Path-Dependent Optimal Stochastic Control and Viscosity Solution of Associated Bellman Equations

In this paper we study the optimal stochastic control problem for a path-dependent stochastic system under a recursive path-dependent cost functional, whose associated Bellman equation from dynamic programming principle is a path-dependent fully nonlinear partial differential equation of second order. A novel notion of viscosity solutions is introduced. Using Dupire's functional It\^o calculus, we characterize the value functional of the optimal stochastic control problem as the unique viscosity solution to the associated path-dependent Bellman equation.


Introduction
Since the initial investigation of Krylov [26], it has been the subject of many studies to verify that the value function of an optimal stochastic control problem should be the unique solution of the associated Bellman equation from the dynamic programming principle for the optimal stochastic control problem. Nowadays, the notion of viscosity solution invented in 1983 by Crandall and Lions [10] has become a universal tool to study such a broad fundamental subject. For detailed exposition of such a tool and the related general dynamic programming theory on optimal stochastic control, see among others the survey paper of Crandall, Ishii & Lions [9] and the monographs of Fleming & Soner [20] and Yong & Zhou [51].
Such a theme has also been developing in terms of backward stochastic differential equations (BSDEs). For instance, a BSDE depending on a Markovian diffusion in a Markovian way via the generator and the terminal condition, is associated to a secondorder partial differential equation (PDE), and a fully coupled forward and backward stochastic differential equation (FBSDE) is associated to a quasi-linear PDE. For relevant details, see Pardoux & Peng [37], Ma, Proter & Yong [35], and Pardoux & Tang [38]. These developments appear quite natural in view of the close relation between BSDEs and minimax problems as exposed by Tang [48]. Furthermore, the second-order BSDE (2BSDE) is associated to a fully nonlinear PDE, and for such a relation see Cheritdito, Soner, Touzi & Victoir [6] and Soner, Touzi & Zhang [45]. More generally, Peng [39] shows that an optimal stochastic control problem-where the coefficients of both the system and the cost functional depend on the history path of the underlying Brownian motion-should be associated to a fully nonlinear backward stochastic PDE as the underlying Bellman equation.
Recently Dupire [14] introduced horizontal and vertical derivatives in the path space of a path functional, non-trivially generalized the classical Itô formula to a functional Itô formula (see Cont & Fournie [7,8] for a more general and systematic research), and provided a functional extension of the classical Feynman-Kac formula. His insightful work is becoming a foundation to stochastic analysis of path functionals, and has stimulated extensions of some above-mentioned developments to the functional case. In fact, he has shown that a path functional in C 1,2 (Λ) solves a linear path-dependent PDE (PPDE) if its composition with a Brownian motion generates a martingale. In the plenary lecture at the International Congress of Mathematicians of the year 2010, Peng [41] pointed out that a non-Markovian BSDE is a PPDE. In view of Dupire's functional Itô formula, it is very natural to associate a BSDE to a "semi-linear" PPDE, and a stochastic optimal control problem of BSDE to a path-dependent Bellman equation. Peng & Wang [42] studied the former relation and give some sufficient conditions for a semi-linear PPDE to admit a classical solution. However, a PPDE, even for the simplest heat equation, rarely has a classical solution, and a path-dependent Bellman equation, even in the simpler state-dependent case, also appeals to a generalized solution in many occasions. Therefore, generalized solution of general PPDEs are demanding, and it has to be developed.
This paper incorporates Dupire's functional Itô calculus to discuss the optimal stochastic control problem for a path-dependent stochastic system under a recursive path-dependent cost functional. The associated Bellman equation from the dynamic programming principle in this general setting is a path-dependent fully nonlinear partial differential equation of second order, and we are concerned with its viscosity solution.
The familiar optimal control of SDEs with delay can be addressed within our current framework.
In the classical theory of viscosity solution to PPDEs (see Crandall, Ishii & Lions [9] and Fleming & Soner [20]), local compactness of the state space and smoothness of the norm play a crucial role. The main difficulty for our path-dependent case lies in both facts that the path space Λ is an infinite dimensional Banach space and lacks of a local compactness, and that the maximal norm · 0 is not smooth. The arguments for the case of Hilbert space introduced by Lions (see [31,29]) contain a limiting procedure based on the structure of Hilbert space, and are difficult to be adapted to our pathdependent case where we have to work in a subspace C α (which has a local compactness, but whose α-Hölder norm is not smooth).
