R
n), then
ϕt(t, x) + F (t, x,Dϕ(t, x), ϕ(t, ·)) ≥ 0. (4.2)
A bounded lower semicontinuous function u : (0, T ]× Rn → R is a viscosity superso-
lution of (4.1) if u(T, x) ≥ g(x) on Rn and whenever u − ϕ has a global minimum at a
point (t, x) for a test function ϕ ∈ C1,2b ((0, T ) × R
n), then
ϕt(t, x) + F (t, x,Dϕ(t, x), ϕ(t, ·)) ≤ 0. (4.3)
A viscosity solution of (4.1) is a function which is both a viscosity subsolution and a
viscosity supersolution.
It is well known (e.g. [8, 25, 26, 39]) that the above definition is equivalent to the
following “localized” definition of viscosity solution.
For 0 < r < 1, (t, x, p, v, u) ∈ (0, T ) × Rn × R
n × C2(Rn) × B(Rn), we set
F−r (t, x, p, v, u) = sup
w∈Winfz∈Z
〈b(t, x, w, z), p〉 + l(t, x, w, z)
+
∫
‖y‖<r
(v(x+ γ(t, x, w, z, y))− v(x) − 〈γ(t, x, w, z, y), Dv(x)〉)ν(dy) (4.4)
+
∫
‖y‖≥r
(
u(x+ γ(t, x, w, z, y)) − u(x) − 〈γ(t, x, w, z, y), p〉1‖y‖<1
)
ν(dy)
.
The function F+r is defined in the same way after we replace supw∈W infz∈Z by infz∈Z supw∈W .
We set Fr to be either F−r or F+
r .
Definition 4.2. A bounded upper semicontinuous function u : (0, T ] × Rn → R is a
viscosity subsolution of (4.1) in the sense of Definition 4.2 if u(T, x) ≤ g(x) on Rn
8
and whenever u − ϕ has a global maximum at a point (t, x) for a test function ϕ ∈C1,2((0, T ) × R
n), then for every 0 < r < 1
ϕt(t, x) + Fr(t, x,Dϕ(t, x), ϕ(t, ·), u(t, ·)) ≥ 0.
A bounded lower semicontinuous function u : (0, T ]× Rn → R is a viscosity superso-
lution of (4.1) in the sense of Definition 4.2 if u(T, x) ≥ g(x) on Rn and whenever u− ϕ
has a global minimum at a point (t, x) for a test function ϕ ∈ C1,2((0, T ) × Rn), then for
every 0 < r < 1
ϕt(t, x) + Fr(t, x,Dϕ(t, x), ϕ(t, ·), u(t, ·)) ≤ 0.
5 Estimates for stochastic differential equations
Let 0 ≤ t < T,W ∈ M(t), Z ∈ N(t), and let ξ be Ft-measurable and such that
IE|ξ|2 < +∞. The existence of a unique (up to a modification) predictable solution
X(s) := X(s; t, ξ) of SDE (1.3) such that X(t) = ξ is standard (see [4, 33]). The so-
lution is unique within the class of all predictable processes such that
supt≤s≤T
IE‖X(s; t, ξ)‖2 < +∞.
The solution has a cadlag modification and thus from now on we will always assume that
the solution is cadlag. We can then write (1.3) in the form
dX(s) = b(s,X(s),W (s), Z(s))ds+∫
Rm\0γ(s,X(s−),W (s), Z(s), y)π(ds, dy)
X(t) = x.(5.1)
If X is an Ft-adapted, cadlag solution of (5.1), then X(s) = X(s−) is a predictable
solution of (1.3) and both processes are equivalent. The same results apply to solutions
of SDE
dY (s) = b(s, Y (s),W (s), Z(s))ds
+
∫
Rm\0
γ(s, Y (s−),W (s), Z(s), y)π(ds, dy) +√
2µdB(s), (5.2)
where B is the Wiener process from assumption (A5).
In this section we collect continuous dependence estimates for solutions of (1.3)-(5.1).
Proposition 5.1. Let 0 ≤ t < T, µ ≥ 0,W ∈M(t), Z ∈ N(t), and let ξ be Ft-measurable
and such that IE‖ξ‖2 < +∞. Let Y be the solution of (5.2) with Y (t) = ξ. We have:
(i)
IE
[
supt≤s≤T
‖Y (s)‖2
]
≤ C(T )(1 + IE‖ξ‖2). (5.3)
9
(ii) Let X(s) be the solution of (1.3)-(5.1) (i.e. (5.2) with µ = 0) with initial condition
X(t) = ξ. Then
IE
[
supt≤s≤T
‖Y (s) −X(s)‖2
]
≤ C(T )µ. (5.4)
(iii) For all t ≤ s1 < s2 ≤ T
IE
[
sups1≤τ≤s2
‖Y (s2) − Y (s1)‖2
]
≤ C(T, IE‖ξ‖2)(s2 − s1). (5.5)
Proof. We will only show (ii) and (iii) as the proof of (i) is standard and similar.
To show (5.4) we notice that
Y (s) −X(s)
=
∫ s
t
(b(τ, Y (τ),W (τ), Z(τ)) − b(τ,X(τ),W (τ), Z(τ)))dτ (5.6)
+
∫ s
t
∫
Rm\0
(γ(τ, Y (τ−),W (τ), Z(τ), y) − γ(τ,X(τ−),W (τ), Z(τ), y))π(dτ, dy)
+
∫ s
t
∫
Rm
(γ(τ, Y (τ−),W (τ), Z(τ), y) − γ(τ,X(τ−),W (τ), Z(τ), y))1‖y‖≥1ν(dy)dt
+√
2µ(B(s) −B(t)).
