Representation formulas for solutions of Isaacs integro-pde

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1 Representation formulas for solutions of Isaacs integro-pde Shigeaki Koike Mathematical Institute, Tôhoku University Aoba, Sendai, Miyagi , Japan AND Andrzej Świe ch School of Mathematics, Georgia Institute of Technology Atlanta, GA 30332, U.S.A. Abstract We prove sub- and super-optimality inequalities of dynamic programming for viscosity solutions of Isaacs integro-pde associated with two-player, zero-sum stochastic differential game driven by a Lévy type noise. This implies that the lower and upper value functions of the game satisfy the dynamic programming principle and they 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 to smooth sub- and super-solutions of slightly perturbed equations. Keywords: viscosity solutions, integro-pde, Isaacs equation, stochastic differential equation, Lévy process, stochastic differential game Mathematics Subject Classification: 35R09, 49L25, 90D15, 90D25. 1 Stochastic differential game Throughout this paper (Ω, F, P) will be a complete probability space and F t, t 0, will be a normal filtration (i.e. F t is right continuous and F 0 contains all null sets of P). Supported by Grant-in-Aid for Scientific Research (No ) of JSPS. Supported by NSF grant DMS

2 Let T > 0, and L be a Lévy process on [0, T] in R m with cádlág (right continuous with left limits) sample paths, which is of the form where L 0, L 1 are independent Lévy processes, t L 0 (t) = yˆπ(ds, dy), L 1 (t) = 0 0< y <1 L(t) = L 0 (t) + L 1 (t), (1.1) t 0 y 1 yπ(ds, dy), where π is the Poisson random measure of jumps of L and ˆπ is the compensated Poisson random measure of jumps: π([0, t], B) = 1 B (L(s) L(s )), B B(R m \ {0), L(s ) = lim L(t), t s 0<s t ˆπ(dt, dy) = π(dt, dy) dt ν(dy). The measure ν, called the Lévy measure of L or the jump intensity measure of L, is a non-negative measure on (R m \ {0, B(R m \ {0)), where B(R m \ {0) is the Borel σ-field, for which y 2 ν(dy) + ν(dy) < +. (1.2) 0< y <1 y 1 π and ˆπ are random measures on B([0, T]) B(R m \ {0)). We extend ν to a measure on (R m, B(R m )) by setting ν({0) = 0. We will assume in addition that L is an F t -Lévy process, i.e. that: (i) L is F t -adapted; (ii) L(s) L(t), i = 0, 1 is independent of F t for all 0 t < s T. The process L 0 is then a square integrable F t -martingale. L 1 is a compound Poisson process. According to the Lévy-Ito decomposition (see [4], Theorem ), (1.1) is a general form of a Lévy process of pure jump type. Moreover, for every B B(R m \ {0), ˆN(t, B) := ˆπ([0, t], B) is a martingale. We will denote π(dt, dy) = { ˆπ(dt, dy) if y < 1 π(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 F s, and {0 A, A F 0. 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

3 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 Z 1, Z 2 N(t) on [t, s] if Z 1 = Z 2, dt dp a.e. on [t, s] Ω. We will then write Z 1 = Z 2 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 Z 1 = Z 2 (resp., W 1 = W 2 ) on [t, s] then α[z 1 ] = α[z 2 ] (resp., β[w 1 ] = β[w 2 ]) 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 cádlág and F s -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 cádlág and F s -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 + γ(s, X(s), W(s), Z(s), y) π(ds, dy) R m \{0 (1.3) X(t) = x, where b : [0, T] R n W Z R n, γ : [0, T] R n W Z R m R n. The pay-off functional is given by { T J(t, x; W( ), Z( )) = IE l(s, X(s), W(s), Z(s))ds + h(x(t)) t (1.4) for some functions l : [0, T] R n 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

4 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) = inf sup β (t) W M(t) U(t, x) = sup inf α Γ(t) Z N(t) J(t, x; W( ), β[w]( )) (lower value of the game), (1.5) 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, F s, P, L) is fixed and does not change when we define M(t) and N(t). If the filtration F s was generated by the Lévy 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 F t. 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 ( )) = 0 (1.7) V (T, x) = h(x), where the lower value Hamiltonian F : [0, T] R n R n Cb 2(Rn ) R is defined by { F (t, x, p, ϕ) = sup inf b(t, x, w, z), p + l(t, x, w, z) (1.8) w W z Z ( + ϕ(x + γ(t, x, w, z, y)) ϕ(x) 1{ y <1 γ(t, x, w, z, y), Dϕ(x) ) ν(dy). R m The upper value function U should satisfy the upper Isaacs integro-pde { Ut + F + (t, x, DU, U( )) = 0 U(T, x) = h(x), (1.9) where the upper value Hamiltonian F + : [0, T] R n R n Cb 2(Rn ) R is defined by { F + (t, x, p, ϕ) = inf sup b(t, x, w, z), p + l(t, x, w, z) (1.10) z Z w W ( + ϕ(x + γ(t, x, w, z, y)) ϕ(x) 1{ y <1 γ(t, x, w, z, y), Dϕ(x) ) ν(dy). R m Equations (1.7) and (1.9) will be understood in the viscosity sense. Since sup w W inf z Z inf z Z sup w W, it is always true that F F +. Thus, if V is a viscosity solution of 4