In the generalization of Kim's i-smooth theory [25], Luyakonov [34] developed a theory of viscosity solution to fully non-linear path-dependent (also called functional in literature) Hamilton-Jacobi (HJ) equations of first order-which include conventionally called Bellman and Isaacs equations arising from deterministic optimal control problems and differential games for time-delayed ordinary differential equations. He used the so-called co-invariant derivatives (it is Clio-derivative in Kim's terminology), which coincide with the restriction of Dupire's derivatives on continuous paths though their definitions appear different. A deterministic functional differential system, subject to some proper conditions, starting at a uniformly Lipschitz continuous path, actually evolves forever in the locally compact space C 0,1 of all uniformly Lipschitz continuous paths. This property of deterministic dynamical systems allows Luyakonov [34] to define the jet functionals and to localize the application of dynamical programming all in C 0,1 (in fact in a sequence of expanding compact subsets) of paths. Due to these conveniences, his proof of existence and uniqueness of viscosity solutions appears fairly straightforward. In our stochastic case, the situation changes in a dramatic manner. Even starting at a uniformly Lipschitz continuous path, due to the essential diffusion nature, our dynamic system could not live in C 0,1 anymore, and it is impossible in general to enclose under proper conditions our dynamical system within any given compact space. We have to choose an α-Hölder continuous paths space C α (∀α ∈ (0, 1 2 )) to substitute C 0,1 , and also define our jet functional on a family of expanding compact subsets {C α µ : µ is sufficiently large} in C α . We give an example to illustrate the following phenomenon in our stochastic dynamic system (see Remark 5.2 for details): starting at the boundary of C α µ , our stochastic system might leave away from the set C α µ with probability one within an arbitrary small time. This essential nature prevents us from starting the dynamic programming at the boundary of C α µ to show the Bellman equation holds there for the value functional. More precisely, to show that the value functional is a viscosity solution, we could not follow the conventional way to start the dynamic programming directly at the minimum/maximum path of the difference between the value functional and a jet functional since the extremum path might be at the boundary. To get around the difficulty, we construct a specific perturbation γ ε t ∈ C α µ (around the extremum path γ t ) where we can start the dynamic programming and use the exit timeτ ε from C α µ (see the definition ofτ ε in the existence proof of Section 5; the probability for our dynamical system starting from γ ε t at time t to stay within C α µ up to time t + δ is shown to converge to one, uniformly with respect to sufficiently large µ, as δ → 0+) to localize our dynamic programming within the compact subset C α µ . It seems to be the first here for us to defineτ ε in association to the α-Hölder modulus of the system's history path. We finally prove that the value functional is a viscosity solution in Section 5. In the proof of the uniqueness, we adapt the smoothing (and viscosity vanishing for the degenerate case) methodology of Lions [30] to our path-dependent case, and also use the natural approximating arguments of parameterized state-dependent PDEs. In the passage to the limit, our a priori maximal and Hölder estimates on the second-order derivatives of the solutions to the approximate PPDEs play a crucial role. Our methodology is expected to be used to study the path-dependent Isaacs equation arising from stochastic differential games (see [49]) and other related fully nonlinear PPDEs. We note that using Gateaux or Frechet derivatives, Goldys & Gozzi [22] and Fuhrman, Masiero & Tessitore [21] considered time-delay stochastic optimal control problems with the diffusion being independent of the control variable, and studied the mild solutions to the associated semi-linear functional Bellman equations on Hilbert space and Banach space, respectively.
Defining the "semi-jets" on non compact subsets in terms of a nonlinear expectation, Ekren et al. [15] studied in a quite different way viscosity solution to second order path-dependent PDE. They prove the existence and uniqueness of viscosity solutions (in their sense) for a semi-linear PPDE by Peron's approach. In the subsequent works, Ekren, Touzi, and Zhang (see for details [16,17,18]) use their previous notion of viscosity solution to study the fully nonlinear PPDE, Pham and Zhang (see [43]) discuss path-dependent Bellman-Isaacs equations. However, their relevant results on the path-dependent Bellman equation require stronger conditions: Their Assumption 2.8 requires, as they have noted, the diffusion coefficient σ to be path-invariant (see [18, page 8] for details), and, in the degenerate case, their Assumption 7.1 (i) requires further approximating structures (see [18, page 29] for details). Our Theorems 4.6 (on the non-degenerate case) and 4.7 (on the degenerate case) give more general uniqueness results on path-dependent Bellman equation. Furthermore, Ekren, Touzi, and Zhang [16,17,18] directly work with an abstract fully nonlinear PPDE, and use a more complicated definition of super-and sub-jets in their notion of viscosity solution, in particular their definitions involve the unnatural and advanced notion of nonlinear expectation.
The rest of the paper is organized as follows. In Section 2, we introduce the calculus for path functionals of [7,8] and [14], and preliminary results on BSDEs. In Section 3, we formulate the path-dependent stochastic optimal control problem and discuss the dynamic programming principle, which is crucial in the proof of the existence of a viscosity solution. In Section 4, we define classical and viscosity solutions to our pathdependent Bellman equation, state our main results, and prove a verification theorem in the context of classical solutions. In Section 5, we prove that the value functional is a viscosity solution, which implies our existence of a viscosity solution to the pathdependent Bellman equations. Finally in Section 6 we prove the uniqueness of viscosity solutions for the path-dependent Bellman equations.