The term
K(s) =
∫ s
t
∫
Rm\0
(γ(τ, Y (τ−),W (τ), Z(τ), y) − γ(τ,X(τ−),W (τ), Z(τ), y))π(dτ, dy)
is a square integrable martingale with cadlag trajectories. Thus, by (2.11), Burkholder-
Davis-Gundy inequality [35], Ito’s isometry [33], and (3.1), (3.3),
IE supt≤τ≤s
‖K(τ)‖2 ≤ C1IE‖K(s)‖2
≤ C2IE
∫ s
t
∫
Rm
‖Y (τ−) −X(τ−)‖2(ρ(y))2ν(dy)dτ
≤ C3
∫ s
t
IE supt≤r≤τ
‖Y (r) −X(r)‖2dτ.
(5.7)
Also√
2µ(B(s) − B(t)) is a martingale and
IE supt≤τ≤s
‖√
2µ(B(τ) −B(t))‖2 ≤ Cµ(s− t). (5.8)
Therefore, squaring both sides of (5.6), taking the sup and the expectation of both sides,
and then using (3.1)-(3.3), (5.7) and (5.8) we obtain
IE supt≤τ≤s
‖Y (s) −X(s)‖2 ≤ C(T )µ+ C4
∫ s
t
IE supt≤r≤τ
‖Y (r) −X(r)‖2dτ. (5.9)
10
The result now follows from Gronwall’s inequality.
As regards (5.5), in light of (5.3), it is enough to show it for s1 = t. We have
‖Y (s) − ξ‖2 = 4
∥
∥
∥
∥
∫ s
t
b(τ, Y (τ),W (τ), Z(τ))dτ
∥
∥
∥
∥
2
+4
∥
∥
∥
∥
∫ s
t
∫
Rm\0
γ(τ, Y (τ−),W (τ), Z(τ), y)π(dτ, dy)
∥
∥
∥
∥
2
+4
∥
∥
∥
∥
∫ s
t
∫
Rm
γ(τ, Y (τ−),W (τ), Z(τ), y)1‖y‖≥1ν(dy)dt
∥
∥
∥
∥
2
+4‖√
2µ(B(s) −B(t))‖2. (5.10)
Again, the process
K(s) =
∫ s
t
∫
Rm\0
γ(τ, Y (τ−),W (τ), Z(τ), y)π(dτ, dy)
is a square integrable martingale with cadlag trajectories, and exactly like in (5.7) we
estimate (now using (3.6))
IE supt≤τ≤s
‖K(τ)‖2 ≤ C
∫ s
t
∫
Rm
(ρ(y))2ν(dy)dτ ≤ C(s− t). (5.11)
It thus follows from (5.10), upon using (3.1), (A2), (3.6), (5.8), (5.11), and Holder’s
inequality, that
IE supt≤τ≤s
‖Y (s) − ξ‖2 ≤ C(s− t)2 + C(s− t)
+ C(s− t)
∫ s
t
∫
Rm
(ρ(y))21‖y‖≥1ν(dy)dt ≤ C(T, IE‖ξ‖2)(s− t).
6 Regularization
Let u : (0, T ] × Rn be bounded and continuous, and let 0 < ǫ, β ≤ 1. We define the
sup-convolution of u by
uǫ,β(t, x) := sup(s,y)
u(s, y)− ‖x− y‖2
2ǫ− (t− s)2
2β
and the inf-convolution of u by
uǫ,β(t, x) := inf(s,y)
u(s, y) +‖x− y‖2
2ǫ+
(t− s)2
2β
.
11
It is well known that the functions uǫ,β, uǫ,β are Lipschitz continuous,
limβ→0
uǫ,β(t, x) = uǫ(t, x) := supy
u(t, y) − ‖x− y‖2
2ǫ
,
limβ→0
uǫ,β(t, x) = uǫ(t, x) := infy
u(t, y) +‖x− y‖2
2ǫ
uniformly on every set [γ, T ] × Rn, and
limǫ→0
uǫ = limǫ→0
uǫ = u
uniformly on every set [γ, T ] ×BR, for every 0 < γ < T,R > 0. Moreover
uǫ,β(t, x) +‖x‖2
2ǫ+t2
2β
is convex, and
uǫ,β(t, x) −‖x‖2
2ǫ− t2
2β
is concave.
In the next lemma we prove that sup-convolutions (respectively, inf-convolutions) of
viscosity subsolutions (respectively, supersolutions) of Isaacs integro-PDE are viscosity
subsolutions (respectively, supersolutions) of slightly perturbed equations. This fact is
well known for PDE. The use of sup- and inf-convolutions for equations with nonlocal
terms is also well established [8, 15, 26], so Lemma 6.1 may not be totally new, however
since the exact statement does not seem to be available in the literature we provide it
with a full proof.
Lemma 6.1. Let u be a continuous viscosity subsolution (respectively, supersolution) of
(4.1). Let 0 < γ < T . There exists a nonnegative, bounded function ργ : [0,∞) × (0, 1] ×(0, 1] → [0,+∞), such that ργ(·, ǫ, β) is continuous and nondecreasing for every ǫ, β, and
for every R > 0
lim supǫ→0
lim supβ→0
ργ(R, ǫ, β) = 0, (6.1)
such that for sufficiently small β, uǫ,β (respectively, uǫ,β) is a viscosity subsolution of
vt + F (t, x,Dv, v(t, ·)) = −ργ(‖x‖, ǫ, β) in (γ, T − γ) × Rn (6.2)
(respectively, viscosity supersolution of
vt + F (t, x,Dv, v(t, ·)) = ργ(‖x‖, ǫ, β) in (γ, T − γ) × Rn). (6.3)
12
Proof. We will only do the proof for the lower Isaacs equation (1.7). Fix ϕ ∈ C1,2b ((0, T )×
Rn). Let (t, x) ∈ (0, T )×R
n, ‖x‖ ≤ R, t ∈ (γ, T − γ) be a point where uǫ,β −ϕ attains its
global maximum. Let (t′, x′) ∈ [0, T ] × Rn be such that
uǫ,β(t, x) = u(t′, x′) − ‖x− x′‖2
2ǫ− (t− t′)2
2β, (6.4)
i.e.
u(s, y)− ‖x− y‖2
2ǫ− (t− s)2
2β≤ u(t′, x′) − ‖x− x′‖2
2ǫ− (t− t′)2
2β, (6.5)
for all (s, y) ∈ (0, T ) × Rn. We also have
u(t′, x′)− ‖x− x′‖2
2ǫ− (t− t′)2
2β−ϕ(t, x) ≥ u(s′, y′)− ‖y − y′‖2
2ǫ− (s− s′)2
2β−ϕ(s, y) (6.6)
for all (s, y), (s′, y′) ∈ (0, T ) × Rn.