5 (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 stochastic 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 deterministic 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 representation 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 subject 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

6 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 Lévy processes and stochastic differential equations with Lévy noise. 2 Notation For a real valued, cádlág, 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 = IEK 2 (T). (2.11) For an interval I R, we will be using the following function spaces. B(R n ) = {u : R n R : u is Borel measurable and bounded, C 2 (R n ) = {u : R n R : u, Du, D 2 u are continuous, C 2 b (Rn ) = {u C 2 (R n ) : u is bounded, C 1,2 (I R n ) = {u : I R n R : u, u t, Du, D 2 u are continuous, C 1,2 b (I Rn ) = {u C 1,2 (I R n ) : u is bounded, For a metric space Z we will denote by B(Z) its Borel σ-field and by d Z its metric. We will write B r (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 Lévy measure satisfying (1.2). 6

7 (A1) There exists a Borel measurable, locally bounded function ρ on R m, such that inf { y >r ρ(y) > 0 for every r > 0, and R m (ρ(y)) 2 ν(dy) < +. (3.1) (A2) b : [0, T] R n W Z R n is bounded, uniformly continuous, and such that b(s 1, x 1, w, z) b(s 2, x 2, w, z) C x 1 x 2 + σ( s 1 s 2 ), (3.2) γ : [0, T] R n W Z R m R n is such that γ(,,,, y) is uniformly continuous on [0, T] R n W Z for every y, γ is Borel measurable with respect to y, and γ(s 1, x 1, w, z, y) γ(s 2, x 2, w, z, y) Cρ(y) ( x 1 x 2 + σ( s 1 s 2 )) (3.3) for all x 1, x 2 R n, s 1, s 2 [0, T], w W, z Z, y R m for some modulus σ. (A3) l : [0, T] R n W Z R, h : R n R are bounded, uniformly continuous and such that l(s 1, x 1, w, z) l(s 2, x 2, w, z) ω( x 1 x 2 + s 1 s 2 ), (3.4) h(x 1 ) h(x 2 ) ω( x 1 x 2 ) (3.5) for all x 1, x 2 R n, s 1, s 2 [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 R n, w W, z Z, y R m. (A5) The filtered probability space (Ω, F, F t, P) is such that there exists an F t -standard Wiener process B in R n 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 defined on a probability space (Ω 1, F 1, Ft 1, P 1 ). We take a complete filtered probability space (Ω 2, F 2, Ft 2, P 2 ) and an Ft 2 -standard Wiener process B in R n. We define a product space (Ω 1 Ω 2, F 1 F 2, Ft 1 F2 t, P1 P 2 ), where we augmented the sigma field and the filtration by the P 1 P 2 null sets. Finally to ensure that the filtration is right continuous we can take F t = s>t (Fs 1 F2 s ). The processes L, B are defined on this new probability space in a natural way and are independent. 7

8 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) R n, (4.1) u(t, x) = g(x), 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] R n R is a viscosity subsolution of (4.1) if u(t, x) g(x) on R n and whenever u ϕ has a global maximum at a point (t, x) for a test function ϕ C 1,2 b ((0, T) Rn ), then ϕ t (t, x) + F(t, x, Dϕ(t, x), ϕ(t, )) 0. (4.2) A bounded lower semicontinuous function u : (0, T] R n R is a viscosity supersolution of (4.1) if u(t, x) g(x) on R n and whenever u ϕ has a global minimum at a point (t, x) for a test function ϕ C 1,2 b ((0, T) Rn ), then ϕ t (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) R n R n C 2 (R n ) B(R n ), we set F r (t, x, p, v, u) = sup inf { b(t, x, w, z), p + l(t, x, w, z) w W z Z + (v(x + γ(t, x, w, z, y)) v(x) γ(t, x, w, z, y), Dv(x) )ν(dy) (4.4) + { y <r { 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 sup w W inf z Z by inf z Z sup w W. We set F r to be either F r or F + r. Definition 4.2. A bounded upper semicontinuous function u : (0, T] R n R is a viscosity subsolution of (4.1) in the sense of Definition 4.2 if u(t, x) g(x) on R n 8

9 and whenever u ϕ has a global maximum at a point (t, x) for a test function ϕ C 1,2 ((0, T) R n ), then for every 0 < r < 1 ϕ t (t, x) + F r (t, x, Dϕ(t, x), ϕ(t, ), u(t, )) 0. A bounded lower semicontinuous function u : (0, T] R n R is a viscosity supersolution of (4.1) in the sense of Definition 4.2 if u(t, x) g(x) on R n and whenever u ϕ has a global minimum at a point (t, x) for a test function ϕ C 1,2 ((0, T) R n ), then for every 0 < r < 1 ϕ t (t, x) + F r (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 F t -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 solution is unique within the class of all predictable processes such that sup IE X(s; t, ξ) 2 < +. t s T The solution has a cádlág modification and thus from now on we will always assume that the solution is cádlág. We can then write (1.3) in the form { dx(s) = b(s, X(s), W(s), Z(s))ds + γ(s, X(s ), W(s), Z(s), y) π(ds, dy) R m \{0 (5.1) X(t) = x. If X is an F t -adapted, cádlág 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 + γ(s, Y (s ), W(s), Z(s), y) π(ds, dy) + 2µdB(s), (5.2) R m \{0 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 F t -measurable and such that IE ξ 2 < +. Let Y be the solution of (5.2) with Y (t) = ξ. We have: (i) [ ] IE sup Y (s) 2 C(T)(1 + IE ξ 2 ). (5.3) t s T 9

