Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I
aa r X i v : . [ m a t h . P R ] S e p Viscosity Solutions of Fully Nonlinear ParabolicPath Dependent PDEs: Part I
Ibrahim
Ekren ∗ Nizar
Touzi † Jianfeng
Zhang ‡ September 15, 2014
Abstract
The main objective of this paper and the accompanying one [12] is to provide anotion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Ourdefinition extends our previous work [10], focused on the semilinear case, and is cruciallybased on the nonlinear optimal stopping problem analyzed in [11]. We prove that ournotion of viscosity solutions is consistent with the corresponding notion of classicalsolutions, and satisfies a stability property and a partial comparison result. The latteris a key step for the wellposedness results established in [12]. We also show that thevalue processes of path-dependent stochastic control problems are viscosity solutions ofthe corresponding path-dependent dynamic programming equations.
Key words:
Path dependent PDEs, second order Backward SDEs, nonlinear expectation,viscosity solutions, comparison principle.
AMS 2000 subject classifications: ∗ University of Southern California, Department of Mathematics, [email protected]. † CMAP, Ecole Polytechnique Paris, [email protected]. Research supported by the Chair
Financial Risks of the
Risk Foundation sponsored by Soci´et´e G´en´erale, and the Chair
Finance and SustainableDevelopment sponsored by EDF and Calyon. ‡ University of Southern California, Department of Mathematics, [email protected]. Research supportedin part by NSF grant DMS 10-08873. Introduction
The objective of this paper is to introduce a notion of viscosity solution of the followingfully nonlinear path-dependent partial differential equation: − ∂ t u ( t, ω ) − G (cid:0) t, ω, u ( t, ω ) , ∂ ω u ( t, ω ) , ∂ ωω u ( t, ω ) (cid:1) = 0 , ≤ t < T, ω ∈ Ω , (1.1)where the unknown u is a progressively measurable process on the canonical space Ω := { ω ∈ C ([0 , T ] , R d ) : ω = } , and the nonlinearity G : [0 , T ] × Ω × R × R d × S d → R isprogressively measurable, satisfies convenient Lipschitz and continuity assumptions, and isdegenerate elliptic.The above equation attracted our attention after the point raised by Peng in [26] thatthis would be an alternative approach to the theory of backward stochastic differentialequations, introduced by the seminal paper of Pardoux and Peng [22].The semilinear case, corresponding to the case where G is linear in the ∂ ωω u − variable,was addressed in [10], where existence and uniqueness results are established for a newnotion of viscosity solution. The main difficulty is related to the fact that the canonicalspace fails to be locally compact, so that many tools from the standard theory of viscositysolutions do not apply to the present context. The main contribution of [10] is to replace thepointwise extremality in the standard definition of viscosity solutions by the correspondingextremality in the context of an optimal stopping problem under a nonlinear expectation E . More precisely, we introduce a set of smooth test processes ϕ which are tangent fromabove or from below to the processes of interest u in the sense of the following nonlinearoptimal stopping problemssup τ E [( ϕ − u ) τ ] , inf τ E [( ϕ − u ) τ ] , where E := sup P ∈P E P , E := inf P ∈P E P ,τ ranges over a convenient set of stopping times, and P is a weakly compact collectionof probability measures, motivated by a convenient linearization of the nonlinearity F .Consequently, in the particular semilinear case of [10], the family P consists of equivalentprobability measures.In this paper, together with the accompanying ones [11, 12], we extend the notionof viscosity solutions to the fully nonlinear case. As in [10], in order to avoid the localcompactness issue of canonical space, we shall still use the optimal stopping problem todefine viscosity solutions. However, in this fully nonlinear context, the family P of prob-ability measures consists of nondominated mutually singular measures, so as to cover allthe measures induced by certain linearization of the generator G . The analysis under the2orresponding nonlinear expectation E P is the major difficulty, mainly due to the failure ofthe dominated convergence theorem under E P . To overcome this difficulty, one needs somestrong regularity of the involved processes which requires rather sophisticated estimates.The corresponding optimal stopping problem is solved in [11], and the major result, thecomparison principle of viscosity solutions, will be proved in [12].In this paper we focus on the definition of viscosity solutions and its basic properties.We first prove that our definition of viscosity solutions is consistent with the correspondingnotion of classical solutions. Next we show that our viscosity solution satisfies a stabilityproperty similar to the finite-dimensional context. Finally, we establish the partial com-parison result, namely for any pair of viscosity subsolution u and supersolution u with u T ≤ u T on Ω, we have u ≤ u on [0 , T ] × Ω whenever either one of them is smooth. Thisresult is crucial for the well-posedness results established in our accompanying paper [12].We remark that Peng [27] also investigated the comparison principle for fully nonlinearPPDEs by using a different approach.We will investigate the connection between our viscosity solution and some other equa-tions in the literature, which will be very helpful for the applications of our results. Inparticular, we show that the value function of path-dependent stochastic control problemsas well as second order backward stochastic differential equations [4, 32] are naturally viscos-ity solutions of the corresponding path-dependent partial differential equation. This extendsthe context of backward stochastic differential equations of [10]. See also the closely relatedworks [25, 17] in terms of the G -expectation. Our PPDE can go beyond stochastic control,see Pham-Zhang [30] for an application in stochastic differential games. Moreover, back-ward stochastic partial differential equations, which can be viewed as the value function ofstochastic control with random coefficients (in contrast with path dependent coefficients)can also be viewed as a PPDE. See also [19] and [21] for applications of BSPDEs.While the wellposedness of semilinear PPDEs has been achieved in [10], the approachthere for the comparison principle does not seem to work in fully nonlinear case. We shallrevisit the semilinear case by providing a new approach which, modulus all the technicality,will be extended to the fully nonlinear case in [12]. Moreover, our context covers first orderpath-dependent PDEs, which has been studied by Lukoyanov [18] by using compactnessarguments.The rest of the paper is organized as follows. In Section 2, we introduce the generalframework, and define a notion of classical differentiability which is weaker than that of[9]. In Section 3, we introduce our notion of viscosity solution of fully nonlinear PPDE,and provide various remarks which highlight the analogy with the properties of viscosity3olutions in finite dimensional spaces. We prove the consistency with the notion of classicalsolution. In Section 4, we provide several examples and show that natural ones as thevalue function of path dependent stochastic control problems, or solutions of second orderbackward stochastic differential equations, are viscosity solutions of the corresponding path-dependent PDEs. Section 5 contains our stability and partial comparison results. Section 6shows that our framework includes backward stochastic PDE by a convenient augmentationof the canonical space. Section 7 revisits the semilinear case and provides an alternative andsimpler well-posedness argument to that of our previous paper [10] which will be extended tothe fully nonlinear case in our accompanying paper [12]. Finally, in Section 8 we investigatethe first order PPDEs . Let Ω := (cid:8) ω ∈ C ([0 , T ] , R d ) : ω = (cid:9) , the set of continuous paths starting from the origin, B the canonical process, F the natural filtration generated by B , P the Wiener measure,and Λ := [0 , T ] × Ω. Here and in the sequel, for notational simplicity, we use to denotevectors, matrices, or paths with appropriate dimensions whose components are all equal to0. Let S d denote the set of d × d symmetric matrices, and x · x ′ := P di =1 x i x ′ i for any x, x ′ ∈ R d , γ : γ ′ := tr [ γγ ′ ] for any γ, γ ′ ∈ S d . We define a seminorm on Ω and a pseudometric on Λ as follows: for any ( t, ω ) , ( t ′ , ω ′ ) ∈ Λ, k ω k t := sup ≤ s ≤ t | ω s | , d ∞ (cid:0) ( t, ω ) , ( t ′ , ω ′ ) (cid:1) := | t − t ′ | + (cid:13)(cid:13) ω . ∧ t − ω ′ . ∧ t ′ (cid:13)(cid:13) T . (2.1)Then (Ω , k · k T ) is a Banach space and (Λ , d ∞ ) is a complete pseudometric space. Infact, the subspace { ( t, ω ·∧ t ) : ( t, ω ) ∈ Λ } is a complete metric space under d ∞ . We shalldenote by L ( F T ) and L (Λ) the collection of all F T -measurable random variables and F -progressively measurable processes, respectively. Let C (Λ) (resp. U C (Λ)) be the subset of L (Λ) whose elements are continuous (resp. uniformly continuous) in ( t, ω ) under d ∞ , and C b (Λ) (resp. U C b (Λ)) be the subset of C (Λ) (resp U C (Λ)) whose elements are bounded.Finally, L (Λ , R d ) denote the space of R d -valued processes with entries in L (Λ), and wedefine similar notations for the spaces C , C b , U C , and