If ǫ, β are small enough we can assume that (t′x′) ∈ (γ/2, T −γ/2)×BR+1(0). Denote
by ωR,γ the modulus of continuity of u on (γ/2, T −γ/2)×BR+1(0). Setting (s, y) = (t, x′)
and (s, y) = (t′, x) in (6.5), we obtain
(t− t′)2
2β≤ u(t′, x′) − u(t, x′) ≤ min(ωR,γ(|t− t′|), 2‖u‖∞), (6.7)
‖x− x′‖2
2ǫ≤ u(t′, x′) − u(t′, x) ≤ min(ωR,γ(‖x− x′‖), 2‖u‖∞). (6.8)
Therefore it follows from (6.7) and (6.8) that
(t− t′)2
β≤ ωR,γ(2
√
‖u‖∞β), (6.9)
‖x− x′‖2
ǫ≤ ωR,γ(2
√
‖u‖∞ǫ). (6.10)
Next, if we set (s′, y′) = (t′, x′) in (6.6), it is easy to verify that
(ϕt(t, x), Dϕ(t, x)) =
(
t′ − t
β,x′ − x
ǫ
)
. (6.11)
We define
ψ(s, y) =〈x− x′, x′ − y〉
ǫ+
‖x′ − y‖2
2ǫ+
(t− t′)(t′ − s)
β+
|t′ − s|22β
.
It follows from (6.5) that u− ψ has a global maximum at (t′, x′), and
(ψs(t′, x′), Dψ(t′, x′)) =
(
t′ − t
β,x′ − x
ǫ
)
. (6.12)
13
Therefore, by Definition 4.2,
0 ≤ t′ − t
β+ sup
w∈Winfz∈Z
〈b(t′, x′, w, z), x′ − x
ǫ〉 + l(t′, x′, w, z)
+
∫
‖y‖<r
(
ψ(t′, x′ + γ(t′, x′, w, z, y)) − ψ(t′, x′) − 〈γ(t′, x′, w, z, y), x′ − x
ǫ〉)
ν(dy)
+
∫
r≤‖y‖<1
(
u(t′, x′ + γ(t′, x′, w, z, y))− u(t′, x′) − 〈γ(t′, x′, w, z, y), x′ − x
ǫ〉)
ν(dy)
+
∫
‖y‖≥1
(u(t′, x′ + γ(t′, x′, w, z, y))− u(t′, x′)) ν(dy)
. (6.13)
Using (3.2) and (3.4) we have
〈b(t′, x′, w, z), x′ − x
ǫ〉 + l(t′, x′, w, z) ≤ 〈b(t, x, w, z), x
′ − x
ǫ〉 + l(t, x, w, z)
+C‖x− x′‖2
ǫ+ ω(‖x− x′‖ + |t− t′|) + σ(|t− t′|)‖x− x′‖
ǫ(6.14)
for all w, z. By the definition of ψ and (3.6) we obtain
∫
‖y‖<r
(
ψ(t′, x′ + γ(t′, x′, w, z, y))− ψ(t′, x′) − 〈γ(t′, x′, w, z, y), x′ − x
ǫ〉)
ν(dy)
≤ C
ǫ
∫
0<‖y‖<r
(ρ(y))2ν(dy) → 0 as r → 0 (6.15)
uniformly for all w, z. Moreover, since by the semiconvexity of uǫ,β (or directly by (6.4))
−‖x− y‖2
2ǫ≤ uǫ,β(t, y) − uǫ,β(t, x) − 〈x
′ − x
ǫ, y − x〉
for all y, using (3.6) it follows that
∫
‖y‖<r
(
uǫ,β(t, x+ γ(t, x, w, z, y)) − uǫ,β(t, x) − 〈γ(t, x, w, z, y), x′ − x
ǫ〉)
ν(dy)
≥ −Cǫ
∫
0<‖y‖<r
(ρ(y))2ν(dy) → 0 as r → 0 (6.16)
uniformly for all w, z. Now, the definition of uǫ,β yields
uǫ,β(t, x+ γ(t, x, w, z, y)) ≥ u(t′, x′ + γ(t′, x′, w, z, y))
−‖x− x′ + γ(t, x, w, z, y)− γ(t′, x′, w, z, y)‖2
2ǫ− (t− t′)2
2β,
14
for all w, z which, together with (6.4), implies
u(t′, x′ + γ(t′, x′, w, z, y))− u(t′, x′) − 〈γ(t′, x′, w, z, y), x′ − x
ǫ〉
≤ uǫ,β(t, x+ γ(t, x, w, z, y)) − uǫ,β(t, x) − 〈γ(t, x, w, z, y), x′ − x
ǫ〉
+‖γ(t, x, w, z, y)− γ(t′, x′, w, z, y)‖2
2ǫ
≤ uǫ,β(t, x+ γ(t, x, w, z, y)) − uǫ,β(t, x) − 〈γ(t, x, w, z, y), x′ − x
ǫ〉
+C(‖x− x′‖ + σ(|t− t′|))2
ǫ(ρ(y))2, (6.17)
where we have used (3.3) to estimate the last line of (6.17). Therefore∫
r≤‖y‖<1
(
u(t′, x′ + γ(t′, x′, w, z, y)) − u(t′, x′) − 〈γ(t′, x′, w, z, y), x′ − x
ǫ〉)
ν(dy)
≤∫
r≤‖y‖<1
(
uǫ,β(t, x+ γ(t, x, w, z, y)) − uǫ,β(t, x) − 〈γ(t, x, w, z, y), x′ − x
ǫ〉)
ν(dy)
+C1(‖x− x′‖ + σ(|t− t′|))2
ǫ(6.18)
for all w, z. Moreover, (3.3) and (6.17) also imply
u(t′, x′ + γ(t′, x′, w, z, y)) − u(t′, x′)
≤ uǫ,β(t, x+ γ(t, x, w, z, y)) − uǫ,β(t, x)
+C(‖x− x′‖ + σ(|t− t′|))2
ǫ(ρ(y))2 + C(‖x− x′‖ + σ(|t− t′|))‖x− x′‖
ǫρ(y),
which, recalling condition (A1), gives∫
‖y‖≥1
(u(t′, x′ + γ(t′, x′, w, z, y))− u(t′, x′)) ν(dy)
≤∫
‖y‖≥1
(
uǫ,β(t, x+ γ(t, x, w, z, y))− uǫ,β(t, x))
ν(dy)
+C2(‖x− x′‖ + σ(|t− t′|))2
ǫ(6.19)
for all w, z. Therefore, using estimates (6.14), (6.15), (6.16), (6.18), (6.19) in (6.13), send-
ing r → 0, and then using (6.9) and (6.10), we obtain that there exists ργ satisfying the
required conditions such that
t′ − t
β+ sup
w∈Winfz∈Z
〈b(t, x, w, z), x′ − x
ǫ〉 + l(t, x, w, z)
+
∫
Rm
(
uǫ,β(t, x+ γ(t, x, w, z, y)) − uǫ,β(t, x) − 〈γ(t, x, w, z, y), x′ − x
ǫ〉1‖y‖<1
)
ν(dy)
≥ −ργ(R, ǫ, β). (6.20)
15
This completes the proof.