Preliminaries
For convenience, define for 0 ≤ t <t ≤ T , andγt, γ t ∈Λ, We define the quasi-norm and metric inΛ as follows: for each 0 ≤ t ≤t ≤ T and γ t ,γt ∈Λ T (R n ), Here | · | is the standard metric of the Euclid space, d p is called parabolic metric. It is easy to verify that (Λ t (R n ), · 0 ) is a Banach space, (Λ, d p ) is a complete metric spaces.
if all the limits exist, with e i , i = 1, · · · , n, being coordinate unit vectors of R n . If (2.2) is well-defined for all γ t , the map D x v := (D 1 v, · · · , D n v) :Λ(R n ) → R n is called the vertical derivative of v. We define the Hessian D xx v(γ t ) in an obvious way. Then D xx v is an S(n)-valued functional defined onΛ(R n ), where S(n) is the space of all n × n symmetric matrices.
(Horizontal derivative). The horizontal derivative at γ t ∈Λ of a functional v :Λ → R is defined as if v together with all its derivatives are bounded, and v ∈ C j,k l (Λ) if v ∈ C j,k (Λ) and v grows in a linear way.
which shows the coincidence of Dupire's derivatives with the classical ones.

Space of continuous paths
Let Λ t (R n ) := C 0 ([0, t], R n ) be the set of all continuous R n -valued functions defined over [0, t] which vanish at time zero, and Λ(R n ) := t∈[0,T ] Λ t (R n ). In the sequel, for notational simplicity, we use 0 to denote γ 0 or vectors and matrices whose components are all zero. Clearly, Λ(R n ) ⊂Λ(R n ). (Λ t (R n ), · 0 ) is a Banach space, and (Λ(R n ), d p ) is a complete metric space. We write Λ for Λ(R n ) if there is no confusion. Let E be a Banach space.v :Λ → E and v : Λ → E are called consistent on Λ if v is the restriction ofv on Λ.
Definition 2.7. (Hölder continuity). For α ∈ (0, 1], we say that a functional v defined on Q ⊂ Λ is Hölder continuous on Q with exponent α if the quantity