U C b .We next introduce the shifted spaces. Let 0 ≤ t ≤ s ≤ T .4 Let Ω t := (cid:8) ω ∈ C ([ t, T ] , R d ) : ω t = (cid:9) be the shifted canonical space; B t the shiftedcanonical process on Ω t ; F t the shifted filtration generated by B t , P t the Wiener measureon Ω t , and Λ t := [ t, T ] × Ω t .- Define k · k ts on Ω t and d t ∞ on Λ t in the spirit of (2.1), and the sets L (Λ t ) etc. in anobvious way.- For ω ∈ Ω t and ω ′ ∈ Ω s , define the concatenation path ω ⊗ s ω ′ ∈ Ω t by:( ω ⊗ s ω ′ )( r ) := ω r [ t,s ) ( r ) + ( ω s + ω ′ r ) [ s,T ] ( r ) , for all r ∈ [ t, T ] . - Let ξ ∈ L ( F tT ), and X ∈ L (Λ t ). For ( s, ω ) ∈ Λ t , define ξ s,ω ∈ L ( F sT ) and X s,ω ∈ L (Λ s ) by: ξ s,ω ( ω ′ ) := ξ ( ω ⊗ s ω ′ ) , X s,ω ( ω ′ ) := X ( ω ⊗ s ω ′ ) , for all ω ′ ∈ Ω s . It is clear that, for any ( t, ω ) ∈ Λ and any u ∈ C (Λ), we have u t,ω ∈ C (Λ t ). The otherspaces introduced before enjoy the same property.We shall use the following type of regularity, which is slightly stronger than the rightcontinuity of a process u in standard sense (that is, for any fixed ω , the mapping t u ( t, ω )is right continuous). Definition 2.1
We say a process u ∈ L (Λ) is right continuous in ( t, ω ) under d ∞ if: forany ( t, ω ) ∈ Λ and any ε > , there exists δ > such that, for any ( t ′ , ω ′ ) ∈ Λ t satisfying d t ∞ (( t ′ , ω ′ ) , ( t, )) ≤ δ , we have | u t,ω ( t ′ , ω ′ ) − u ( t, ω ) | ≤ ε . Definition 2.2 By U , we denote the collection of all processes u ∈ L (Λ) such that- u is bounded from above and right continuous in ( t, ω ) under d ∞ ;- there exists a modulus of continuity function ρ such that for any ( t, ω ) , ( t ′ , ω ′ ) ∈ Λ : u ( t, ω ) − u ( t ′ , ω ′ ) ≤ ρ (cid:16) d ∞ (cid:0) ( t, ω ) , ( t ′ , ω ′ ) (cid:1)(cid:17) whenever t ≤ t ′ . (2.2) By U we denote the set of all processes u such that − u ∈ U . Remark 2.3 (i) The progressive measurability of u implies that u ( t, ω ) = u ( t, ω ·∧ t ), andit is clear that U ∩ U = U C b (Λ). We also recall from [11] Remark 3.2 that Condition (2.2)implies that u has left-limits and u t − ≤ u t for all t ∈ (0 , T ]. Moreover, under (2.2), u isright continuous in ( t, ω ) under d ∞ if and only if it is right continuous in t for every ω .(ii) In finite dimensional case, a continuous function is at least locally uniformly continuous.This is not true anymore in the infinite dimensional case, so it is important to distinguish C (Λ) and U C (Λ) in this paper. 5inally, we denote by T the set of F -stopping times, and H ⊂ T the subset of thosehitting times h of the form h := inf { t ≥ B t ∈ O c } ∧ t , (2.3)for some 0 < t ≤ T , and some open and convex set O ⊂ R d containing with O c := R d \ O .The set H will be important for our optimal stopping problem, which is crucial for thecomparison and the stability results, see Remark 2.7. We note that h = t when O = R d ,and for any h ∈ H ,0 < h ε ≤ h for ε small enough, where h ε := inf { t ≥ | B t | = ε } ∧ ε. (2.4)Moreover, h : Ω → [0 , T ] is lower semicontinuous, and h ∧ h ∈ H for any h , h ∈ H . Define T t and H t in the same spirit. For any τ ∈ T (resp. h ∈ H ) and any ( t, ω ) ∈ Λ suchthat t < τ ( ω ) (resp. t < h ( ω )), it is clear that τ t,ω ∈ T t (resp. h t,ω ∈ H t ). For every constant
L >
0, we denote by P L the collection of all continuous semimartingalemeasures P on Ω whose drift and diffusion characteristics are bounded by L and √ L ,respectively. To be precise, let ˜Ω := Ω be an enlarged canonical space, ˜ B := ( B, A, M )be the canonical processes, and ˜ ω = ( ω, a, m ) ∈ ˜Ω be the paths. P ∈ P L means that thereexists an extension Q of P on ˜Ω such that: B = A + M, A is absolutely continuous, M is a martingale , | α P | ≤ L, tr (( β P ) ) ≤ L, where α P t := dA t dt , β P t := q d h M i t dt , Q -a.s. (2.5)Similarly, for any t ∈ [0 , T ), we may define P tL on Ω t .As in Denis, Hu and Peng [8], the set P tL induces the following capacity: C Lt [ A ] := sup P ∈P tL P [ A ] , for all A ∈ F tT . (2.6)We denote by L ( F tT , P tL ) the set of ξ ∈ L ( F tT ) satisfying sup P ∈P tL E P [ | ξ | ] < ∞ . Thefollowing nonlinear expectation will play a crucial role: E Lt [ ξ ] := sup P ∈P tL E P [ ξ ] and E Lt [ ξ ] := inf P ∈P tL E P [ ξ ] = −E Lt [ − ξ ] for all ξ ∈ L ( F tT , P tL ) . (2.7)6e remark that E L [ ξ ] can be viewed as the solution of a Second Order BSDE (2BSDE, forshort) in the sense of [32], or a conditional G -expectation in the sense of [25]. See Section4 for more details. The following result will be important for us. Lemma 2.4
For any h ∈ H and any L > , we have E L [ h ] > . Proof
By (2.4), we may assume h ε ≤ h for some ε >
0. For any P ∈ P L and 0 < δ ≤ ε ,we have P ( h ≤ δ ) ≤ P ( h ε ≤ δ ) = P ( k B k δ ≥ ε ) ≤ ε − E P [ k B k δ ] ≤ CL ε − δ . (2.8)This implies that, for δ := ε √ CL ∧ ε , E P [ h ] ≥ δ P ( h > δ ) = δ (cid:16) − P ( h ≤ δ ) (cid:17) ≥ δ (cid:16) − CL ε − δ ) ≥ δ . Thus E L [ h ] ≥ δ > Definition 2.5
Let X ∈ L (Λ) satisfy X t ∈ L ( F t , P L ) for all ≤ t ≤ T . We say that X is an E L − supermartingale (resp. submartingale, martingale) if, for any ( t, ω ) ∈ Λ and any τ ∈ T t , E Lt [ X t,ωτ ] ≤ (resp. ≥ , = ) X t ( ω ) . We now state an important result for our subsequent analysis. Given a bounded process X ∈ L (Λ), consider the nonlinear optimal stopping problem S Lt [ X ]( ω ) := sup τ ∈T t E Lt (cid:2) X t,ωτ (cid:3) and S Lt [ X ]( ω ) := inf τ ∈T t E Lt (cid:2) X t,ωτ (cid:3) , ( t, ω ) ∈ Λ . (2.9)By definition, we have S L [ X ] ≥ X and S LT [ X ] = X T . The following nonlinear Snell envelopecharacterization is proved in [11]. Theorem 2.6
Let X ∈ U be bounded, h ∈ H , and set b X t := X t { t< h } + X h − { t ≥ h } . Define Y := S L (cid:2) b X (cid:3) and τ ∗ := inf { t ≥ Y t = b X t } . Then Y τ ∗ = b X τ ∗ , Y is an E L -supermartingale on [0 , h ] , and an E L -martingale on [0 , τ ∗ ] .Consequently, τ ∗ is an optimal stopping time. Remark 2.7 (i) We emphasize that the maturity of the above nonlinear optimal stoppingproblem is restricted to be a hitting time in H . This requirement is due to technical aspectsin the proof of Theorem 2.6 reported in [11]. The difficulty is related to the regularity ofthe Snell envelope Y and to some limiting arguments under nonlinear expectation.(ii) Y is continuous in [0 , h ) and has a left limit at h . However, in general Y may havea negative jump at h . 7 .3 The derivatives We define the path derivatives via the functional Itˆo formula, which is initiated by Dupire[9] and plays an important role in our paper. Denote P t ∞ := [ L> P tL , t ∈ [0 , T ] . Definition 2.8
We say u ∈ C , (Λ) if u ∈ C (Λ) and there exist ∂ t u ∈ C (Λ) , ∂ ω u ∈ C (Λ , R d ) , ∂ ωω u ∈ C (Λ , S d ) such that, for any P ∈ P ∞ , u is a P -semimartingale satisfying: du = ∂ t udt + ∂ ω u · dB t + 12 ∂ ωω u : d h B i t , ≤ t ≤ T, P -a.s. (2.10)We remark that the above ∂ t u , ∂ ω u and ∂ ωω u , if they exist, are unique. Indeed, firstconsidering P , , the probability measure corresponding to α = , β = in (2.5), togetherwith the required regularity ∂ t u ∈ C (Λ) we obtain ∂ t u ( t, ω ) = lim h ↓ h [ u (cid:0) t + h, ω ·∧ t (cid:1) − u (cid:0) t, ω (cid:1) ] . (2.11)Next, considering P such that α P = 1, β P = 0 we obtain the uniqueness of ∂ ω u . Finally,considering P = P we see that ∂ ωω u is also unique. Consequently, we call them the timederivative, first order and second order space derivatives of u , respectively. We define C , (Λ t ) similarly. It is clear that, for any ( t, ω ) and u ∈ C , (Λ), we have u t,ω ∈ C , (Λ t ),and ∂ t ( u t,ω ) = ( ∂ t u ) t,ω , ∂ ω ( u t,ω ) = ( ∂ ω u ) t,ω , ∂ ωω ( u t,ω ) = ( ∂ ωω u ) t,ω . Remark 2.9 (i) In Markovian case, namely u ( t, ω ) = v ( t, ω t ), if v ∈ C , ([0 , T ] × R d ), thenby the standard Itˆo formula we see immediately that u ∈ C , (Λ) with ∂ t u ( t, ω ) = ∂ t v ( t, ω t ) , ∂ ω u ( t, ω ) = Dv ( t, ω t ) , ∂ ωω u ( t, ω ) = D v ( t, ω t ) . Here D and D denote the standard gradient and hessian of v with respect to x .(ii) The typical case that the path derivatives exist is that u is smooth in Dupire’s sense [9](more precisely, the space C , b in Cont and Fournie [5]), and in that case our time derivativeand space derivatives agree with the horizontal and vertical derivatives introduced therein,respectively, due to their functional Itˆo formula. Therefore, any smooth function in the senseof Duprie’s calculus is also smooth in the sense of Definition 2.8, namely our space C , (Λ)is a priori larger than the space C , b in [5]. In particular, Definition 2.8 is different fromthe corresponding definition in our previous paper [10], which uses Dupire’s derivatives.(iii) The main advantage of our definition is that all derivatives are defined within thecontinuous path space Ω. In Dupire [9], one has to extend the process u (and the generator8s well as the terminal condition of our PPDE later) to a larger domain [0 , T ] × D ([0 , T ]),where D is the set of c`adl`ag paths. This is not natural in many situations, and is notnecessary for our purpose, as it turns out that what we need is exactly the functional Itˆoformula, rather than the precise form of the derivatives.(iv) Moreover, compared to [9], our definition does not require all the derivatives to bebounded, and we do not need (2.10) to hold true for all semimartinagle measures P . How-ever, under our definition we do not require ∂ ωω u = ∂ ω ( ∂ ω u ). When ∂ ω u is indeed differ-entiable, typically we should have ∂ ωω u = (cid:2) ∂ ω ( ∂ ω u ) + [ ∂ ω ( ∂ ω u )] T (cid:3) . For the last point seemore details in [2].(v) As explained in Cont and Fournie [5], when u is smooth enough in both senses, it holdsthat ∂ ω u ( t, ω ) = D t u ( t, ω ), where D t denotes the Malliavin derivative. We emphasize that,unlike the Malliavian derivative D t ξ which involves the perturbation of ξ over the wholepath ω , ∂ ω u ( t, ω ) involves the perturbation of u only at the current time t . In particular, ∂ ω u is F -adapted. Remark 2.10
We shall also remark that, in our proof of comparison principle for PPDEsin our accompanying paper [12], we actually uses only piecewise Markovian test functionsand thus the standard Itˆo formula is sufficient. So technically speaking, we can prove boththe existence and uniqueness of viscosity solutions without using the path derivatives andthe functional Itˆo formula. However, it is more natural to consider truly path dependenttest functions in this framework. In particular, it is more natural to define classical solutionsfor PPDEs by using path derivatives.