Remark 6.2. Lemma 6.1 would still be true if the Hamiltonian F contained a purely
second order term Tr(σ(t, x, w, z)σ∗(t, x, w, z)D2u). The proof however would be much
more complicated and it would involve a combination of the above proof with the use of
the so called non-local maximum principle, see [8, 26, 41].
We need to regularize further the functions uǫ,β and uǫ,β. This will be done using
sup-inf and inf-sup-convolutions of Lasry and Lions [29] and adapting the method of [18].
Recall that for a function v and δ > 0 we denoted by vδ its sup-convolution in the space
variable only, i.e.
vδ(t, x) = supy
v(t, y) − ‖x− y‖2
2δ
,
and by vδ its corresponding inf-convolution in the space variable. It is easy to see from
the definitions that vǫ+δ = (vǫ)δ and v ≤ (vδ)δ.
For u from Theorem 6.1 and ǫ, β, δ > 0 we denote
u := (uǫ+δ,β)δ, u := (uǫ+δ,β)δ.
It is well known that u (respectively, u) converge uniformly on every set [γ, T ] × Rn as
δ → 0 to uǫ,β (respectively, uǫ,β), and that for every t, u(t, ·), u(t, ·) ∈W 2,∞(Rn). Moreover
we have
u(t, x) − ‖x‖2
2δis concave, u(t, x) +
‖x‖2
2δis convex,
and by Proposition 4.5 of [18],
u(t, x) +‖x‖2
2ǫis convex, u(t, x) − ‖x‖2
2ǫis concave. (6.21)
It also follows that
Du,Du are continuous (6.22)
and Lipschitz continuous in x, uniformly in t. Let us show (6.22) for Du. It is enough to
prove that if tn → t then Du(tn, x) → Du(t, x) for every x. Since u is semiconvex, for every
n there exists an such that (an, Du(tn, x)) ∈ D−u(tn, x), the generalized subdifferential
of u at (tn, x). Since u is Lipschitz continuous (with the same Lipschitz constant as uǫ,β),
we can assume that (an, Du(tn, x)) → (a, p) as n → +∞. By semiconvexity of u we have
(a, p) ∈ D−u(t, x). Since Du(t, x) exists, this implies that p = Du(t, x) and the claim
follows.
Proposition 6.3. Let u, F be from Lemma 6.1, let ǫ, β, δ > 0 and β be so that the
conclusion of Lemma 6.1 is satisfied. Let µ > 0. Then there exists δ0 = δ0(ǫ, β, µ) such
that for 0 < δ < δ0,
ut+µTr(D2u)+F (t, x,Du, u(t, ·)) ≥ −ργ(‖x‖, ǫ, β)−Cµ
ǫa.e. in (γ, T−γ)×R
n (6.23)
16
(respectively,
ut+µTr(D2u)+F (t, x,Du, u(t, ·)) ≤ ργ(‖x‖, ǫ, β)+Cµ
ǫa.e. in (γ, T−γ)×R
n). (6.24)
Proof. We will only prove (6.23). We will show that (6.23) holds at every point (t, x)
where the function u is twice differentiable. Let (t, x) be such point.
If u(t, x) > uǫ,β(t, x), then by Proposition 4.4 of [18], D2u(t, x) has 1/δ as one of its
eigenvalues. Thus an easy computation using (6.21), the fact that u has the same Lipschitz
constant as uǫ,β, and the definition of F , yields that (6.23) holds if δ is small enough.
If u(t, x) = uǫ,β(t, x) the statement is almost obvious since in this case u acts like a
test function. It is easy to see that there exists a test function ϕ ∈ C1,2b ((0, T )×R
n) such
that ϕt(t, x) = (u)t(t, x), Dϕ(t, x) = Du(t, x), D2ϕ(t, x) ≥ D2u(t, x) ≥ −1ǫI, and uǫ,β − ϕ
has a global maximum at (t, x). We then use Definition 4.2 and let r → 0 to obtain (6.23).
Remark 6.4. It can be proved that (6.23) and (6.24) being satisfied a.e. is equivalent to
them being satisfied by u and u in the viscosity sense.