10 (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 sup Y (s) X(s) 2 C(T)µ. (5.4) t s T (iii) For all t s 1 < s 2 T [ ] IE sup Y (s 2 ) Y (s 1 ) 2 C(T, IE ξ 2 )(s 2 s 1 ). (5.5) s 1 τ s 2 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 s t s t The term K(s) = (b(τ, Y (τ), W(τ), Z(τ)) b(τ, X(τ), W(τ), Z(τ)))dτ (5.6) (γ(τ, Y (τ ), W(τ), Z(τ), y) γ(τ, X(τ ), W(τ), Z(τ), y))ˆπ(dτ, dy) R m \{0 (γ(τ, Y (τ ), W(τ), Z(τ), y) γ(τ, X(τ ), W(τ), Z(τ), y))1 { y 1 ν(dy)dt R m + 2µ(B(s) B(t)). s t R m \{0 (γ(τ, Y (τ ), W(τ), Z(τ), y) γ(τ, X(τ ), W(τ), Z(τ), y))ˆπ(dτ, dy) is a square integrable martingale with cádlág trajectories. Thus, by (2.11), Burkholder- Davis-Gundy inequality [35], Ito s isometry [33], and (3.1), (3.3), IE sup K(τ) 2 C 1 IE K(s) 2 t τ s s C 2 IE Y (τ ) X(τ ) 2 (ρ(y)) 2 ν(dy)dτ t R m C 3 s t IE sup Y (r) X(r) 2 dτ. t r τ (5.7) Also 2µ(B(s) B(t)) is a martingale and IE sup 2µ(B(τ) B(t)) 2 Cµ(s t). (5.8) t τ s 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 sup t τ s Y (s) X(s) 2 C(T)µ + C 4 s 10 t IE sup Y (r) X(r) 2 dτ. (5.9) t r τ

11 The result now follows from Gronwall s inequality. As regards (5.5), in light of (5.3), it is enough to show it for s 1 = t. We have Again, the process s 2 Y (s) ξ 2 = 4 b(τ, Y (τ), W(τ), Z(τ))dτ t s 2 +4 γ(τ, Y (τ ), W(τ), Z(τ), y)ˆπ(dτ, dy) t R m \{0 s 2 +4 γ(τ, Y (τ ), W(τ), Z(τ), y)1 { y 1 ν(dy)dt t R m +4 2µ(B(s) B(t)) 2. (5.10) K(s) = s t R m \{0 γ(τ, Y (τ ), W(τ), Z(τ), y)ˆπ(dτ, dy) is a square integrable martingale with cádlág trajectories, and exactly like in (5.7) we estimate (now using (3.6)) s IE sup K(τ) 2 C (ρ(y)) 2 ν(dy)dτ C(s t). (5.11) t τ s t R m It thus follows from (5.10), upon using (3.1), (A2), (3.6), (5.8), (5.11), and Hölder s inequality, that IE sup Y (s) ξ 2 C(s t) 2 + C(s t) t τ s s + C(s t) (ρ(y)) 2 1 { y 1 ν(dy)dt C(T, IE ξ 2 )(s t). t R m 6 Regularization Let u : (0, T] R n be bounded and continuous, and let 0 <, β 1. We define the sup-convolution of u by u,β x y 2 (t s)2 (t, x) := sup {u(s, y) (s,y) 2 2β and the inf-convolution of u by u,β (t, x) := inf {u(s, y) + (s,y) x y (t s)2. 2β 11

12 It is well known that the functions u,β, u,β are Lipschitz continuous, lim β 0 u,β (t, x) = u (t, x) := sup {u(t, y) y lim u,β(t, x) = u (t, x) := inf {u(t, y) + β 0 y uniformly on every set [γ, T] R n, and lim 0 u = lim u = u 0 x y 2, 2 x y 2 2 uniformly on every set [γ, T] B R, for every 0 < γ < T, R > 0. Moreover u,β (t, x) + x t2 2β is convex, and u,β (t, x) x 2 2 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 ρ γ : [0, ) (0, 1] (0, 1] [0, + ), such that ρ γ (,, β) is continuous and nondecreasing for every, β, and for every R > 0 lim sup 0 lim sup ρ γ (R,, β) = 0, (6.1) β 0 such that for sufficiently small β, u,β (respectively, u,β ) is a viscosity subsolution of v t + F(t, x, Dv, v(t, )) = ρ γ ( x,, β) in (γ, T γ) R n (6.2) (respectively, viscosity supersolution of v t + F(t, x, Dv, v(t, )) = ρ γ ( x,, β) in (γ, T γ) R n ). (6.3) 12