Example 2.11
Let d = 1. As highlighted by Cont and Fournie [5], a simple example ofnon-differentiable process is the running maximum process: u ( t, ω ) := ω t := max ≤ s ≤ t ω s ,( t, ω ) ∈ Λ. Indeed, if it is differentiable, by (2.11) it is obvious that ∂ t u ( t, ω ) = 0 for all( t, ω ) ∈ Λ. Then by (2.10) one must have ∂ ω u = 0, and ∂ ωω udt = dB t , which is impossibleunder P . In terms of the Dupire’s vertical derivatives, we have ∂ ω u ( t, ω ) = 0 whenever ω t < ω t , and ∂ + ω u ( t, ω ) = 1 and ∂ − ω u ( t, ω ) = 0 whenever ω t = ω t , where ∂ + ω and ∂ − ω denote the right and left space derivatives in the sense of Dupire. Hencethe process u is not differentiable on { ω t = ω t } .9 Fully nonlinear path dependent PDEs
In this paper we study the following fully nonlinear parabolic path-dependent partial dif-ferential equation (PPDE, for short): L u ( t, ω ) := {− ∂ t u − G ( ., u, ∂ ω u, ∂ ωω u ) } ( t, ω ) = 0 , ≤ t < T, ω ∈ Ω , (3.1)where the generator G : Λ × R × R d × S d → R satisfies the following standing assumptions: Assumption 3.1
The nonlinearity G satisfies: (i) For fixed ( y, z, γ ) , G ( · , y, z, γ ) ∈ L (Λ) and | G ( · , , , ) | ≤ C . (ii) G is elliptic, i.e. nondecreasing in γ . (iii) G is uniformly Lipschitz continuous in ( y, z, γ ) , with a Lipschitz constant L . (iv) For any ( y, z, γ ) , G ( · , y, z, γ ) is right continuous in ( t, ω ) under d ∞ , in the sense ofDefinition 2.1. Remark 3.2
In the Markovian case, namely G ( t, ω, . ) = g ( t, ω t , . ) and u ( t, ω ) = v ( t, ω t ),the PPDE (3.1) reduces to the following PDE: recalling Remark 2.9 (i), L v ( t, x ) := {− ∂ t v − g ( ., v, Dv, D v ) } ( t, x ) = 0 , t ∈ [0 , T ) , x ∈ R d . (3.2)Namely, u is a solution (classical or viscosity as we will introduce soon) of PPDE (3.1)corresponds to that v is a solution of PDE (3.2). However, slightly different from the PDEliterature but consistent with (2.11), here we should interpret ∂ t v as the right derivative ofthe function v in t . Definition 3.3
Let u ∈ C , (Λ) . We say u is a classical solution (resp. sub-solution,super-solution) of PPDE (3.1) if L u ( t, ω ) = ( resp. ≤ , ≥ ) 0 for all ( t, ω ) ∈ [0 , T ) × Ω . It is clear that, in the Markovian setting as in Remark 3.2 with smooth v , u is a classicalsolution (resp. sub-solution, super-solution) of PPDE (3.1) if and only if v is a classicalsolution (resp. sub-solution, super-solution) of PDE (3.2). Example 3.4
Let d = 1 and u ( t, ω ) := E P t (cid:2) R T B t dt (cid:3) ( ω ) = R t ω s ds + ( T − t ) ω t , ( t, ω ) ∈ Λ.Then u ∈ C , (Λ), and is a classical solution of the path dependent heat equation − ∂ t u − ∂ ωω u = 0 (3.3)with terminal condition u ( T, ω ) = R T ω t dt . 10 xample 3.5 Let d = 1 and u ( t, ω ) := E P t h ( ω ⊗ t B t ) T i , ( t, ω ) ∈ Λ with the notation ofExample 2.11. Then one can easily check that u ( t, ω ) = v ( t, ω t , ω t ), where v is a determin-istic function defined by: v ( t, x, y ) := E P t h y ∨ ( x + ( B t ) T ) i = x + √ T − tψ ( y − x √ T − t ) , x ≤ yψ ( z ) := E P h z ∨ B i = E P h z ∨ | B | i = z [2Φ( z ) −
1] + √ π e − z / , z ≥ , (3.4)and Φ denotes the cdf of the standard normal distribution. We note that v is smooth for t < T , and D y v ( t, x, x ) = 0. Since the support of dB t is in { B t = B t } , it follows that D y v ( t, B t , B t ) dB t = 0. This implies that du ( t, ω ) = dv ( t, ω t , ω t ) = ∂ t vdt + D x vdB t + 12 D xx vd h B i t . By (2.11) it is clear that ∂ t u ( t, ω ) = ∂ t v ( t, ω t , ω t ). Then by (2.10) we see that ∂ ω u ( t, ω ) = D x v ( t, ω t , ω t ) and ∂ ωω u ( t, ω ) = D xx v ( t, ω t , ω t ). Thus u ∈ C , (Λ).Finally, it is straightforward to check that u is a classical solution to the path dependentheat equation (3.3) with terminal condition u ( T, ω ) = B T . Remark 3.6
We shall remark that, unlike a standard heat equation which always hasclassical solution in [0 , T ), a path dependent one may not have a classical solution in [0 , T ).One simple example is the equation (3.3) with terminal condition u ( T, ω ) = B t ( ω ) for some0 < t < T . Then clearly u ( t, ω ) = B t ∧ t ( ω ), and thus ∂ ω u ( t, ω ) = [0 ,t ] ( t ) is discontinuous.Following Proposition 4.4 below and our accompanying paper [12] (or Section 7 belowunder a slight reformulation), and weakening the boundedness assumption as pointed outin Remark 3.8 below, one can easily see u is the unique viscosity solution of the equation (3.3)with terminal condition u ( T, ω ) = B t ( ω ). We refer to Peng and Wang [28] for sufficientconditions of existence of classical solutions for more general semilinear PPDEs. We next introduce our notion of viscosity solutions. Recall the nonlinear Snell envelopenotation (2.9). For any u ∈ L (Λ), ( t, ω ) ∈ [0 , T ) × Ω, and
L >
0, define A L u ( t, ω ) := n ϕ ∈ C , (Λ t ) : ( ϕ − u t,ω ) t = 0 = S Lt (cid:2) ( ϕ − u t,ω ) ·∧ h (cid:3) for some h ∈ H t o , A L u ( t, ω ) := n ϕ ∈ C , (Λ t ) : ( ϕ − u t,ω ) t = 0 = S Lt (cid:2) ( ϕ − u t,ω ) ·∧ h (cid:3) for some h ∈ H t o . (3.5)11 efinition 3.7 (i) Let
L > . We say u ∈ U (resp. U ) is a viscosity L -subsolution (resp. L -supersolution) of PPDE (3.1) if, for any ( t, ω ) ∈ [0 , T ) × Ω and any ϕ ∈ A L u ( t, ω ) (resp. ϕ ∈ A L u ( t, ω ) ): L t,ω ϕ ( t, ) := (cid:8) − ∂ t ϕ − G t,ω ( ., ϕ, ∂ ω ϕ, ∂ ωω ϕ ) (cid:9) ( t, ) ≤ ( resp. ≥ ) 0 . (ii) u ∈ U (resp. U ) is a viscosity subsolution (resp. supersolution) of PPDE (3.1) if u isviscosity L -subsolution (resp. L -supersolution) of PPDE (3.1) for some L > . (iii) u ∈ U C b (Λ) is viscosity solution of PPDE (3.1) if it is viscosity sub- and supersolution. Remark 3.8
For technical simplification, in this paper and the accompanying one [12], weconsider only bounded viscosity solutions. By some more involved estimates one can extendour theory to viscosity solutions satisfying certain growth conditions. We shall leave thisfor future research, however, in some examples below we may consider unbounded viscositysolutions as well.