Filtration and localization
Now we introduce the filtration of Λ T . Let G T := B(Λ T ), the smallest Borel σ-field generated by metric space and σ means the smallest σ-field generated by the underlying class of subsets. G := {G t , t ∈ [0, T ]} is a filtration. G t is just the smallest σ-algebra generated by the collection of finite-dimensional cylinder sets of the form where, for all i = 1, · · · , k, t i ∈ [0, t]. For more details, see the monograph of Stroock and Varadhan [46, Section 1.3]. Define π t : Λ T → Λ t as follows: It is easy to observe that π −1 t (G T ) = σ(π −1 t (G T )) = B(Λ t ), the smallest Borel σ-field generated by metric space (Λ t , · 0 ).
A map H : [0, T ] × Λ T → E is called a functional process. Moreover, we say a process H is adapted to the filtration G , if H(t, ·) is G t -measurable for any t ∈ [0, T ]. Obviously an adapted process H has the property that, for any γ 1 , γ 2 ∈ Λ T satisfying γ 1 (s) = γ 2 (s) for all s ∈ [0, t], H(t, γ 1 ) = H(t, γ 2 ). Hence H(t, γ T ) can be view as H(γ t ), that is to say a functional adapted process equals a path functional defined on Λ. Let B = {B(t), t ∈ [0, T ]} be the canonical process on Λ, i.e. B(t, γ) = γ(t). Define B t := {B(s), s ∈ [0, t]}, and it is G t -measurable. For any continuous map v : Λ → E, we know that {v(B t ), t ∈ [0, T ]} is an adapted functional process.
Let P 0 be the Wiener measure of space (Λ T , G ), under which the canonical process B (i.e. B(t, γ) = γ(t)) is a standard Brownian motion. For any integrable G T -measurable variable ξ, denote P 0 [ξ] the integration of ξ under measure P 0 .
Let X := {X(t), t ∈ [0, T ]} be an n-dimensional adapted continuous stochastic process on the probability space (Ω, F , (F t ) 0≤t≤T , P ). For t ∈ [0, T ], X(t) is the value of X at time t and X t is the path of X up to time t, i.e., X t := {X(r), r ∈ [0, t]}. X can be viewed as a map from Ω to Λ T , the continuity and adaption of X imply that The following functional Itô formula was initiated by Dupire [14] and later extended to a more general context by Cont & Fournie [8].

Space shift
For any fixed γ t ∈Λ and s ∈ [t, T ], we introduce the shifted spaces of cadlag and continuous paths.
For any map v :Λ γt → R, define the derivatives in the spirit of Definition 2.2, and define the spaces } be a G -progressively measurable functional process, and ξ be a G T measurable functional variable, let γ t ∈Λ. Define the process H γt on [0, T ] × Λ γt T and the variable ξ γt on Λ γt T , as the restriction on Λ γt of H and ξ, respectively; that is, T ]} is a G γt -progressively measurable functional process, and ξ γt is a G γt T -measurable functional variable.
µ has a convergence subsequence and the limit lies in C α µ .
Let Q be the set of rational numbers, and {r 1 , r 2 , · · · } be an enumeration of Q∩[0, t). By (2.9), we can choose a subsequence {γ k ] for all k = 1, 2, · · · and {γ converges to a limit γ t (r 2 ). Continue this process, and then let For any r, s ∈ Q ∩ [0, t), where K is a sufficiently large integer such that r, s ∈ [t,t k ] for all k ≥ K. Setting n → ∞, we have Hence γ t has a continuous extension on [0, t], still denoted by γ t , and it lies in C α µ . It remains to show the limit lim k→∞ d p (γ k t k , γ t ) = 0. In fact, for any ε > 0, define where ⌊s⌋ denotes the greatest integer less than or equal to s.
which leads to lim k→∞ d p (γ k t k , γ t ) = 0. Define the following (random) time for the path to oscillate beyond a given α-Hölder modulus: This kind of exit time will play a crucial role in the subsequent proof of the existence of a viscosity solution.