13 Proof. We will only do the proof for the lower Isaacs equation (1.7). Fix ϕ C 1,2 b ((0, T) R n ). Let (t, x) (0, T) R n, x R, t (γ, T γ) be a point where u,β ϕ attains its global maximum. Let (t, x ) [0, T] R n be such that u,β (t, x) = u(t, x ) x x 2 2 (t t ) 2, (6.4) 2β i.e. u(s, y) x y 2 2 (t s)2 2β for all (s, y) (0, T) R n. We also have u(t, x ) x x 2 2 (t t ) 2, (6.5) 2β u(t, x ) x x 2 2 (t t ) 2 2β ϕ(t, x) u(s, y ) y y 2 2 (s s ) 2 2β ϕ(s, y) (6.6) for all (s, y), (s, y ) (0, T) R n. If, β are small enough we can assume that (t x ) (γ/2, T γ/2) B R+1 (0). Denote by ω R,γ the modulus of continuity of u on (γ/2, T γ/2) B R+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(ω R,γ ( t t ), 2 u ), (6.7) x x 2 u(t, x ) u(t, x) min(ω R,γ ( x x ), 2 u ). (6.8) 2 Therefore it follows from (6.7) and (6.8) that (t t ) 2 β ω R,γ (2 u β), (6.9) x x 2 ω R,γ (2 u ). (6.10) Next, if we set (s, y ) = (t, x ) in (6.6), it is easy to verify that ( ) t t (ϕ t (t, x), Dϕ(t, x)) = β, x x. (6.11) We define ψ(s, y) = x x, x y + x y (t t )(t s) β + t s 2. 2β It follows from (6.5) that u ψ has a global maximum at (t, x ), and ( ) t (ψ s (t, x ), Dψ(t, x t )) = β, x x. (6.12) 13

14 Therefore, by Definition 4.2, { 0 t t + sup inf b(t, x, w, z), x x + l(t, x, w, z) β w W z Z ( ) + ψ(t, x + γ(t, x, w, z, y)) ψ(t, x ) γ(t, x, w, z, y), x x ν(dy) { y <r ( ) + u(t, x + γ(t, x, w, z, y)) u(t, x ) γ(t, x, w, z, y), x x ν(dy) {r y <1 + (u(t, x + γ(t, x, w, z, y)) u(t, x )) ν(dy). (6.13) { y 1 Using (3.2) and (3.4) we have b(t, x, w, z), x x +C x x 2 + l(t, x, w, z) b(t, x, w, z), x x + l(t, x, w, z) + ω( x x + t t ) + σ( t t ) x x (6.14) for all w, z. By the definition of ψ and (3.6) we obtain ( ) ψ(t, x + γ(t, x, w, z, y)) ψ(t, x ) γ(t, x, w, z, y), x x ν(dy) { y <r C (ρ(y)) 2 ν(dy) 0 as r 0 (6.15) {0< y <r uniformly for all w, z. Moreover, since by the semiconvexity of u,β (or directly by (6.4)) x y 2 2 u,β (t, y) u,β (t, x) x x, y x for all y, using (3.6) it follows that ( ) u,β (t, x + γ(t, x, w, z, y)) u,β (t, x) γ(t, x, w, z, y), x x ν(dy) { y <r C (ρ(y)) 2 ν(dy) 0 as r 0 (6.16) {0< y <r uniformly for all w, z. Now, the definition of u,β yields u,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 2 (t t ) 2, 2β 14

15 for all w, z which, together with (6.4), implies {r y <1 u(t, x + γ(t, x, w, z, y)) u(t, x ) γ(t, x, w, z, y), x x u,β (t, x + γ(t, x, w, z, y)) u,β (t, x) γ(t, x, w, z, y), x x + γ(t, x, w, z, y) γ(t, x, w, z, y) 2 2 u,β (t, x + γ(t, x, w, z, y)) u,β (t, x) γ(t, x, w, z, y), x x + C( x x + σ( t t )) 2 (ρ(y)) 2, (6.17) where we have used (3.3) to estimate the last line of (6.17). Therefore ( ) u(t, x + γ(t, x, w, z, y)) u(t, x ) γ(t, x, w, z, y), x x ν(dy) {r y <1 ( ) u,β (t, x + γ(t, x, w, z, y)) u,β (t, x) γ(t, x, w, z, y), x x ν(dy) + C 1( x x + σ( t t )) 2 for all w, z. Moreover, (3.3) and (6.17) also imply u(t, x + γ(t, x, w, z, y)) u(t, x ) u,β (t, x + γ(t, x, w, z, y)) u,β (t, x) + C( x x + σ( t t )) 2 (ρ(y)) 2 + C( x x + σ( t t )) x x ρ(y), which, recalling condition (A1), gives (u(t, x + γ(t, x, w, z, y)) u(t, x )) ν(dy) { y 1 ( u,β (t, x + γ(t, x, w, z, y)) u,β (t, x) ) ν(dy) { y 1 (6.18) + C 2( x x + σ( t t )) 2 (6.19) for all w, z. Therefore, using estimates (6.14), (6.15), (6.16), (6.18), (6.19) in (6.13), sending r 0, and then using (6.9) and (6.10), we obtain that there exists ρ γ satisfying the required conditions such that { t t + sup inf b(t, x, w, z), x x + l(t, x, w, z) β w W z Z ( ) + u,β (t, x + γ(t, x, w, z, y)) u,β (t, x) γ(t, x, w, z, y), x x 1 { y <1 ν(dy) R m ρ γ (R,, β). (6.20) 15