Remark 3.9
Since our PPDE is backward, in (3.5) the test functions ϕ are defined onlyafter t . By this nature, both the viscosity solution u and the generator G are required onlyto be right continuous in ( t, ω ) under d ∞ . To prove the comparison principle, however, wewill assume some stronger regularity of G , see our accompanying paper [12].We next provide an intuitive justification of our Definition 3.7 which shows how the abovenonlinear optimal stopping problems S and S appear naturally.Let u ∈ C , (Λ) be a classical supersolution of PPDE (3.1), ( t ∗ , ω ∗ ) ∈ [0 , T ) × Ω, and ϕ ∈ C , (Λ t ∗ ). Then: 0 ≤ L u ( t ∗ , ω ∗ ) = L t ∗ ,ω ∗ ϕ ( t ∗ , ) + R ( t ∗ , ) (3.6)where R ( t, ω ) = ∂ t ( ϕ − u t ∗ ,ω ∗ )( t, ω ) + ˆ α · ∂ ω ( ϕ − u t ∗ ,ω ∗ )( t, ω ) + ˆ β : ∂ ωω ( ϕ − u t ∗ ,ω ∗ )( t, ω )for ( t, ω ) ∈ Λ t ∗ , ˆ α := G z ( t ∗ , ω ∗ , u ( t ∗ , ω ∗ ) , ˆ z, ˆ γ ) , and ˆ β := (cid:0) G γ ( t ∗ , ω ∗ , u ( t ∗ , ω ∗ ) , ˆ z, ˆ γ ) (cid:1) / are constant drift and diffusion coefficients, and (ˆ z, ˆ γ ) are some convex combination of( ∂ ω u, ∂ ωω u )( t ∗ , ω ∗ ) and ( ∂ ω ϕ, ∂ ωω ϕ )( t ∗ , ).The question is how to choose the test process ϕ so as to deduce from (3.6) that L t ∗ ,ω ∗ ϕ ( t ∗ , ≥
0. A natural sufficient condition is R ( t ∗ , ) ≥
0. To achieve that, ourcrucial observation is that d ( ϕ − u t ∗ ,ω ∗ )( t, ω ) = R ( t, ω ) dt + ∂ ω ( ϕ − u t ∗ ,ω ∗ )( t, ω ) · ˆ βd ˆ W t , ˆ P − a.s.12here ˆ W is a Brownian motion under the probability measure ˆ P ∈ P t ∗ L defined by thepair ( ˆ α, ˆ β ), and L is the Lipschitz constant of the nonlinearity G . Therefore, in order toensure R ( t ∗ , ) ≤
0, we have to choose the test process ϕ so that the difference ( ϕ − u t ∗ ,ω ∗ )has a nonpositive ˆ P − drift locally at the right hand-side of t ∗ . This essentially means that( ϕ − u t ∗ ,ω ∗ ) is a ˆ P − supermartingale on some right-neighborhood [ t ∗ , h ] of t ∗ , and therefore( ϕ − u t ∗ ,ω ∗ ) t ∗ ≥ E ˆ P [( ϕ − u t ∗ ,ω ∗ ) τ ∧ h ] for any stopping time τ . Since the probability measureˆ P is imposed by the above calculation, we must choose the test process ϕ so that ( ϕ − u t ∗ ,ω ∗ ) t ∗ ≥ E Lt ∗ (cid:2) ( ϕ − u t ∗ ,ω ∗ ) τ ∧ h (cid:3) for all stopping time τ . Finally, since τ = t ∗ is a legitimatestopping rule, we arrive at( ϕ − u t ∗ ,ω ∗ ) t ∗ = S Lt ∗ (cid:2) ( ϕ − u t ∗ ,ω ∗ ) τ ∧ h (cid:3) , which corresponds exactly to our definition of A L u ( t ∗ , ω ∗ ).Conversely, if the pair (cid:0) ( t ∗ , ω ∗ ) , ϕ (cid:1) satisfies the last equality, then it follows from theSnell envelope characterization of Theorem 2.6 that ( ϕ − u t ∗ ,ω ∗ ) t ∗ ≥ E Lt ∗ (cid:2) S Lτ ∧ h [ ϕ − u t ∗ ,ω ∗ ] (cid:3) ≥E Lt ∗ (cid:2) ( ϕ − u t ∗ ,ω ∗ ) τ ∧ h (cid:3) , for all stopping time τ . By the right-continuity, this implies that R ( t ∗ , ) ≤
0. Hence our definition of the set of test processes A L u ( t ∗ , ω ∗ ) is essentiallynecessary and sufficient for the inequality R ( t ∗ , ) ≤ Remark 3.10
From the last intuitive justification of our definition, we see that for a semi-linear path-dependent PDE, ˆ β is a constant matrix. Then, in agreement with our previouspaper [10], also see Section 7 below, it is not necessary to vary the coefficient β in thedefinition of the operator E L .Similarly, in the context of a linear PPDE, both coefficients ˆ α and ˆ β are constant, andwe may define the sets A L u and A L u by means of the linear expectation operator. Finally,for a first order PPDE, we may take the diffusion coefficient β ≡
0, see Section 8.In the rest of this section we provide several remarks concerning our definition of viscositysolutions. In most places we will comment on the viscosity subsolution only, but obviouslysimilar properties hold for the viscosity supersolution as well.
Remark 3.11
As standard in the literature on viscosity solutions of PDEs:(i) The viscosity property is a local property in the following sense. For any ( t, ω ) ∈ [0 , T ) × Ωand any ε >
0, define as in (2.4), h tε := inf n s > t : | B ts | ≥ ε o ∧ ( t + ε ) and thus h ε = h ε . (3.7)13t is clear that h tε ∈ H t . To check the viscosity property of u at ( t, ω ), it suffices to knowthe value of u t,ω on [ t, h ε ] for an arbitrarily small ε >
0. In particular, since u and ϕ arelocally bounded, there is no integrability issue in (3.5). Moreover, for any ϕ ∈ A L u ( t, ω )with corresponding h ∈ H t , by (2.4) we have h tε ≤ h when ε is small enough.(ii) The fact that u is a viscosity solution does not mean that the PPDE must hold withequality at some ( t, ω ) and ϕ in some appropriate set. One has to check viscosity subsolutionproperty and viscosity supersolution property separately.(iii) In general A L u ( t, ω ) could be empty. In this case automatically u satisfies the viscositysubsolution property at ( t, ω ). Remark 3.12 (i) Consider the Markovian setting in Remark 3.2. One can easily checkthat u is a viscosity subsolution of PPDE (3.1) in the sense of Definition 3.7 implies that v is a viscosity subsolution of PDE (3.2) in the standard sense, see e.g. [6] or [14]. However,the opposite direction is in general not true. We shall point out though, when the PDEis wellposed, by uniqueness our definition of viscosity solution of PPDE (3.1) is consistentwith the viscosity solution of PDE (3.2) in the standard sense. Moreover, we emphasizethat our definition involves a richer set of test functions which in principle opens the hopefor an easier proof of uniqueness.(ii) Definition 3.7 does not reduce to the definition introduced in the semilinear contextof [10] (or Section 7 below) either, because we are using a different nonlinear expectation E L here. It is obvious that any viscosity subsolution in the sense of [10] is also a viscositysubsolution in the sense of this paper, but the opposite direction is in general not true.However, the definitions of viscosity solutions are actually equivalent for semilinear PPDEs,in view of the uniqueness result of our accompanying paper [12]. See also Remark 3.10. Remark 3.13
For 0 < L < L , obviously P tL ⊂ P tL , E L t ≤ E L t , and A L u ( t, ω ) ⊂A L u ( t, ω ). Then one can easily check that a viscosity L -subsolution must be a viscosity L -subsolution. Consequently, u is a viscosity subsolution if and only ifthere exists an L ≥ L ′ ≥ L , u is a viscosity L ′ -subsolution.We next report the following result whose proof follows exactly the lines of Remark 3.9(i) in [10]. Proposition 3.14
Let Assumption 3.1 hold true, and let u be a viscosity subsolution ofPPDE (3.1) . For λ ∈ R , the process ˜ u t := e λt u t is a viscosity subsolution of: ˜ L ˜ u := − ∂ t ˜ u − ˜ G ( t, ω, ˜ u, ∂ ω ˜ u, ∂ ωω ˜ u ) ≤ , (3.8)14 here ˜ G ( t, ω, y, z, γ ) := − λy + e λt G ( t, ω, e − λt y, e − λt z, e − λt γ ) . Remark 3.15
Under Assumption 3.1, we are not able to prove a more general change ofvariable formula. However, this will be achieved under stronger assumptions, see Proposi-tion 4.5 and Theorem 4.6 of our accompanying paper [12].