Remark 6.5. We do not know if it can be shown that the functions u and u are sub-
and supersolutions of perturbed equations without the uniformly elliptic term µTr(D2u),
or if this term is replaced by some kind of non-degeneracy condition on the integral term
or by adding an integral term with non-degenerate kernel. For second order PDE without
integral terms, the functions u and u are sub- and supersolutions of perturbed equations
only if the PDE is uniformly elliptic/parabolic (see [18]), and hence if the equation is
degenerate, the addition of the µTr(D2u) term is necessary.
For η > 0 let ψη be standard mollifiers with compact support in Rn+1. We denote
uη = ψη ∗ u, uη = ψη ∗ u, i.e.
uη(t, x) =
∫
Rn+1
ψη(t− s, x− y)u(s, y)dsdy, uη(t, x) =
∫
Rn+1
ψη(t− s, x− y)u(s, y)dsdy.
The functions uη and uη are well defined on (γ, T − γ) × Rn if η is small enough, are
smooth, and they converge uniformly on (γ, T − γ) × Rn to u and u respectively.
Proposition 6.6. Let the assumptions of Proposition 6.3 be satisfied and let u, u be from
Proposition 6.3. Then uη satisfies
(uη)t + µTr(D2uη) + F (t, x,Duη, uη(t, ·)) ≥ −ργ(‖x‖, ǫ, β, η) −Cµ
ǫin (γ, T − γ) × R
n
(6.25)
17
and uη satisfies
(uη)t + µTr(D2uη) + F (t, x,Duη, uη(t, ·)) ≤ ργ(‖x‖, ǫ, β, η) +Cµ
ǫin (γ, T − γ) × R
n,
(6.26)
where ργ(‖x‖, ǫ, β, η) is bounded, ργ(·, ǫ, β, η) is continuous, and for every fixed ǫ, β, µ, δ
like in Proposition 6.3,
limη→0
ργ(‖x‖, ǫ, β, η) = ργ(‖x‖, ǫ, β) (6.27)
locally uniformly.
Proof. We will only show (6.25) as the proof of (6.26) is similar. We set f(t, x) =
F (t, x,Du(t, x), u(t, ·)). Applying the mollification to (6.23) we obtain
(uη)t(t, x) + µTr(D2uη(t, x)) + ψη ∗ f(t, x) ≥ −ψη ∗ ργ(‖x‖, ǫ, β) − Cµ
ǫ. (6.28)
The function f is bounded. We will show that f is continuous. To do this it is enough to
prove that for every w, z, the function
fw,z(t, x) = 〈b(t, x, w, z), Du(t, x)〉 + l(t, x, w, z) (6.29)
+
∫
Rm
(
u(t, x+ γ(t, x, w, z, y)) − u(t, x) − 1‖y‖<1〈γ(t, x, w, z, y), Du(t, x)〉)
ν(dy)
has a modulus of continuity independent of w, z on every bounded set. Let (t, x1), (s, x2)
be such that ‖x1‖, ‖x2‖ ≤ R. We break the integral in (6.29) into
∫
Rm
=
∫
‖y‖<τ
+
∫
‖y‖≥τ
≤ ω1(τ) +
∫
‖y‖≥τ
for some modulus ω1, which follows from the fact that u(t, ·) ∈ W 2,∞(Rn). Therefore,
using the continuity properties of u,Du and assumptions (A1) − (A4), we obtain
|fw,z(t, x1) − fw,z(s, x2)| ≤ C‖x1 − x2‖ + σ(|t− s|) + 2ω1(τ)
+
∫
‖y‖≥τ
|u(t, x1 + γ(t, x1, w, z, y))− u(s, x2 + γ(s, x2, w, z, y))|ν(dy)
+
∫
‖y‖≥τ
|u(t, x1) − u(s, x2)|ν(dy)
+
∫
τ≤‖y‖<1
|〈γ(t, x1, w, z, y), Du(t, x1)〉 − 〈γ(s, x2, w, z, y), Du(s, x2)〉|ν(dy)
≤ C‖x1 − x2‖ + σ(|t− s|) + 2ω1(τ)
+
∫
‖y‖≥τ
C(σR(|t− s|) + ‖x1 − x2‖)(1 + ρ(y))ν(dy)
≤ 2ω1(τ) + Cτ (σR(|t− s|) + ‖x1 − x2‖)
18
for some modulus σR and constant Cτ . This gives a local modulus of continuity of fw,z
independent of w, z, and hence we obtain that there exists a modulus ω and for every
R > 0 a modulus ωR such that
|f(t, x1)− f(s, x2)| ≤ ωR(|t− s|) + ω(‖x1 − x2‖) for all t, s ∈ (γ, T − γ), ‖x1‖, ‖x2‖ ≤ R.
(6.30)
Thus it follows that
|ψη ∗ f(t, x) − f(t, x)| ≤ ρ(η, ‖x‖), (6.31)
for some bounded local modulus ρ. Let fη(t, x) = F (t, x,Duη(t, x), uη(t, ·)). Since
supt
‖uη(t, ·)‖W 2,∞(Rn) ≤ C,
uη → u uniformly, and Duη → Du locally uniformly as η → 0, it follows that fη → f
locally uniformly as η → 0, i.e. that
|fη(t, x) − f(t, x)| ≤ ρ1(η, ‖x‖), (6.32)
for some bounded local modulus ρ1. Combining (6.31) and (6.32) we thus obtain
|ψη ∗ f(t, x) − fη(t, x)| ≤ ρ(η, ‖x‖) + ρ1(η, ‖x‖),
which, together with (6.28), yields the required claim.
Remark 6.7. A version of Proposition 6.6 would still be true if the Hamiltonian F
contained a purely second order term Tr(σ(t, x, w, z)σ∗(t, x, w, z)D2u), however the term
ργ(‖x‖, ǫ, β, η) would have to be replaced by a term ργ(x, ǫ, β, η) for which the convergence
(6.27) would hold only pointwise. This would be a major obstacle in the proof of Step 2
of (iii) of Theorem 7.1 as we would have to know estimates on distribution of stochastic
integrals. Such results are known for diffusions without jumps (see [28]) and were used in
[40]. They are basically probabilistic versions of Aleksandrov-Bakelman-Pucci (ABP) type
maximum principles, for which little is known for integro-PDE. For existing ABP type
results for integro-PDE we refer to [15, 22]. This is why we have to restrict the dynamic
of our stochastic differential game (1.3) to the noise of pure jump type.