16 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)d 2 u). 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 u,β and u,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 v δ its sup-convolution in the space variable only, i.e. v δ (t, x) = sup {v(t, y) y x y 2, 2δ and by v δ its corresponding inf-convolution in the space variable. It is easy to see from the definitions that v +δ = (v ) δ and v (v δ ) δ. For u from Theorem 6.1 and, β, δ > 0 we denote u := (u +δ,β ) δ, u := (u +δ,β ) δ. It is well known that u (respectively, u) converge uniformly on every set [γ, T] R n as δ 0 to u,β (respectively, u,β ), and that for every t, u(t, ), u(t, ) W 2, (R n ). Moreover we have u(t, x) x 2 is concave, u(t, x) + x 2 is convex, 2δ 2δ and by Proposition 4.5 of [18], u(t, x) + x 2 is convex, u(t, x) x 2 is concave. (6.21) 2 2 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 t n t then Du(t n, x) Du(t, x) for every x. Since u is semiconvex, for every n there exists a n such that (a n, Du(t n, x)) D u(t n, x), the generalized subdifferential of u at (t n, x). Since u is Lipschitz continuous (with the same Lipschitz constant as u,β ), we can assume that (a n, Du(t n, 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, β, δ > 0 and β be so that the conclusion of Lemma 6.1 is satisfied. Let µ > 0. Then there exists δ 0 = δ 0 (, β, µ) such that for 0 < δ < δ 0, u t +µtr(d 2 u)+f(t, x, Du, u(t, )) ρ γ ( x,, β) Cµ 16 a.e. in (γ, T γ) R n (6.23)

17 (respectively, u t +µtr(d 2 u)+f(t, x, Du, u(t, )) ρ γ ( x,, β)+ Cµ a.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) > u,β (t, x), then by Proposition 4.4 of [18], D 2 u(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 u,β, and the definition of F, yields that (6.23) holds if δ is small enough. If u(t, x) = u,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 C 1,2 b ((0, T) Rn ) such that ϕ t (t, x) = (u) t (t, x), Dϕ(t, x) = Du(t, x), D 2 ϕ(t, x) D 2 u(t, x) 1 I, and u,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 suband supersolutions of perturbed equations without the uniformly elliptic term µtr(d 2 u), 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(d 2 u) term is necessary. For η > 0 let ψ η be standard mollifiers with compact support in R n+1. We denote u η = ψ η u, u η = ψ η u, i.e. u η (t, x) = ψ η (t s, x y)u(s, y)dsdy, u η (t, x) = ψ η (t s, x y)u(s, y)dsdy. R n+1 R n+1 The functions u η and u η are well defined on (γ, T γ) R n if η is small enough, are smooth, and they converge uniformly on (γ, T γ) R n 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 u η satisfies (u η ) t + µtr(d 2 u η ) + F(t, x, Du η, u η (t, )) ρ γ ( x,, β, η) Cµ in (γ, T γ) R n (6.25) 17

18 and u η satisfies (u η ) t + µtr(d 2 u η ) + F(t, x, Du η, u η (t, )) ρ γ ( x,, β, η) + Cµ in (γ, T γ) R n, (6.26) where ρ γ ( x,, β, η) is bounded, ρ γ (,, β, η) is continuous, and for every fixed, β, µ, δ like in Proposition 6.3, lim ρ γ( x,, β, η) = ρ γ ( x,, β) (6.27) η 0 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 (u η ) t (t, x) + µtr(d 2 u η (t, x)) + ψ η f(t, x) ψ η ρ γ ( x,, β) Cµ. (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 f w,z (t, x) = b(t, x, w, z), Du(t, x) + l(t, x, w, z) (6.29) ( + 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) R m has a modulus of continuity independent of w, z on every bounded set. Let (t, x 1 ), (s, x 2 ) be such that x 1, x 2 R. We break the integral in (6.29) into + ω 1 (τ) + R m = { y <τ { y τ { y τ for some modulus ω 1, which follows from the fact that u(t, ) W 2, (R n ). Therefore, using the continuity properties of u, Du and assumptions (A1) (A4), we obtain f w,z (t, x 1 ) f w,z (s, x 2 ) C x 1 x 2 + σ( t s ) + 2ω 1 (τ) + u(t, x 1 + γ(t, x 1, w, z, y)) u(s, x 2 + γ(s, x 2, w, z, y)) ν(dy) { y τ + u(t, x 1 ) u(s, x 2 ) ν(dy) { y τ + γ(t, x 1, w, z, y), Du(t, x 1 ) γ(s, x 2, w, z, y), Du(s, x 2 ) ν(dy) {τ y <1 C x 1 x 2 + σ( t s ) + 2ω 1 (τ) + C(σ R ( t s ) + x 1 x 2 )(1 + ρ(y))ν(dy) { y τ 2ω 1 (τ) + C τ (σ R ( t s ) + x 1 x 2 ) 18