Backward stochastic differential equations
Let (Ω, F , (F t ) 0≤t≤T , P ) be a probability space with the usual condition (see Karatzas and Shreve [23]), and {W (t), t ∈ [0, T ]} be a d-dimensional standard Brownian motion. Let N be the collection of all P -null sets in Ω, for any and the following two conditions are satisfied: ( We recall the following comparison theorem on BSDEs (see El Karoui, Peng and Quenez [19]) Lemma 2.11. Let two BSDEs of data (ξ 1 , f 1 ) and (ξ 2 , f 2 ), satisfy all the assumptions of Lemma 2.10. Denote by (Y 1 , Z 1 ) and (Y 2 , Z 2 ) their respective adapted solutions. Then we have: (2) (Strict monotonicity). If, in addition to (1), we also P {ξ 1 > ξ 2 }>0, then 3 Formulation of the path-dependent optimal stochastic control problem and dynamic programming principle Let the set of admissible control processes U be the set of all F -progressively measurable process valued in some compact metric space U.
We make the following assumption.
(H1) There exists a constant C > 0 such that, for all (t, γ T , y, z, u), : Ω → Λ t and admissible control u ∈ U, consider the following SDE: The constant C p only depends on the Lipschitz constant of b and σ in (γ t , t).
Combing Lemmas 2.10 and 3.1, we have , and u ∈ U, P -a.s., For the particular case of a deterministic Γ t , i.e. Γ t = γ t ∈ Λ t : Given the control process u ∈ U, we introduce the following cost functional: The value functional of the optimal control is defined bỹ for details). For the convenience of reader we recall the notion of esssup of random variables. Given a family of real-valued random variables η α , α ∈ I, a random variable η is said to be ess sup α∈I η α , if 1. η ≤ η α , P -a.s., for any α ∈ I; 2. if there is another random variable ξ such that ξ ≤ η α , P -a.s., for any α ∈ I, then ξ ≤ η, P -a.s..
The existence of ess sup α∈I η α is well known. Under Assumption (H1), the random variableṽ(γ t ) ∈ L p (Ω) is F t -measurable. We have Proposition 3.4. The value functionalṽ is deterministic. (3.9) In view of [24,Theorem A.3], it is sufficient to prove that, for any u 1 , u 2 ∈ U, we have By the uniqueness of solution of BSDE, we have . Since Y γt,u is continuous in u ∈ U, we suppose without lost of generality that u i (·) takes the following form: Like (3.11) and (3.12), we know that It is easy to prove that J(γ t , u ij (·)) is deterministic. Without lost of generality, we suppose .
From (3.3) and (3.4), we have the following estimates on functionalṽ.
To formulate the DPP for the optimal control problem (3.6), (3.7) and (3.8), we define the family of backward semi-groups generated by BSDE (3.7) in the spirit of of Peng [40].
Given the initial path γ t ∈ Λ, an F -stopping timeτ ≥ t, an admissible control process u ∈ U, and a real-valued random variable η ∈ L 2 (Ω, Fτ , P ; R), we put where the pair (Ỹ γt,u ,Z γt,u ) solves the following BSDE of the terminal timeτ : with X γt,u being the solution to SDE (3.6). Then, obviously, for the solution (Y γt,u , Z γt,u ) of BSDE (3.7), the uniqueness of the BSDE yields The following dynamic programming principle (DPP) is adapted from the Markovian case, by mimicking the method of Peng [39,40].
Theorem 3.6. Let Assumption (H1) be satisfied. Then for any δ ∈ (0, T −t), the value functionalṽ obeys the following: Our proof requires the following lemma Proof. Sinceṽ is continuous in γ t ∈ Λ t and Y γt,u t is continuous in (γ t , u) ∈ Λ t × U, it is sufficient to prove (3.16) for the following class of Γ t and u: Here, N is a positive integer, u i is F t -adapted, and We obtain the first assertion (3.16).
In a similar way, we prove (3.17). Obviously there existsΓ t ∈ L 2 (Ω, F t ; Λ t , B(Λ t )) of the formΓ such that where C is the constant in Lemmas 3.1 and 3.2, By Lemmas 3.1 and 3.2, we have for any u ∈ U, P -a.e., Then for any γ i t , we can choose an F t -adapted admissible control Combining (3.18), we have The proof is complete.
In Lemma 3.5, the value functionalṽ is Lipschitz continuous in Λ t , uniformly in t. Theorem 3.6 implies the following continuity in t.
Lemma 3.8. Let Assumption (H1) be satisfied. There is a constant C such that for Proof. Suppose that t ≤ t ′ . From Theorem 3.6, we see that for any ε > 0, there is u ∈ U such that Sinceṽ is uniformly continuous in γ t , we have from Lemmas 3.1 and 3.2, for a positive constant C being independent of u. By the definition of G γt,u t,t ′ , we have Since ε is arbitrary, we have (3.19).
From Lemmas 3.5 and 3.8, we have the regularity for the value functionalṽ.
Theorem 3.9. Let Assumption (H1) be satisfied. There is a constant C > 0 such that 4 Associated path-dependent Bellman equation