7 Sub- and super-optimailty inequalities of dynamic
programming
Theorem 7.1. (i) If u is a (bounded) continuous viscosity subsolution of (1.9) then for
every 0 < t0 ≤ h ≤ T
u(t0, x0) ≤ supα∈Γ(t0)
infZ∈N(t0)
IE
∫ h
t0
l(X(s), α[Z](s), Z(s))ds+ u(h,X(h))
,
19
where X(s) = X(s; t0, x0) is the solution of (1.3) with W = α[Z] for Z ∈ N(t0).
(ii) If u is a (bounded) continuous viscosity supersolution of (1.9) then for every 0 < t0 ≤h ≤ T
u(t0, x0) ≥ supα∈Γ(t0)
infZ∈N(t0)
IE
∫ h
t0
l(X(s), α[Z](s), Z(s))ds+ u(h,X(h))
,
where X(s) = X(s; t0, x0) is the solution of (1.3) with W = α[Z] for Z ∈ N(t0).
(iii) If u is a (bounded) continuous viscosity subsolution of (1.7) then for every 0 < t0 ≤h ≤ T
u(t0, x0) ≤ infβ∈∆(t0)
supW∈M(t0)
IE
∫ h
t0
l(X(s),W (s), β[W ](s))ds+ u(h,X(h))
, (7.1)
where X(s) = X(s; t0, x0) is the solution of (1.3) with Z = β[W ] for W ∈M(t0).
(iv) If u is a (bounded) continuous viscosity supersolution of (1.7) then for every 0 < t0 ≤h ≤ T
u(t0, x0) ≥ infβ∈∆(t0)
supW∈M(t0)
IE
∫ h
t0
l(X(s),W (s), β[W ](s))ds+ u(h,X(h))
, (7.2)
where X(s) = X(s; t0, x0) is the solution of (1.3) with Z = β[W ] for W ∈M(t0).
Proof. We will only show (iii) and (iv) since the proofs of (i) and (ii) are similar.
Proof of (iii): Let t0 < h ≤ T, x0 ∈ Rn.
Step 1. (Smooth case, µ > 0.) We will first show (iii) when µ > 0 and u ∈ C1,2((0, T )×R
n), u, ut, Du,D2u are bounded and Lipschitz continuous, and u is a viscosity subsolution
of
ut + µTr(D2u) + F−(t, x,Du, u(·)) = 0 in (0, T ) × Rn.
Let m ≥ 1 and we set r = (h− t0)/m, ti = t0 + ir, i = 0, ..., m. Define
Λ(t, x, w) = ut(t, x) + µTr(D2u(t, x))
+ infz∈Z
〈b(t, x, w, z), Du(t, x)〉 + l(t, x, w, z)
+
∫
Rm
(
u(t, x+ γ(t, x, w, z, y)) − u(t, x) − 1‖y‖<1〈γ(t, x, w, z, y), Du(t, x)〉)
ν(dy)
.
It is not difficult to show, that the functions Λ(t, ·, ·) are uniformly continuous on
20
Rn ×W, uniformly in t. To see it for the integral term we notice that for every z ∈ Z∣
∣
∣
∣
∫
Rm
(
u(t, x1 + γ(t, x1, w1, z, y)) − u(t, x1) − 1‖y‖<1〈γ(t, x1, w1, z, y), Du(t, x1)〉)
ν(dy)
−∫
Rm
(
u(t, x2 + γ(t, x2, w2, z, y)) − u(t, x2) − 1‖y‖<1〈γ(t, x2, w2, z, y), Du(t, x2)〉)
ν(dy)
∣
∣
∣
∣
≤ C
∫
‖y‖<κ
(ρ(y))2ν(dy)
+
∫
‖y‖≥κ
(
|u(t, x1 + γ(t, x1, w1, z, y)) − u(t, x2 + γ(t, x2, w2, z, y))| + |u(t, x1) − u(t, x2)|
+1‖y‖<1|〈γ(t, x1, w1, z, y), Du(t, x1)〉 − 〈γ(t, x2, w2, z, y), Du(t, x2)〉|)
ν(dy)
≤ ω1(κ) + Cκ‖x1 − x2‖ +
∫
‖y‖≥κ
‖γ(t, x2, w1, z, y) − γ(t, x2, w2, z, y)‖ν(dy)
≤ ω1(κ) + Cκ‖x1 − x2‖ +
∫
‖y‖≥κ
min(ωy(dW(w1, w2)), Cρ(y))ν(dy)
≤ ω1(κ) + Cκ‖x1 − x2‖ + ωκ(dW(w1, w2)) (7.3)
for some moduli ω1, ωκ independent of z and t. We have used assumptions (A1), (A2), (A4)
and the Lebesgue dominated convergence theorem to estimate the last three lines. (Above
ωy is the modulus of continuity of γ(·, ·, ·, ·, y).)Therefore, since W is separable, for every i = 0, ..., m − 1, we can find a sequence
wij∞j=1 and a family and balls Bi
j∞j=1 covering Rn, such that
Λ(ti, x, wij) ≥ −r if x ∈ Bi
j.
We now define maps ψi : Rn → W, i = 0, ..., m− 1 by
ψi(x) = wik if x ∈ Bi
k \k−1⋃
j=1
Bij .