19 for some modulus σ R and constant C τ. This gives a local modulus of continuity of f w,z independent of w, z, and hence we obtain that there exists a modulus ω and for every R > 0 a modulus ω R such that f(t, x 1 ) f(s, x 2 ) ω R ( t s ) + ω( x 1 x 2 ) for all t, s (γ, T γ), x 1, x 2 R. (6.30) Thus it follows that ψ η f(t, x) f(t, x) ρ(η, x ), (6.31) for some bounded local modulus ρ. Let f η (t, x) = F(t, x, Du η (t, x), u η (t, )). Since sup u η (t, ) W 2, (R n ) C, t u η u uniformly, and Du η Du locally uniformly as η 0, it follows that f η f locally uniformly as η 0, i.e. that f η (t, x) f(t, x) ρ 1 (η, x ), (6.32) for some bounded local modulus ρ 1. Combining (6.31) and (6.32) we thus obtain ψ η f(t, x) f η (t, x) ρ(η, x ) + ρ 1 (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)d 2 u), however the term ρ γ ( x,, β, η) would have to be replaced by a term ρ γ (x,, 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 < t 0 h T { h u(t 0, x 0 ) sup inf IE l(x(s), α[z](s), Z(s))ds + u(h, X(h)), α Γ(t 0 ) Z N(t 0 ) t 0 19

20 where X(s) = X(s; t 0, x 0 ) is the solution of (1.3) with W = α[z] for Z N(t 0 ). (ii) If u is a (bounded) continuous viscosity supersolution of (1.9) then for every 0 < t 0 h T { h u(t 0, x 0 ) sup inf IE l(x(s), α[z](s), Z(s))ds + u(h, X(h)), α Γ(t 0 ) Z N(t 0 ) t 0 where X(s) = X(s; t 0, x 0 ) is the solution of (1.3) with W = α[z] for Z N(t 0 ). (iii) If u is a (bounded) continuous viscosity subsolution of (1.7) then for every 0 < t 0 h T { h u(t 0, x 0 ) inf sup IE l(x(s), W(s), β[w](s))ds + u(h, X(h)), (7.1) β (t 0 ) W M(t 0 ) t 0 where X(s) = X(s; t 0, x 0 ) is the solution of (1.3) with Z = β[w] for W M(t 0 ). (iv) If u is a (bounded) continuous viscosity supersolution of (1.7) then for every 0 < t 0 h T { h u(t 0, x 0 ) inf sup IE l(x(s), W(s), β[w](s))ds + u(h, X(h)), (7.2) β (t 0 ) W M(t 0 ) t 0 where X(s) = X(s; t 0, x 0 ) is the solution of (1.3) with Z = β[w] for W M(t 0 ). Proof. We will only show (iii) and (iv) since the proofs of (i) and (ii) are similar. Proof of (iii): Let t 0 < h T, x 0 R n. Step 1. (Smooth case, µ > 0.) We will first show (iii) when µ > 0 and u C 1,2 ((0, T) R n ), u, u t, Du, D 2 u are bounded and Lipschitz continuous, and u is a viscosity subsolution of u t + µtr(d 2 u) + F (t, x, Du, u( )) = 0 in (0, T) R n. Let m 1 and we set r = (h t 0 )/m, t i = t 0 + ir, i = 0,..., m. Define Λ(t, x, w) = u t (t, x) + µtr(d 2 u(t, x)) { + inf b(t, x, w, z), Du(t, x) + l(t, x, w, z) z Z ( + 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). R m It is not difficult to show, that the functions Λ(t,, ) are uniformly continuous on 20

21 R n W, uniformly in t. To see it for the integral term we notice that for every z Z ( u(t, x1 + γ(t, x 1, w 1, z, y)) u(t, x 1 ) 1 { y <1 γ(t, x 1, w 1, z, y), Du(t, x 1 ) ) ν(dy) R m ( u(t, x2 + γ(t, x 2, w 2, z, y)) u(t, x 2 ) 1 { y <1 γ(t, x 2, w 2, z, y), Du(t, x 2 ) ) ν(dy) R m C (ρ(y)) 2 ν(dy) { y <κ ( + u(t, x 1 + γ(t, x 1, w 1, z, y)) u(t, x 2 + γ(t, x 2, w 2, z, y)) + u(t, x 1 ) u(t, x 2 ) { y κ ) +1 { y <1 γ(t, x 1, w 1, z, y), Du(t, x 1 ) γ(t, x 2, w 2, z, y), Du(t, x 2 ) ν(dy) ω 1 (κ) + C κ x 1 x 2 + γ(t, x 2, w 1, z, y) γ(t, x 2, w 2, z, y) ν(dy) { y κ ω 1 (κ) + C κ x 1 x 2 + min(ω y (d W (w 1, w 2 )), Cρ(y))ν(dy) { y κ ω 1 (κ) + C κ x 1 x 2 + ω κ (d W (w 1, w 2 )) (7.3) for some moduli ω 1, ω κ 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 ω y is the modulus of continuity of γ(,,,, y).) Therefore, since W is separable, for every i = 0,..., m 1, we can find a sequence {w i j j=1 and a family and balls {Bi j j=1 covering Rn, such that Λ(t i, x, w i j ) r if x Bi j. We now define maps ψ i : R n W, i = 0,..., m 1 by ψ i (x) = w i k k 1 if x Bk i \ j=1 B i j. They are B(R n )/B(W) measurable maps, and by construction Λ(t i, x, ψ i (x)) r for every x R n, i = 0,..., m 1. (7.4) We now fix Z N(t 0 ) and define a control W m M(t 0 ) inductively. We set W m (s) = ψ 0 (x 0 ) s [t 0, t 1 ], (7.5) and let X be the solution of (5.2) on [t 0, t 1 ] with the controls W m and Z. Suppose now that W m and X have been defined on [t 0, t i ], i = 0,..., m 1. We then set W m (s) = ψ i (X(t i )) s (t i, t i+1 ], 21