Path-dependent Bellman equation and viscosity solution
Define the Hamiltonian H : Consider the following path-dependent Bellman equation: For (κ, ι) ∈ (0, ∞) × (0, T ) and γ t ∈ Λ with t ∈ [0, T − ι), define the cylinder Throughout the rest of this paper, we fix α ∈ (0, 1 2 ) and β ∈ (0, 1). We now generalize the classical notions of semi-jets (see Crandall, Ishii, and Lions [9]). For γ t ∈ C α with t ∈ [0, T ), (µ, κ) ∈ ( γ t 0 , ∞) × (0, T − t), and v ∈ C (Λ), define the super-jet of v at γ t sliced by the double index of Hölder modulus (µ, κ): and the sub-jet of v at γ t sliced by the double index of Hölder modulus (µ, κ): Remark 4.2. Both J + µ,κ (γ t , v) and J − µ,κ (γ t , v) may be empty. Our notion of viscosity solutions is defined as follows.    (2) In the classical uniqueness proof of viscosity solution to state-dependent PDEs in an unbounded domain (which is locally compact), a conventional technique is to construct an auxiliary smooth function decaying outside a compact domain. In our path-dependent case, we find it difficult to construct such a smooth functionals. For the sake of the uniqueness proof, our new notion of jets is enlarged to be defined only on C α µ , which is compact in Λ. However, at a cost, our modification leads to additional difficulty in the existence proof.
(3) Assume that all the coefficients of Bellman equation (4.3) and terminal condition (4.4) are state-dependent. Let state-dependent function u be a viscosity solution to (4.3) as a path-dependent functional. Then u is also a classical viscosity solution as a function of time and state.

Main results
Our main results on the existence and the uniqueness of the viscosity solution to the path-dependent Bellman equation  The uniqueness is given on both non-degenerate and degenerate cases. we first address the non-degenerate case.
We make the following assumption, extending our previous assumption (H1) to the larger path spaceΛ.
Moreover, σ satisfies the non-degenerate condition The uniqueness of viscosity solutions of (4.3) is an immediate consequence of the following representation theorem. In the degenerate case of σσ T ≥ 0, we have the following extra smooth conditions on the coefficients.
We have the following representation theorem. Remark 4.8. Analogous to the proof of Lemma 3.5 and Theorem 3.9, from the bounded and Lipschitz assumption on the coefficients in (H2) or (H3), the value functionalṽ can be shown to be bounded and to satisfy which impliesṽ ∈ C b (Λ) ∩ C u (Λ). Note that the non-degeneracy condition (4.12) is not needed here.
In view of the comparison theorem of BSDEs, we immediately have the following comparison theorem. Corollary 4.9. Suppose that either (H2) or (H3) holds. Let v 1 , v 2 ∈ C b (Λ) satisfy (4.13), and g 1 , g 2 ∈ C b (Λ T ) satisfy g 1 ≤ g 2 . Furthermore, let v 1 and v 2 be viscosity solutions to the path-dependent Bellman equation (4.3) with the terminal conditions: For an initial path γ t ∈ Λ, define We have Then we have the following two assertions: (ii) If the following holds for an admissible control u * ∈ U t : for every γ t ∈ Λ, then v(γ t ) = J(γ t ) for any γ t ∈ Λ.
Proof. For u ∈ U t , since v is a classical solution, we have (L u) (X γt,u s , u(s)) ≤ 0.

From [19, Proposition 2.2], we have
, u(r))dr ≥ 0, and the equality holds for u = u * . This proves Assertions (i) and (ii). The proof is complete.