They are B(Rn)/B(W) measurable maps, and by construction
Λ(ti, x, ψi(x)) ≥ −r for every x ∈ Rn, i = 0, ..., m− 1. (7.4)
We now fix Z ∈ N(t0) and define a control Wm ∈M(t0) inductively. We set
Wm(s) = ψ0(x0) s ∈ [t0, t1], (7.5)
and let X be the solution of (5.2) on [t0, t1] with the controls Wm and Z. Suppose now
that Wm and X have been defined on [t0, ti], i = 0, ..., m− 1. We then set
Wm(s) = ψi(X(ti−)) s ∈ (ti, ti+1],
21
and hence we can define the solution X of (5.2) on [t0, ti+1] with controls Wm and Z. We
notice that Wm is predictable. Using this notation we denote
Au(t) = ut(t, X(t)) + 〈b(t, X(t),Wm(t), Z(t)), Du(t, X(t))〉
+
∫
Rm
(
u(t, X(t−) + γ(t, X(t−),Wm(t), Z(t), y))− u(t, X(t−))
−1‖y‖<1〈γ(t, X(t−),Wm(t), Z(t), y), Du(t, X(t−))〉)
ν(dy) + µTr(D2u(t, X(t))).
Estimate (5.5) gives
P
(
supti≤s≤ti+1
|X(s) −X(ti)| ≥ r1/4
)
≤ Cr1
2 , i = 0, ..., m− 1, (7.6)
for some constant C1 independent of Z. Therefore, using Ito’s formula, definition of Wm,
assumptions (A1)− (A4), (7.6), and arguing similarly as in (7.3), we obtain that for every
κ > 0, i = 0, ..., m− 1,
u(ti, X(ti)) = IE
∫ ti+1
ti
−Au(s)ds+ u(ti+1, X(ti+1))
≤ IE
∫ ti+1
ti
(−Λ(ti, X(ti−), ψi(X(ti−))) + l(ti, X(ti−),Wm(s), Z(s)))ds
+u(ti+1, X(ti+1))
+ Cr(κ+ ωκ(r))
≤ IE
∫ ti+1
ti
l(s,X(s),Wm(s), Z(s))ds+ u(ti+1, X(ti+1))
+ Cr(κ+ ωκ(r)),
where C and moduli ωκ are independent of i, Z and only depend on u and various constants
and moduli in (A1) − (A4). Adding the above inequalities for i = 0, ..., m − 1 we thus
obtain
u(t0, x0) ≤ IE
∫ h
t0
l(s,X(s),Wm(s), Z(s))ds+ u(h,X(h))
+ Cr(κ+ ωκ(r))m
= IE
∫ h
t0
l(s,X(s),Wm(s), Z(s))ds+ u(h,X(h))
+ C(h− t0)
(
κ+ ωκ
(
h− t0m
))
.
(7.7)
We now define a strategy αm ∈ Γ(t0) by setting αm[Z](s) = Wm(s). Rewriting (7.7)
slightly it follows that for every κ > 0 there exists a modulus ωκ such that
u(t0, x0) ≤ IE
∫ h
t0
l(s,X(s), αm[Z](s), Z(s))ds+ u(h,X(h))
+ κ+ ωκ(1
m). (7.8)
Following [40] we claim that for every β ∈ ∆(t0) there exist Z ∈ N(t0), W ∈ M(t0) such
that
αm[Z] = W , and β[W ] = Z on [t0, h]. (7.9)
22
They are constructed inductively in the following way. We set W|[t0,t1] = ψ0(x0) and
Z|[t0,t1] = β[W ]|[t0,t1]. We remark here that to do this we have to extend W|[t0,t1] to the
whole interval [t0, T ] but since strategies are non-anticipating, Z|[t0,t1] does not depend on
the extension and we will thus omit this technical detail. If we know W , Z on [t0, ti] we
also know the solution of (5.2) X on [t0, ti], and we extend W to [t0, ti+1] by W|[t0,ti+1] =
αm[Z]|[t0,ti+1] (since αm[Z]|[t0,ti+1] only depends on Z|[t0,ti]). We then extend Z to [t0, ti+1]
by setting Z|[t0,ti+1] = β[W ]|[t0,ti+1]. This is an extension since β is non-anticipating. It is
clear from the construction that after m iterations we produce controls Z, W satisfying
(7.9).
Applying (7.8) to any β ∈ ∆(t0) and Z = Z we thus have
u(t0, x0) ≤ IE
∫ h
t0
l(s,X(s), W (s), β[W ](s))ds+ u(h,X(h))
+ κ+ ωκ(1
m)
≤ supW∈M(t0)
IE
∫ h
t0
l(s,X(s),W (s), β[W ](s))ds+ u(h,X(h))
+ κ + ωκ(1
m).
and taking the infimum above over all strategies yields
u(t0, x0) ≤ infβ∈∆(t0)
supW∈M(t0)
IE
∫ h
t0
l(s,X(s),W (s), β[W ](s))ds+ u(h,X(h))
+κ+ ωκ(1
m).
(7.10)
We now let m→ +∞ and then κ→ 0 in (7.10) to obtain (7.1).
Step 2. (General case, µ = 0.) Let u be now as in Theorem 7.1-(iii). By continuity it
is enough to show the result for t0 < h < T . We set γ := min(t0, T −h)/2. For sufficiently
small ǫ, β, µ, δ, η > 0 (i.e. δ < δ0(ǫ, β, µ) from Proposition 6.3), let uη and ργ be from
Proposition 6.6. Then uη satisfies the assumptions of Step 1 on (γ, T − γ) × Rn with
l(t, x, w, z) = l(t, x, w, z) + ργ(‖x‖, ǫ, β, η) +Cµ
ǫ.
Therefore, by Step 1, we obtain
uη(t0, x0) ≤ infβ∈∆(t0)
supW∈M(t0)
IE
∫ h
t0
(l(s,Xµ(s),W (s), β[W ](s)) + ργ(‖Xµ(s)‖, ǫ, β, η))ds
+ uη(h,Xµ(h))
+Cµ
ǫ(h− t0),
(7.11)
where Xµ is the solution of (5.2) with controls W,β[W ].
We have limη→0 uη = u and limδ→0 u = uǫ,β uniformly on (γ, T−γ)×Rn, limβ→0 u
ǫ,β =
uǫ uniformly on (γ, T − γ) × Rn, and limǫ→0 u
ǫ = u uniformly on (γ, T − γ) × BR(0) for
23
every R > 0. Moreover all functions are uniformly bounded for all 0 < ǫ, β, δ, η < 1.