Theorem 3.16
Let Assumption 3.1 hold and u ∈ C , (Λ) ∩ U C b (Λ) . Then u is a classicalsolution (resp. subsolution, supersolution) of PPDE (3.1) if and only if it is a viscositysolution (resp. subsolution, supersolution). Proof
We prove the subsolution property only. Assume u is a viscosity L -subsolution.For any ( t, ω ), since u ∈ C , (Λ), we have u t,ω ∈ C , (Λ t ) and thus u t,ω ∈ A L u ( t, ω ) with h := T . By definition of viscosity L -subsolution we see that L u ( t, ω ) ≤ u is a classical subsolution. If u is not a viscosity subsolution,then it is not a viscosity L -subsolution. Thus there exist ( t, ω ) ∈ Λ and ϕ ∈ A L u ( t, ω )such that 2 c := L ϕ ( t, ) >
0. Without loss of generality, we set t := 0 and, by Remark 3.11(i), let h = h ε ∈ H defined in (2.4) for some small constant ε > A L u (0 , ). Now recall (2.5) and let P ∈ P L corresponding to someconstants α ∈ R d and β ∈ S d which will be determined later. Then0 ≤ E L (cid:2) ( ϕ − u ) h (cid:3) ≤ E P (cid:2) ( ϕ − u ) h (cid:3) . Applying functional Itˆo’s formula (2.10) and noticing that ( ϕ − u ) = 0, we have( ϕ − u ) h = Z h h ∂ t ( ϕ − u ) s + 12 ∂ ωω ( ϕ − u ) s : β + ∂ ω ( ϕ − u ) s · α i ds + Z h ∂ ω ( ϕ − u ) s · βdW P s . Taking expected values, this leads to0 ≤ E P h Z h (cid:16) ∂ t ( ϕ − u ) s + 12 ∂ ωω ( ϕ − u ) s : β + ∂ ω ( ϕ − u ) s · α (cid:17) ds i = E P h Z h ( ˜ L ϕ − ˜ L u ) s ds i , where ˜ L ϕ s := −L ϕ s − G ( · , ϕ, ∂ ω ϕ, ∂ ωω ϕ ) s + ( ∂ ωω ϕ ) s : β + ( ∂ ω ϕ ) s · α . Since ˜ L ϕ and ˜ L u are continuous, for ε small enough we have | ˜ L ϕ s − ˜ L ϕ | + | ˜ L u s − ˜ L u | ≤ c on [0 , h ]. Then0 ≤ E P h ( ˜ L ϕ − ˜ L u + c ) h i . (3.9)Note that L u ≤ L ϕ = 2 c , and ϕ = u . Thus˜ L ϕ − ˜ L u ≤ − c + 12 ∂ ωω ( ϕ − u ) : β + ∂ ω ( ϕ − u ) · α − [ G (cid:0) ., u, ∂ ω ϕ, ∂ ωω ϕ (cid:1) − G (cid:0) ., u, ∂ ω u, ∂ ωω u (cid:1) ] .
15y Assumption 3.1 (iii), there exist α and β such that P ∈ P L and G (cid:0) ., u, ∂ ω ϕ, ∂ ωω ϕ (cid:1) − G (cid:0) ., u, ∂ ω u, ∂ ωω u (cid:1) = 12 ∂ ωω ( ϕ − u ) : β + ∂ ω ( ϕ − u ) · α. Then ˜ L ϕ − ˜ L u ≤ − c , and (3.9) leads to 0 ≤ E P [ − c h ] <
0, contradiction.
In this section, we study several special PPDEs which have (semi-)explicit viscosity solu-tions, for example via backward SDEs or second order BSDEs. These solutions provideprobabilistic representations for the PPDEs and thus can be viewed as path dependentnonlinear Feynman-Kac formula. More importantly, as value functions of some stochasticcontrol problems, these examples illustrate how to check the viscosity properties of pro-cesses arising in applied problems. As in the viscosity theory of PDEs, the main tools arethe regularity of the processes in ( t, ω ) and the dynamic programming principle.
Example 4.1
Suppose that u ( t, ω ) = v ( ω t ) for all ( t, ω ) ∈ Λ, where v : R d → R is boundedand continuous. Then by (2.11) we should have ∂ t u = 0. We now verify that u is a viscositysolution of the equation − ∂ t u = 0.Indeed, for ϕ ∈ A L u ( t, ω ), it follows from our definition that, for some h ∈ H t :( ϕ − u t,ω ) t = 0 ≥ E P , (cid:2) ( ϕ − u t,ω ) ( t + δ ) ∧ h (cid:3) for all δ > . where P , is again the probability measure corresponding to α = , β = in (2.5). Noticethat under P , , the canonical process ω is frozen to its value at time t . Then h = T , P , -a.s. and thus, for δ < T − t , ϕ ( t, ) − v ( ω t ) = ( ϕ − u t,ω ) t ≥ E P , (cid:2) ( ϕ − u t,ω ) ( t + δ ) ∧ h (cid:3) = ϕ ( t + δ, ) − v ( ω t ) . This implies that ∂ t ϕ ( t, ) ≤
0. A similar argument shows that ∂ t ϕ ( t, ) ≥ ϕ ∈ A L u ( t, ω ). Example 4.2
Let d = 1 and use the notations in Example 2.11. We check that u ( t, ω ) :=2 B t − B t is a viscosity solution of the first order equation: − ∂ t u − | ∂ ω u | + 1 = 0 . (4.1)16y Example 2.11, u is not smooth, so it is a viscosity solution but not a classical solution.When ω t < ω t , it is clear that u is smooth with ∂ t u ( t, ω ) = 0 , ∂ ω u ( t, ω ) = − ω t = ω t . Without loss ofgenerality, we check it at ( t, ω ) = (0 , A L u (0 ,
0) is empty for L ≥
1, and thus u is a viscosity subsolution.Indeed, assume ϕ ∈ A L u (0 ,
0) with corresponding h ∈ H . By Remark 3.11 (i), without lossof generality we may assume h = h ε for some small ε >
0, and thus ∂ t ϕ, ∂ ωω ϕ are boundedon [0 , h ]. Note that P ∈ P L . By definition of A L we have, for any 0 < δ < ε ,0 ≤ E P h ( ϕ − u ) δ ∧ h i = E P h Z δ ∧ h ( ∂ t ϕ + ∂ ωω ϕ )( t, ω ) ds − B δ ∧ h i ≤ C E P [ δ ∧ h ] − E P [ B δ ∧ h ] ≤ Cδ − E P [ B δ ] + 2 E P [ B δ { h ≤ δ } ] ≤ Cδ − c √ δ + C p P ( h ≤ δ ) ≤ C [ δ + ε − δ ] − c √ δ, where c := 2 E P [ B ] > δ is small enough. Therefore, A L u (0 , ) is empty.(ii) We next check the viscosity supersolution property. Assume to the contrary that − c := − ∂ t ϕ (0 , − | ∂ ω ϕ (0 , | + 1 < ϕ ∈ A L u (0 ,
0) and L ≥
1. Let α := sgn ( ∂ ω ϕ (0 , β := 0, and P ∈ P L be determined by (2.5). When α = 1, we have B t = t, B t = t , P -a.s. When α = −
1, we have B t = − t , B t = 0, P -a.s.In both cases, it holds that u ( t, ω ) = t , h ε = ε , P -a.s. By choosing h = h ε and ε smallenough, we may assume | ∂ t ϕ ( t, B ) − ∂ t ϕ (0 , | + | ∂ ω ϕ ( t, B ) − ∂ ω ϕ (0 , | ≤ c for t ≤ h ε . Bythe definition of A L u (0 ,
0) we get0 ≥ E P h ( ϕ − u ) h ε i = E P h Z ε ( ∂ t ϕ + α∂ ω ϕ ) t dt − ε i ≥ E P h Z ε (cid:16) ∂ t ϕ + α∂ ω ϕ − c (cid:17) dt i − ε = E P h Z ε (cid:16) ∂ t ϕ + | ∂ ω ϕ | − c (cid:17) dt i − ε = Z ε (cid:16) c − c (cid:17) dt − ε = 12 cε > . This is the required contradiction, and thus u is a viscosity supersolution of (4.1). We now consider the following semi-linear PPDE: − ∂ t u − σ ( t, ω ) : ∂ ωω u − F (cid:0) t, ω, u, σ ( t, ω ) ∂ ω u (cid:1) = 0 , u ( T, ω ) = ξ ( ω ) , (4.2)where σ ∈ L ( F , S d ), ξ ∈ L ( F T ), and F is F -progressively measurable in all variables. Wenote that [10] studied the case σ = I d for simplicity. We shall assume17 ssumption 4.3 (i) σ , F ( t, ω, , ) , and ξ are bounded by C , and σ > . (ii) σ is uniformly Lipschitz continuous in ω and F is uniformly Lipschitz contin. in ( y, z ) . (iii) F and ξ are uniformly continuous in ω , and the common modulus of continuity function ρ has polynomial growth. (iv) σ and F ( · , y, z ) are right continuous in ( t, ω ) under d ∞ for any ( y, z ) , in the sense ofDefinition 2.1. The boundedness in Assumption 4.3 (i) is just for simplification, and can be weakenedto some growth condition. The assumption σ > and that F depends on the gradientterm through the special form σ ( t, ω ) ∂ ω u are mainly needed for the subsequent BSDErepresentation.For any ( t, ω ) ∈ Λ, consider the following decoupled FBSDE on [ t, T ]: X s = Z st σ t,ω ( r, X · ) dB tr , Y s = ξ t,ω ( X ) + Z Ts F t,ω ( r, X · , Y r , Z r ) dr − Z Ts Z r · dB tr , P t − a.s. (4.3)Under Assumption 4.3, clearly FBSDE (4.3) has a unique solution ( X t,ω , Y t,ω , Z t,ω ). Alter-natively, we may consider the BSDE in weak formulation: Y t,ωs = ξ t,ω ( B t )+ Z Ts F t,ω ( r, B t · , Y t,ωr , Z t,ωr ) dr − Z Ts Z t,ωr · ( σ t,ω ( r, B t · )) − dB tr , P t,ω -a.s. (4.4)where P t,ω := P t ◦ ( X t,ω ) − denotes the distribution of X t,ω . Then, for any fixed ( t, ω ), Y t,ωt = Y t,ωt and is a constant due to the Blumenthal zero-one law . Proposition 4.4
Under Assumption 4.3, u ( t, ω ) := Y t,ωt = Y t,ωt is a viscosity solution ofPPDE (4.2) . Proof
We proceed in two steps.