22 and hence we can define the solution X of (5.2) on [t 0, t i+1 ] with controls W m and Z. We notice that W m is predictable. Using this notation we denote Au(t) = u t (t, X(t)) + b(t, X(t), W m (t), Z(t)), Du(t, X(t)) ( + u(t, X(t ) + γ(t, X(t ), W m (t), Z(t), y)) u(t, X(t )) R m 1 { y <1 γ(t, X(t ), W m (t), Z(t), y), Du(t, X(t )) ) ν(dy) + µtr(d 2 u(t, X(t))). Estimate (5.5) gives ( ) P sup X(s) X(t i ) r 1/4 Cr 1 2, i = 0,..., m 1, (7.6) t i s t i+1 l(s, X(s), W m (s), Z(s))ds + u(t i+1, X(t i+1 )) for some constant C 1 independent of Z. Therefore, using Ito s formula, definition of W m, assumptions (A1) (A4), (7.6), and arguing similarly as in (7.3), we obtain that for every κ > 0, i = 0,..., m 1, { ti+1 u(t i, X(t i )) = IE Au(s)ds + u(t i+1, X(t i+1 )) { ti+1 t i IE ( Λ(t i, X(t i ), ψ i (X(t i ))) + l(t i, X(t i ), W m (s), Z(s)))ds t i +u(t i+1, X(t i+1 )) + Cr(κ + ω κ (r)) { ti+1 IE + Cr(κ + ω κ (r)), t i where C and moduli ω κ 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 { h u(t 0, x 0 ) IE l(s, X(s), W m (s), Z(s))ds + u(h, X(h)) + Cr(κ + ω κ (r))m t 0 { h ( ( )) h = IE l(s, X(s), W m t0 (s), Z(s))ds + u(h, X(h)) + C(h t 0 ) κ + ω κ. t 0 m (7.7) We now define a strategy α m Γ(t 0 ) by setting α m [Z](s) = W m (s). Rewriting (7.7) slightly it follows that for every κ > 0 there exists a modulus ω κ such that { h u(t 0, x 0 ) IE l(s, X(s), α m [Z](s), Z(s))ds + u(h, X(h)) + κ + ω κ ( 1 ). (7.8) t 0 m Following [40] we claim that for every β (t 0 ) there exist Z N(t 0 ), W M(t 0 ) such that α m [ Z] = W, and β[ W] = Z on [t 0, h]. (7.9) 22

23 They are constructed inductively in the following way. We set W [t0,t 1 ] = ψ 0 (x 0 ) and Z [t0,t 1 ] = β[ W] [t0,t 1 ]. We remark here that to do this we have to extend W [t0,t 1 ] to the whole interval [t 0, T] but since strategies are non-anticipating, Z [t0,t 1 ] does not depend on the extension and we will thus omit this technical detail. If we know W, Z on [t 0, t i ] we also know the solution of (5.2) X on [t 0, t i ], and we extend W to [t 0, t i+1 ] by W [t0,t i+1 ] = α m [ Z] [t0,t i+1 ] (since α m [ Z] [t0,t i+1 ] only depends on Z [t0,t i ]). We then extend Z to [t 0, t i+1 ] by setting Z [t0,t i+1 ] = β[ W] [t0,t i+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 (t 0 ) and Z = Z we thus have { h u(t 0, x 0 ) IE l(s, X(s), W(s), β[ W](s))ds + u(h, X(h)) + κ + ω κ ( 1 t 0 m ) { h sup IE l(s, X(s), W(s), β[w](s))ds + u(h, X(h)) + κ + ω κ ( 1 W M(t 0 ) t 0 m ). and taking the infimum above over all strategies yields { h u(t 0, x 0 ) inf sup IE l(s, X(s), W(s), β[w](s))ds + u(h, X(h)) β (t 0 ) W M(t 0 ) t 0 +κ + ω κ ( 1 (7.10) m ). 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 t 0 < h < T. We set γ := min(t 0, T h)/2. For sufficiently small, β, µ, δ, η > 0 (i.e. δ < δ 0 (, β, µ) from Proposition 6.3), let u η and ρ γ be from Proposition 6.6. Then u η satisfies the assumptions of Step 1 on (γ, T γ) R n with l(t, x, w, z) = l(t, x, w, z) + ργ ( x,, β, η) + Cµ. Therefore, by Step 1, we obtain { h u η (t 0, x 0 ) inf sup IE (l(s, X µ (s), W(s), β[w](s)) + ρ γ ( X µ (s),, β, η))ds β (t 0 ) W M(t 0 ) t 0 + u η (h, X µ (h)) + Cµ (h t 0), (7.11) where X µ is the solution of (5.2) with controls W, β[w]. We have lim η 0 u η = u and lim δ 0 u = u,β uniformly on (γ, T γ) R n, lim β 0 u,β = u uniformly on (γ, T γ) R n, and lim 0 u = u uniformly on (γ, T γ) B R (0) for 23