Example 4.11.
In what follows, we show that the conventional non-Markovian optimal stochastic control problem is included as a particular case of our problem (3.6), (3.7) and (3.8). Under some suitable smooth conditions, the corresponding path-dependent Bellman equation is associated to a backward stochastic Bellman equation via Dupire's functional calculus.
Let {B t , 0 ≤ t ≤ T } be a d-dimensional Winner process on the probability space The non-Markovian stochastic optimal control problem is formulated as follows. For any u ∈ U and (t, x) ∈ [0, T ] × R n , consider the following forward and backward stochastic differential systems: The optimal value fieldv : This problem depends on the Brownian path B t and the state X(t). Now we translate this problem into the path-dependent case. For any (γ t , ξ t ) ∈ Λ(R d )×Λ(R n ) and u ∈ U, f (γ t , ξ t , y, z, u) :=f (γ t , ξ t (t), y, z, u), g(γ T , ξ T ) :=ḡ(γ T , ξ(T )).

Existence of viscosity solutions
In this section we give the solution of the path-dependent Bellman equation (4.3) with the help of FBSDEs (3.6) and (3.7). First, let us perturb a path γ t ∈ C α µ . For µ > 0, ε ∈ (0, µ) and γ t ∈ C α µ , define a perturbation of γ t in the following manner: We have there is a constant C, independent of µ and u ∈ U, such that for some p, p( 1 2 − α) > 1 and for all δ < T − t, Proof. Assertion (i) is obvious. Now we prove Assertion (ii).
The case of x 1 = 0. We have We assert that C 2 ε ≤ C 2 , which implies (5.2) immediately. It is obvious if by the definition (5.1), which together with equality (5.5) gives C 2 − C 2 ε ≥ 0. The case of x 1 > 0 and r 2 ≤ r 1 . We have Therefore, we have x 2 = 0 from the definition (5.1), and thus The case of x 1 > 0 and r 2 > r 1 . We have Then and thus (5.2) holds. The last inequality is deduced from the following fact: if 2α ∈ [0, 1], then a 2α + b 2α ≥ (a + b) 2α for all a > 0, b > 0.
It remains to show Assertion (iii). For any δ < T − t andγ t+δ ∈ Λ such that sup t≤s 1 <s 2 ≤t+δ in view of (5.1) and Assertion (ii), we have γ Assertion (iii) then follows from Proposition 7.1 in the Appendix.
In a symmetric (also easier) way, we show thatṽ is a super-solution to the pathdependent Bellman equation (4.3). The proof is complete.
Remark 5.2. (1) Our existence proof is more complicated than the classical counterpart (for the state-dependent case). The complication arises from the fact that we start the dynamic programming at the perturbation γ ε t instead of directly at the minimum path γ t of ψ −ṽ like the conventional arguments. Since our jets are defined on some compact subset C α µ of Λ, the minimum path γ t might happen to be at the boundary of C α µ , i.e. γ t α = µ. If we started at γ t , BSDE (5.11) would be trivial and nothing from the localized dynamic programming principle could be derived if P { X γt s α ≤ µ, ∃s > t} = 0. (5.24) The following example illustrates that (5.24) might happen, and therefore explains why we have to start the dynamic programming at the perturbation γ ε t . Let W be a one-dimensional standard Brownian Motion and γ t ∈ Λ(R) such that for some Then Since the function µ(| · −t 1 | α − |t − t 1 | α ) ∈ C 1 [t, t + δ], by the law of iterated logarithm (see [23,Theorem 9.23, Chapter 2]), we have P ∃δ > 0, s.t. W γt t+δ α ≤ µ = 0. This example enlightens us to perturb the left µ-Hölder modulus of γ t at time t in (5.1).
(2) The introduction of Q M 0 ,T −κ in Definition 4.3 plays a crucial role in the proof of Theorem 4.5. Otherwise, we only have the following too rough estimate on our perturbation: γ ε t − γ t 0 ≤ Cε, from which and (5.23) only results the following inequality It does not help us, for the relation of δ is required in the estimate (5.22) by Proposition 7.1 and implies that ε β δ −1 increases to ∞ as δ is decreasing to zero.
(3) In the above proof, both parameters µ and M 0 in our definition of viscosity sub-solutions play a key role, while the parameter κ is fixed such that the following associated family of path functionals , µ ≥ µ 0 } for some sufficiently large µ 0 , share a common Hölder modulus, which implies the socalled equi-continuous but with the underlying functionals being considered on varying domains.