Step 1.
In Step 2 below, we will show that u ∈ U C b (Λ) and satisfies the dynamic program-ming principle: for any ( t, ω ) ∈ Λ and τ ∈ T t , Y t,ωs = u t,ω ( τ, B t ) + Z τs F t,ω ( r, B t · , Y t,ωr , Z t,ωr ) dr − Z τs Z t,ωr · ( σ t,ω ( r, B t · )) − dB tr , P t,ω -a.s.(4.5)Let L be a Lipschitz constant of F in z satisfying | σ | ≤ √ L . We now show that u isan L -viscosity solution. Without loss of generality, we prove only the viscosity subsolutionproperty at ( t, ω ) = (0 , ). For notational simplicity we omit the superscript , in the restof this proof. Assume to the contrary that, c := − (cid:8) ∂ t ϕ + 12 σ : ∂ ωω ϕ + F ( · , u, σ∂ ω ϕ ) (cid:9) (0 , ) > ϕ ∈ A L u (0 , ) . h ∈ H be the hitting time corresponding to ϕ in (3.5), and by Remark 3.11 (i), withoutloss of generality we may assume h = h ε for some small ε >
0. Since ϕ ∈ C , (Λ) and u ∈ U C b (Λ), by Assumption 4.3 (iv) and the uniform Lipschitz property of F in ( y, z ), wemay assume ε is small enough such that − (cid:8) ∂ t ϕ + 12 σ : ∂ ωω ϕ + F ( · , u, σ∂ ω ϕ ) (cid:9) ( t, ω ) ≥ c > , t ∈ [0 , h ] . Notice that d h B i t = σ ( t, B · ) dt , P -a.s. Using the dynamic programming principle (4.5),and applying Itˆo’s formula on ϕ , we have:( ϕ − u ) h = ( ϕ − u ) h − ( ϕ − u ) = Z h (cid:0) ∂ ω ϕ − σ − Z (cid:1) ( s, B · ) · dB s + Z h (cid:16) ∂ t ϕ + 12 σ : ∂ ωω ϕ + F ( · , u, Z ) (cid:17) ( s, B · ) ds ≤ Z h (cid:0) σ∂ ω ϕ − Z (cid:1) ( s, B · ) · σ − ( s, B · ) dB s − Z h (cid:16) c F ( · , u, σ∂ ω ϕ ) − F ( · , u, Z ) (cid:17) ( s, B · ) ds = Z h (cid:0) σ∂ ω ϕ − Z (cid:1) ( s, B · ) · σ − ( s, B · ) dB s − Z h h c σ∂ ω ϕ − Z ) · α i ( s, B · ) ds = Z h (cid:0) σ∂ ω ϕ − Z (cid:1) ( s, B · ) · (cid:0) σ − ( s, B · ) dB s − α s ds (cid:1) − c h , P -a.s.where | α | ≤ L . Notice that σ − dB t is a P -Brownian motion. Applying Girsanov Theoremone sees immediately that there exists ˜ P ∈ P L equivalent to P such that σ − dB t − α t dt is a˜ P -Brownian motion. Then the above inequality holds ˜ P -a.s., and by the definition of A L u :0 ≤ E ˜ P (cid:2) ( ϕ − u ) h (cid:3) ≤ − c E ˜ P [ h ] < , which is the required contradiction. Step 2.
We now show the dynamic programming principle together with the followingregularity of u : there exists a modulus of continuity function ρ such that, | u ( t, ω ) | ≤ C and | u ( t, ω ) − u ( t ′ , ω ′ ) | ≤ Cρ (cid:16) d ∞ (cid:0) ( t, ω ) , ( t ′ , ω ) (cid:1)(cid:17) , t ≤ t ′ , ω, ω ′ ∈ Ω . (4.6)Indeed, by standard arguments it is clear that, for any p ≥ E P t h kX t,ω k pT + kY t,ω k pT + (cid:0) Z Tt |Z t,ωs | ds (cid:1) p/ i ≤ C p ; E P t,ω h k B t k pT + k Y t,ω k pT + (cid:0) Z Tt | [ σ t,ω ( s, B t )] − Z t,ωs | ds (cid:1) p/ i ≤ C p ; E P t h kX t,ω − X t,ω ′ k T i ≤ Cρ ( k ω − ω ′ k t ) ;and, since ρ has polynomial growth, E P t h kY t,ω − Y t,ω ′ k T + Z Tt |Z t,ωs − Z t,ω ′ s | ds i ≤ Cρ ( k ω − ω ′ k t ) + C E P t h Cρ ( kX t,ω − X t,ω ′ k T ) i ≤ Cρ ( k ω − ω ′ k t ) , ρ . In particular, this implies that | u ( t, ω ) | ≤ C and | u ( t, ω ) − u ( t, ω ′ ) | ≤ Cρ ( k ω − ω ′ k t ) . (4.7)Given the above regularity, by standard arguments in BSDE theory, we have the followingdynamic programming principle: for any t < t ′ ≤ T , Y t,ωs = u t,ω ( t ′ , B t ) + Z t ′ s F t,ω ( r, B t · , Y t,ωr , Z t,ωr ) dr − Z t ′ s Z t,ωr · ( σ t,ω ( r, B t · )) − dB tr , P t,ω -a.s.(4.8)In particular, Y t,ωs = u t,ω ( s, B t ) for all t ≤ s ≤ T , P t,ω -a.s. That is, Y t,ωs = u ( s, ω ⊗ t B t ) = Y s,ω ⊗ t B t s , P t,ω -a.s.Denote δ := d ∞ (cid:0) ( t, ω ) , ( t ′ , ω ) (cid:1) . Then | u t − u t ′ | ( ω ) = (cid:12)(cid:12)(cid:12) E P t,ω h Y t,ωt − Y t,ωt ′ + u t,ω ( t ′ , B t ) − u ( t ′ , ω ) i(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) E P t,ω h Z t ′ t F t,ω ( r, B t · , Y t,ωr , Z t,ωr ) dr + u t,ω ( t ′ , B t ) − u ( t ′ , ω )] i(cid:12)(cid:12)(cid:12) ≤ E P t,ω h Z t ′ t | F t,ω ( r, B t · , Y t,ωr , Z t,ωr ) | dr + Cρ (cid:16) δ + k B t k t ′ (cid:17)i , (4.9)Notice that E P t,ω h Z t ′ t (cid:12)(cid:12) F t,ω ( r, B t · , Y t,ωr , Z t,ωr ) (cid:12)(cid:12) dr i ≤ C E P t,ω h Z t ′ t (cid:0) | Y t,ωr | + | Z t,ωr | (cid:1) dr i ≤ C √ δ (cid:16) E P t,ω h Z t ′ t (cid:0) | Y t,ωr | + | Z t,ωr ) | (cid:1) dr i(cid:17) / ≤ C √ δ. As for the second term, since ρ has polynomial growth, one can easily see that we mayassume without loss of generality that ρ also has polynomial growth. Note that t ′ − t ≤ δ .Then it is clear that there exists a modulus of continuity function ρ such that E P t,ω h ρ (cid:16) δ + k B t k t ′ (cid:17)i ≤ ρ ( δ ) . Without loss of generality we assume ρ ( δ ) ≥ √ δ . Then, plugging the last estimates into(4.9) and combining with (4.7), we obtain (4.6).Moreover, given the regularity in t , we may extend the dynamic programming principle(4.8) to stopping times, proving (4.5). Remark 4.5
For FBSDE (4.3) with ( t, ω ) = (0 , ), we have Y s := u ( s, X · ) P -a.s. Thisextends the nonlinear Feynman-Kac formula of [23] to the path-dependent case.20 .3 Path dependent HJB equations and 2BSDEs Let K be a measurable space (equipped with some σ -algebra). We now consider the followingpath dependent HJB equation: − ∂ t u − G ( t, ω, u, ∂ ω u, ∂ ωω u ) = 0 , u ( T, ω ) = ξ ( ω ); (4.10)where G ( t, ω, y, z, γ ) := sup k ∈ K h σ ( t, ω, k ) : γ + F ( t, ω, y, σ ( t, ω, k ) z, k ) i , where σ ∈ S d and F are F -progressively measurable in all variables, and ξ is F T -measurable.We shall assume Assumption 4.6 (i) σ , F ( t, ω, , , k ) , and ξ are bounded by C , and σ > . (ii) σ is uniformly Lipschitz continuous in ω , and F is uniformly Lipschitz contin. in ( y, z ) . (iii) F and ξ are uniformly continuous in ω , and the common modulus of continuity function ρ has polynomial growth. (iv) σ ( · , k ) , F ( · , y, z, k ) , and G ( · , y, z ) are right continuous in ( t, ω ) under d ∞ for any ( y, z, k ) , in the sense of Definition 2.1. For each t , let K t denote the set of F t -progressively measurable K -valued processes onΛ t . For any ( t, ω ) ∈ Λ and k ∈ K t , let X t,ω,k denote the solution to the following SDE: X s = Z st σ t,ω ( r, X · , k r ) dB tr , t ≤ s ≤ T, P t -a.s.Denote P t,ω,k := P t ◦ ( X t,ω,k ) − . Since σ > , as discussed in [31] X t,ω,k and B t induce thesame P t -augmented filtration, and thus there exists ˜ k ∈ K t such that ˜ k ( X t,ω,k · ) = k , P t -a.s.Let ( Y t,ω,k , Z t,ω,k ) denote the solution to the following BSDE on [ t, T ]: Y s = ξ t,ω ( B t ) + Z Ts F t,ω ( r, B t · , Y r , Z r , ˜ k r ) dr − Z Ts Z r · ( σ t,ω ( r, B t · , ˜ k r )) − dB tr , P t,ω,k -a.s.We now consider the stochastic control problem: u ( t, ω ) := sup k ∈K t Y t,ω,kt , ( t, ω ) ∈ Λ . We observe that this process u was considered by Nutz [20], in the stochastic control context,and shown to be the solution of a second order BSDE. The next result shows that our notionof viscosity solution is also suitable for this stochastic control problem. Proposition 4.7