24 every R > 0. Moreover all functions are uniformly bounded for all 0 <, β, δ, η < 1. Therefore, using (5.3) and (5.4), it follows that lim lim lim 0 β 0 lim lim sup sup µ 0 δ 0 η 0 β (t 0 ) W M(t 0 ) Moreover, by (6.1), (6.27), (5.3) and (5.4), we obtain { lim sup lim sup lim sup lim sup lim sup 0 β 0 µ 0 δ 0 η 0 sup sup β (t 0 ) W M(t 0 ) IE IE u η (h, X µ (h)) u(h, X(h)) = 0. (7.12) ρ γ ( X µ (s),, β, η)ds = 0. t 0 h (7.13) Therefore, taking lim sup 0 lim sup β 0 lim sup µ 0 lim sup δ 0 lim sup η 0 in (7.11), and combining it with (7.12) and (7.13) yields { h u(t 0, x 0 ) inf sup IE l(s, X(s), W(s), β[w](s)) + u(h, X(h)). β (t 0 ) W M(t 0 ) t 0 Proof of (iv): We will only show (iv) in the smooth case, i.e. when µ > 0, u C 1,2 ((0, T) R n ), u, u t, Du, D 2 u are bounded and Lipschitz continuous, u is a viscosity supersolution of u t + µtr(d 2 u) + F (t, x, Du, u( )) = 0 in (0, T) R n, and t 0 < 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 t 0 )/m, t i = t 0 + ir, i = 0,..., m. Define Λ(t, x, w, z) = u t (t, x) + µtr(d 2 u(t, x)) + b(t, x, w, z), Du(t, x) + l(t, x, w, z) ( + 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). R m The functions Λ(t,,, ) are uniformly continuous on R n W Z, uniformly in t. Since for every (t, x) sup inf Λ(t, x, w, z) 0, w W z Z for every i = 0,..., m 1, we can find a sequence {z i j j=1 and a family of products of balls {B i j j=1 { B i j j=1 covering Rn W, such that Λ(t i, x, w, z i j ) r if (x, w) Bi j B i j. We now define maps ψ i : R n W Z, i = 0,..., m 1 by ψ i (x, w) = z i k if (x, w) (Bk i B k 1 k i ) \ (Bj i B j i ). j=1 24

25 These are B(R n ) B(W)/B(Z) measurable maps, and by construction Λ(t i, x, w, ψ i (x, w)) r for every (x, w) R n W, i = 0,..., m 1. (7.14) We now fix W M(t 0 ) and define a control Z m N(t 0 ) inductively. We set Z m (s) = ψ 0 (x 0, W(s)) s [t 0, t 1 ], (7.15) and take X to be the solution of (5.2) on [t 0, t 1 ] with the controls W and Z m. If Z m and X have been defined on [t 0, t i ], i = 0,..., m 1, we set Z m (s) = ψ i (X(t i ), W(s)) s (t i, t i+1 ], and then we can define the solution X of (5.2) on [t 0, t i+1 ] with controls W and Z m. The control Z m is predictable. Therefore, as in Step 1 of the proof of (iii) we obtain that for every κ > 0, i = 0,..., m 1, u(t i, X(t i )) (7.16) { ti+1 IE l(s, X(s), W(s), Z m (s))ds + u(t i+1, X(t i+1 )) Cr(κ + ω κ (r)), t i where C and moduli ω κ are independent of i, W and only depend on u and various constants and moduli in (A1) (A4). Setting β m [W] = Z m, we see that β m is nonanticipating, and thus β m (t 0 ). Therefore, adding (7.16) for i = 0,..., m 1, we get u(t 0, x 0 ) { h IE l(s, X(s), W(s), β m [W](s))ds + u(h, X(h)) Cr(κ + ω κ (r))m. t 0 for every W M(t 0 ). The above implies u(t 0, x 0 ) inf sup β (t 0 ) W M(t 0 ) { h IE and it remains to let m + and then κ 0. l(s, X(s), W(s), β[w](s))ds + u(h, X(h)) t 0 ( ( )) h t0 C(h t 0 ) κ + ω κ, m 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 programming principle. 25

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