Therefore, using (5.3) and (5.4), it follows that
limǫ→0
limβ→0
limµ→0
limδ→0
limη→0
supβ∈∆(t0)
supW∈M(t0)
IE|uη(h,Xµ(h)) − u(h,X(h))| = 0. (7.12)
Moreover, by (6.1), (6.27), (5.3) and (5.4), we obtain
lim supǫ→0
lim supβ→0
lim supµ→0
lim supδ→0
lim supη→0
supβ∈∆(t0)
supW∈M(t0)
IE
∫ h
t0
ργ(‖Xµ(s)‖, ǫ, β, η)ds
= 0.
(7.13)
Therefore, taking lim supǫ→0 lim supβ→0 lim supµ→0 lim supδ→0 lim supη→0 in (7.11), and
combining it with (7.12) and (7.13) yields
u(t0, x0) ≤ infβ∈∆(t0)
supW∈M(t0)
IE
∫ h
t0
l(s,X(s),W (s), β[W ](s)) + u(h,X(h))
.
Proof of (iv): We will only show (iv) in the smooth case, i.e. when µ > 0, u ∈C1,2((0, T ) × R
n), u, ut, Du,D2u are bounded and Lipschitz continuous, u is a viscosity
supersolution of
ut + µTr(D2u) + F−(t, x,Du, u(·)) = 0 in (0, T ) × Rn,
and t0 < h < T . The general case follows from the smooth one in exactly the same way
as in Step 2 in the proof of (iii).
We let m ≥ 1 and we set r = (h− t0)/m, ti = t0 + ir, i = 0, ..., m. Define
Λ(t, x, w, z) = ut(t, x) + µTr(D2u(t, x)) + 〈b(t, x, w, z), Du(t, x)〉 + l(t, x, w, z)
+
∫
Rm
(
u(t, x+ γ(t, x, w, z, y))− u(t, x) − 1‖y‖<1〈γ(t, x, w, z, y), Du(t, x)〉)
ν(dy).
The functions Λ(t, ·, ·, ·) are uniformly continuous on Rn ×W × Z, uniformly in t. Since
for every (t, x)
supw∈W
infz∈Z
Λ(t, x, w, z) ≤ 0,
for every i = 0, ..., m− 1, we can find a sequence zij∞j=1 and a family of products of balls
Bij∞j=1 × Bi
j∞j=1 covering Rn ×W, such that
Λ(ti, x, w, zij) ≤ r if (x, w) ∈ Bi
j × Bij.
We now define maps ψi : Rn ×W → Z, i = 0, ..., m− 1 by
ψi(x, w) = zik if (x, w) ∈ (Bi
k × Bik) \
k−1⋃
j=1
(Bij × Bi
j).
24
These are B(Rn) ⊗ B(W)/B(Z) measurable maps, and by construction
Λ(ti, x, w, ψi(x, w)) ≤ r for every (x, w) ∈ Rn ×W, i = 0, ..., m− 1. (7.14)
We now fix W ∈M(t0) and define a control Zm ∈ N(t0) inductively. We set
Zm(s) = ψ0(x0,W (s)) s ∈ [t0, t1], (7.15)
and take X to be the solution of (5.2) on [t0, t1] with the controls W and Zm. If Zm and
X have been defined on [t0, ti], i = 0, ..., m− 1, we set
Zm(s) = ψi(X(ti−),W (s)) s ∈ (ti, ti+1],
and then we can define the solution X of (5.2) on [t0, ti+1] with controls W and Zm. The
control Zm is predictable.
Therefore, as in Step 1 of the proof of (iii) we obtain that for every κ > 0, i =
0, ..., m− 1,
u(ti, X(ti)) (7.16)
≥ IE
∫ ti+1
ti
l(s,X(s),W (s), Zm(s))ds+ u(ti+1, X(ti+1))
− Cr(κ+ ωκ(r)),
where C and moduli ωκ are independent of i,W and only depend on u and various
constants and moduli in (A1) − (A4). Setting βm[W ] = Zm, we see that βm is non-
anticipating, and thus βm ∈ ∆(t0). Therefore, adding (7.16) for i = 0, ..., m− 1, we get
u(t0, x0)
≥ IE
∫ h
t0
l(s,X(s),W (s), βm[W ](s))ds+ u(h,X(h))
− Cr(κ+ ωκ(r))m.
for every W ∈M(t0). The above implies
u(t0, x0) ≥ infβ∈∆(t0)
supW∈M(t0)
IE
∫ h
t0
l(s,X(s),W (s), β[W ](s))ds+ u(h,X(h))
−C(h− t0)
(
κ+ ωκ
(
h− t0m
))
,
and it remains to let m→ +∞ and then κ→ 0.
An immediate consequence of Theorem 7.1 is that (under its assumptions) the bounded
viscosity solutions of (1.7) and (1.9) are unique and are equal respectively to the lower
and upper value functions. Moreover the value functions must satisfy the dynamic pro-
gramming principle.
25
We recall that in our formulation of the game the reference probability space (Ω,F ,Fs,
P, L) was fixed. However it is clear from the proof of Theorem 7.1 that what happens be-
fore time t0 is irrelevant to the proofs of (i) − (iv). Therefore the same result would hold
if for every t0 we allowed the reference probability space to change, i.e. if the control
sets M(t0), N(t0) were defined on a space that varied with t0. For instance we could fix
(Ω,F ,P) but take L(s) − L(t0) as our new Levy process on [t0, T ] and take a filtration
on [t0, T ] generated by this new Levy process and the increments B(s) − B(t0) of the
Wiener process B (augmented by the null sets), so that the controls in M(t0), N(t0) were
independent of the original Ft0, i.e. of the past of L and B. The proof of Theorem 7.1
shows that the value functions are the same regardless of the setup and the choice of
reference probability spaces, provided that they support a Wiener process B independent
of L.
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