Under Assumption 4.6, u is a viscosity solution of PPDE (4.10) . roof By Proposition 3.14, without loss of generality we assume G , hence F , is increasing in y . (4.11)Following similar arguments as in Proposition 4.4, we may prove that | u ( t, ω ) | ≤ C, | u ( t, ω ) − u ( t ′ , ω ′ ) | ≤ Cρ (cid:16) d ∞ (cid:0) ( t, ω ) , ( t ′ , ω ) (cid:1)(cid:17) , for any ( t, ω ) , ( t ′ , ω ′ ) ∈ Λ . This regularity, together with the standard arguments, see e.g. [31] or [24], implies furtherthe following dynamic programming principle: u ( t, ω ) = sup k ∈K t Y t,ω,kt ( τ, u t,ω ( τ, · )) , for any ( t, ω ) ∈ Λ , τ ∈ T t , (4.12)where, for any F tτ -measurable random variable η , ( Y , Z ) := ( Y t,ω,k ( τ, η ) , Z t,ω,k ( τ, η )) solvesthe following BSDE on [ t, τ ]: Y s = η ( B t · ) + Z τs F t,ω ( r, B t , Y r , Z r , ˜ k r ) dr − Z τs Z r · ( σ t,ω ( r, B t · , ˜ k r )) − dB tr , P t,ω,k − a.s.We now prove the viscosity property, for the same L as in Proposition 4.4. Again we shallonly prove it at ( t, ω ) = (0 , ) and we will omit the superscript , . However, since in thiscase u is defined through a supremum, we need to prove the viscosity subsolution propertyand supersolution property differently. Viscosity L − subsolution property. Assume to the contrary that, c := − (cid:8) ∂ t ϕ + G ( · , u, ∂ ω ϕ, ∂ ωω ϕ ) (cid:9) (0 , ) > ϕ ∈ A L u (0 , ) . As in Proposition 4.4, let h = h ε ∈ H be the hitting time corresponding to ϕ in (3.5). Since ϕ ∈ C , (Λ), u ∈ U C b (Λ), and by Assumption 4.6 (iv) G is right continuous in ( t, ω ) under d ∞ , we may assume ε is small enough such that − (cid:8) ∂ t ϕ + G ( · , u, ∂ ω ϕ, ∂ ωω ϕ ) (cid:9) ( t, ω ) ≥ c > , t ∈ [0 , h ] . By the definition of G , this implies that, for any t ∈ [0 , h ] and k ∈ K , − (cid:8) ∂ t ϕ + 12 σ ( t, ω, k ) : ∂ ωω ϕ + F ( t, ω, u, σ ( · , k ) ∂ ω ϕ, k ) (cid:9) ( t, ω ) ≥ c > . Now for any k ∈ K , notice that d h B i t = σ ( t, B · , ˜ k t ) dt , P k -a.s. Denote ( Y k , Z k ) :=( Y k ( h , u ( h , · )) , Z k ( h , u ( h , · ))). One can easily see that u ( s, B ) ≥ Y ks , 0 ≤ s ≤ h , P k -a.s. For22ny δ >
0, applying functional Itˆo’s formula on ϕ we see that, :( ϕ − Y k ) − ( ϕ − u ) h ∧ δ ≥ ( ϕ − Y k ) − ( ϕ − Y k ) h ∧ δ = − Z h ∧ δ h ∂ t ϕ + 12 σ : ∂ ωω ϕ + F ( · , Y k , Z k ) i ( s, B · , ˜ k s ) ds − Z h ∧ δ (cid:0) ∂ ω ϕ − σ − Z k (cid:1) ( s, B · , ˜ k s ) · dB s ≥ Z h ∧ δ h c F ( · , u, σ∂ ω ϕ ) − F ( · , Y k , Z k ) i ( s, B · , ˜ k s ) ds − Z h ∧ δ (cid:0) ∂ ω ϕ − σ − Z k (cid:1) ( s, B · , ˜ k s ) · dB s , P k -a.s.Note again that Y ks ≤ u ( s, B · ). Then by (4.11) we have (cid:0) u − Y k (cid:1) − (cid:0) ϕ − u (cid:1) h ∧ δ = (cid:0) ϕ − Y k (cid:1) − (cid:0) ϕ − u (cid:1) h ∧ δ ≥ Z h ∧ δ h c F ( · , u, σ∂ ω ϕ ) − F ( · , u, Z k ) i ( s, B · , ˜ k s ) ds − Z h ∧ δ (cid:0) ∂ ω ϕ − σ − Z k (cid:1) ( s, B · , ˜ k s ) · dB s = Z h ∧ δ h c σ∂ ω ϕ − Z k ) · α i ( s, B · , ˜ k s ) ds − Z h ∧ δ (cid:0) ∂ ω ϕ − σ − Z k (cid:1) ( s, B · , ˜ k s ) · dB s = c h ∧ δ ) − Z h ∧ δ (cid:0) σ∂ ω ϕ − Z k (cid:1) ( s, B · , ˜ k s ) · ( σ − ( s, B · , ˜ k s ) dB s − α s ds ) , P k -a.s.where | α | ≤ L and λ is bounded. As in Proposition 4.4, we may define ˜ P k ∈ P L equivalentto P such that σ − ( t, B · , ˜ k t ) dB t − α t dt is a ˜ P k -Brownian motion. Then the above inequalityholds ˜ P k -a.s., and by the definition of A L u , we have u − Y k ≥ u − Y k − E ˜ P k (cid:2) ( ϕ − u ) h ∧ δ (cid:3) ≥ c E ˜ P k [ h ∧ δ ] ≥ c δ h − ˜ P k [ h ≤ δ ] i . By (2.8), for δ small enough we have u − Y k ≥ c δ h − Cε − δ i ≥ cδ > . This implies that u − sup k ∈K Y k ≥ cδ >
0, which is in contradiction with (4.12).
Viscosity L − supersolution property. Assume to the contrary that, c := n ∂ t ϕ + G ( · , u, ∂ ω ϕ, ∂ ωω ϕ ) i (0 , ) > ϕ ∈ A L u (0 , ) . By the definition of F , there exists k ∈ K such that n ∂ t ϕ + 12 σ ( · , k ) : ∂ ωω ϕ + F ( · , u, σ ( · , k ) ∂ ω ϕ, k ) o (0 , ) ≥ c > h = h ε ∈ H be the hitting time corresponding to ϕ in (3.5), and by the rightcontinuity of σ and F in Assumption 4.6 (iv) we may assume ε is small enough so that n ∂ t ϕ + 12 σ ( · , k ) : ∂ ωω ϕ + F ( · , u, σ ( · , k ) ∂ ω ϕ, k ) o ( t, ω ) ≥ c > , t ∈ [0 , h ] . Consider the constant process k := k ∈ K . It is clear that the corresponding ˜ k = k . Followsimilar arguments as in the subsolution property, we arrive at the following contradiction: u − Y k ≤ − c E ˜ P k [ h ] < . Example 4.8
Assume K := { k ∈ S d : σ ≤ k ≤ σ } , where < σ < σ are constant matrices.Set σ ( t, ω, k ) := k . Then Y t ( ω ) = u ( t, ω ) is the solution to the following second order BSDE,as introduced by [32]: Y t = ξ ( B · ) + Z Tt F ( s, B · , Y s , Z s , ˆ a s ) ds − Z Tt Z s · (ˆ a s ) − dB s − dK t , P -q.s. (4.13)where P := { P ∈ P ∞ : α P = 0 , β P ∈ K } , ˆ a is the universal process such that d h B i t = ˆ a t dt , P -q.s. and K is an increasing process satisfying certain minimum condition. Remark 4.9
By using the zero-sum game, we may also obtain a representation formulafor the viscosity solution of the following path dependent Bellman-Isaacs equation: − ∂ t u − G ( t, ω, u, ∂ ω u, ∂ ωω u ) = 0 , u ( T, ω ) = ξ ( ω ) , (4.14)where G ( t, ω, y, z, γ ) := sup k ∈ K inf k ∈ K h σ ( t, k , k ) : γ + F ( t, ω, y, σ ( t, k , k ) z, k , k ) i = inf k ∈ K sup k ∈ K h σ ( t, k , k ) : γ + F ( t, ω, y, σ ( t, k , k ) z, k , k ) i . See Pham and Zhang [30].