Non-degenerate case
We assume without loss of generality that, there exists a constant K > 0, such that, for all (γ t , p, A, u) ∈ Λ × R n × R n×n × U and r 1 , r 2 ∈ R such that r 1 < r 2 , Then v is a viscosity solution of PHJB equation (4.3) if and only ifv is a viscosity solution of the following PPDE whereH (γ t , r, p, A, u) := −λr + e −λt H(γ t , e λt r, e λt p, e λt A, u).

Assumption (H2) implies the following estimates
Consider the following FBSDE: for any γ t ∈ Λ, t < T , and u ∈ U, Applying Itô formula, BDG and Gronwall inequality, using standard arguments, we have Obviously, (6.5) yields, for any γ t ∈ Λ and positive integer m, Similar to the state-dependent optimal stochastic control problem, v m has a PDE interpretation. For each m and i = 1, . . . , m, define functions B m,i : Obviously, for any γ ∈ Λ T , u ∈ U and t ∈ (t i−1 , Here f (t i +) is the right limit of the function f at time t i . Let V m,i , i = 1, · · · m be the unique viscosity solutions of second order parametrized nonlinear parabolic equations According to the the relationship between viscosity solution of Bellman equations and the optimal control problems, we have, for any , γ t andĤ m,i;ε satisfy the following structure conditions: . Consider the following system of fully nonlinear parameterized state-dependent PDE We have the following key lemma.
Lemma 6.2. Assume (H2). Then the system (6.7) has unique viscosity solutions Moreover, there is some positive constants C which are independent of m, i and ε 1 (ε 1 < m −1 ), such that: where K is the constant in (6.1).
(2) Hölder continuity: for any t ∈ [t i−1 , t i ), − → x i , − → y i ∈ R i×n , and s ∈ [t j−1 , t j ), i ≤ j ≤ m, (3) smoothly approximating rate: Proof. Firstly we prove the existence of viscosity solution. Define It is easy to verify that are respectively viscosity super-and sub-solutions of system (6. are viscosity super-and sub-solutions of system (6.11), respectively, we have (6.12). Then combining the interior C 2,α estimates and interpolation inequality, we have (6.13). Assertion (1) is proved.
In view of Assertion (1), we know |a| + |b| + |c| < C, and Similar to recursive method in Assertion (1), we have In view of the interior Schauder estimate for linear parabolic equation (see Lieberman [28,Theorem 4.9]), we have Incorporating the following interior C 2,α estimate of V m,i;ε 1 ε : we have (6.14).
Both (6.1) and (6.25) imply Note that the constants C in this proof all do not depend on m, ε 1 , µ and λ. Setting m → ∞, ε 1 → 0, and then considering the upper-limit as µ → ∞ on both sides of (6.31), we have from Definition (4.10) the following inequality: 0 ≥ −e −λ (ε + C) + 1 2 Cr 0 , which is a contradiction when λ tends to ∞. The proof is complete.
Remark 6.5. Assertions of Theorems 4.6 and 4.7 are still true if the coefficients in Assumptions (H2) and (H3) are relaxed to grow in a linear way.

Degenerate case
In this subsection, we prove Theorem 4.7 using the vanishing viscosity method (see [30]).
Proof of Theorem 4.7. Similarly as in the non-degenerate case, we assume that H strictly decreases in y ∈ R without loss of generality, i.e., (6.1) holds. We only proveṽ ≥ v, and the reverse inequality can be proved in a symmetric (also easier) way.
Since the constants C and C(θ) do not depend on µ, m, ε 1 and λ, first setting m → ∞, ε 1 → 0 and then considering the upper-limit as µ → ∞ on both sides of (6.42), we have from the definition (4.9) the following inequality which yields a contradiction when sending λ to ∞ and θ to 0. The proof is complete.

Appendix
In this Appendix, we prove the α-Hölder continuity of the path for the following SDE: X(t) =ˆt 0 b(X r )dr +ˆt 0 σ(X r )dW (r), (7.1) where b : Λ → R n and σ : Λ → R n×d are uniformly Lipschitz continuous.
In conclusion, we have   Thus, applying Lemma 7.2, This completes the proof.