The main result of this section is the following extension of Theorem 4.1 in [10], with a prooffollowing the same line of argument. However the present fully nonlinear context makes acrucial use of Theorem 2.6. Denote, for any ( t, y, z, γ ) ∈ [0 , T ) × R × R d × S d and δ > O δ ( t, y, z, γ ) := n ( s, ˜ ω, ˜ y, ˜ z, ˜ γ ) ∈ Λ t × R × R d × S d : d t ∞ (( s, ˜ ω ) , ( t, )) + | ˜ y − y | + | ˜ z − z | + | ˜ γ − γ | ≤ δ o . heorem 5.1 Let
L > , G satisfy Assumption 3.1, and u ∈ U (resp. u ∈ U ). Assume(i) for any ε > , there exist G ε and u ε ∈ U (resp. u ε ∈ U ) such that G ε satisfiesAssumption 3.1 and u ε is a viscosity L -subsolution (resp. L -supersolution) of PPDE (3.1) with generator G ε ;(ii) as ε → , ( G ε , u ε ) converge to ( G, u ) locally uniformly in the following sense: forany ( t, ω, y, z, γ ) ∈ Λ × R × R d × S d , there exists δ > such that, lim ε → sup ( s, ˜ ω, ˜ y, ˜ z, ˜ γ ) ∈ O δ ( t,y,z,γ ) h | ( G ε − G ) t,ω ( s, ˜ ω, ˜ y, ˜ z, ˜ γ ) | + | ( u ε − u ) t,ω ( s, ˜ ω ) | i = 0 . (5.1) Then u is a viscosity L -subsolution (resp. L -supersol.) of PPDE (3.1) with generator G . Proof
Without loss of generality we shall only prove the viscosity subsolution propertyat (0 , ). Let ϕ ∈ A L u (0 , ) with corresponding h ∈ H , δ > h δ ≤ h and lim ε → ρ ( ε, δ ) = 0, where ρ ( ε, δ ) := sup ( t,ω, ˜ y, ˜ z, ˜ γ ) ∈ O δ (0 ,y ,z ,γ ) h | G ε − G | ( t, ω, ˜ y, ˜ z, ˜ γ ) + | u ε − u | ( t, ω ) i , and ( y , z , γ ) := ( ϕ , ∂ ω ϕ , ∂ ωω ϕ ) . For 0 < δ ≤ δ , denote ϕ δ ( t, ω ) := ϕ ( t, ω ) + δt . By (3.5) and Lemma 2.4 we have( ϕ δ − u ) = ( ϕ − u ) = 0 ≤ E L h ( ϕ − u ) h δ i < E L h ( ϕ δ − u ) h δ i . By (5.1), there exists ε δ > ε ≤ ε δ ,( ϕ δ − u ε ) < E L h ( ϕ δ − u ε ) h δ i . (5.2)Denote X := X ε,δ := u ε − ϕ δ ∈ U . Define ˆ X := X [0 , h δ ) + X h δ − [ · δ ,T ] , Y := E L [ ˆ X ], and τ ∗ := inf { t ≥ Y t = ˆ X t } , as in Theorem 2.6. Then all the results in Theorem 2.6 hold.Noticing that X h δ − ≤ X h δ , by (5.2) we have E L [ ˆ X h δ ] ≤ E L [ X h δ ] = −E L h ( ϕ δ − u ε ) h δ i < − ( ϕ δ − u ε ) = X ≤ Y = E L [ Y τ ∗ ] = E L [ ˆ X τ ∗ ] . Then there exists ω ∗ such that t ∗ := τ ∗ ( ω ∗ ) < h δ ( ω ∗ ), and thus h t ∗ ,ω ∗ δ ∈ H t ∗ . We shallremark though that here Y, τ ∗ , ω ∗ , t ∗ all depend on ε, δ . Now define ϕ εδ ( t, ω ) := ϕ t ∗ ,ω ∗ δ ( t, ω ) − ϕ δ ( t ∗ , ω ∗ ) + u ε ( t ∗ , ω ∗ ) , ( t, ω ) ∈ Λ t ∗ . It is straightforward to check that ϕ εδ ∈ A L u ε ( t ∗ , ω ∗ ) with corresponding hitting time h t ∗ ,ω ∗ δ .Since u ε is a viscosity L -subsolution of PPDE (3.1) with generator G ε , we have0 ≥ h − ∂ t ϕ εδ − ( G ε ) t ∗ ,ω ∗ ( · , ϕ εδ , ∂ ω ϕ εδ , ∂ ωω ϕ εδ ) i ( t ∗ , )= h − ∂ t ϕ − δ − G ε ( · , u ε , ∂ ω ϕ, ∂ ωω ϕ ) i ( t ∗ , ω ∗ ) . (5.3)25ote that t ∗ < h δ ( ω ∗ ), then | u ε − u | ( t ∗ , ω ∗ ) ≤ ρ ( ε, δ ) ≤ ρ ( ε, δ ). By (5.1) and Definition2.1, we may set δ small enough and then ε small enough so that ( · , u ε , ∂ ω ϕ, ∂ ωω ϕ )( t ∗ , ω ∗ ) ∈ O δ (0 , y , z , γ ). Thus, (5.3) leads to0 ≥ h − ∂ t ϕ − δ − G ε ( · , u ε , ∂ ω ϕ, ∂ ωω ϕ ) i ( t ∗ , ω ∗ ) ≥ h − ∂ t ϕ − G ( · , u ε , ∂ ω ϕ, ∂ ωω ϕ ) i ( t ∗ , ω ∗ ) − δ − ρ ( ε, δ ) ≥ h − ∂ t ϕ − G ( · , u, ∂ ω ϕ, ∂ ωω ϕ ) i ( t ∗ , ω ∗ ) − δ − ρ ( ε, δ ) − Cρ ( ε, δ ) ≥ L ϕ − C sup ( t,ω ): t< h δ ( ω ) h | u ( t, ω ) − u | + | ∂ ω ϕ ( t, ω ) − ∂ ω ϕ | + | ∂ ωω ϕ ( t, ω ) − ∂ ωω ϕ | i − sup ( t,ω ): t< h δ ( ω ) (cid:12)(cid:12)(cid:12) G ( t, ω, y , z , γ ) − G (0 , , y , z , γ ) (cid:12)(cid:12)(cid:12) − δ − ρ ( ε, δ ) − Cρ ( ε, δ ) , where we used the fact that G satisfies Assumption 3.1. Notice that the right-continuityof G in ( t, ω ) under d ∞ allows us to control the last line. Now by first sending ε → δ → L ϕ ≤
0. Since ϕ ∈ A L u (0 , ) is arbitrary, we see that u is a viscositysubsolution of PPDE (3.1) with generator G at (0 , ) and thus complete the proof. Remark 5.2
Similar to Theorem 4.1 in [10], we need the same L in the proof of Theorem5.1. If u ε is only a viscosity subsolution of PPDE (3.1) with generator G ε , but with possiblydifferent L ε , we are not able to show that u is a viscosity subsolution of PPDE (3.1) withgenerator G . In this section, we prove a partial comparison principle, i.e. a comparison result of a viscositysuper- (resp. sub-) solution and a classical sub- (resp. super-) solution. The proof is alsocrucially based on Theorem 2.6. Moreover, this result is a first key step for our comparisonprinciple in the accompanying paper [12].
Proposition 5.3
Let Assumption 3.1 hold true. Let u ∈ U be a viscosity subsolution and u ∈ U a viscosity supersolution of PPDE (3.1). If u ( T, · ) ≤ u ( T, · ) and either u or u is in C , (Λ) , then u ≤ u on Λ . Proof
We shall only prove u ≤ u . The inequality for general t can be proved similarly.Without loss of generality, we assume u is a viscosity L -subsolution and u ∈ C , (Λ) is aclassical L -supersolution. By Proposition 3.14, we may assume that G is nonincreasing in y. (5.4)26ssume to the contrary that c := T [ u − u ] >
0. Denote X t := ( u − u ) + t + ct, b X t := X t { t
0. Since ( u − u ) t ∗ ,ω ∗ ∈ U t ∗ , there exists h ∈ H t ∗ such that h < T and ( u − u ) t ∗ ,ω ∗ t > t ∈ [ t ∗ , h ], and thus b X t ∗ ,ω ∗ t = X t ∗ ,ω ∗ t = ( u − u ) t ∗ ,ω ∗ t + ct forall t ∈ [ t ∗ , h ].Now observe that ϕ ( t, ω ) := ( u ) t ∗ ,ω ∗ ( t, ω ) − ct + X t ∗ ( ω ∗ ) ∈ C , (Λ t ∗ ), a consequenceof our assumption u ∈ C , (Λ). Moreover, for any τ ∈ T t ∗ , it follows from the E L -supermartingale property of the nonlinear Snell envelope Y that (cid:0) ( u ) t ∗ ,ω ∗ − ϕ (cid:1) t ∗ = 0 = Y t ∗ ( ω ∗ ) − X t ∗ ( ω ∗ ) ≥ E Lt ∗ (cid:2) Y t ∗ ,ω ∗ τ ∧ h (cid:3) − X t ∗ ( ω ∗ ) ≥ E Lt ∗ (cid:2) X t ∗ ,ω ∗ τ ∧ h (cid:3) − X t ∗ ( ω ∗ ) = E Lt ∗ (cid:2)(cid:0) ( u ) t ∗ ,ω ∗ − ϕ (cid:1) τ ∧ h (cid:3) . By the arbitrariness of τ ∈ T t ∗ , and the fact that E L [ · ] = −E L [ −· ], this proves that ϕ ∈A L u ( t ∗ , ω ∗ ), and by the viscosity L -subsolution property of u :0 ≥ (cid:8) − ∂ t ϕ − G ( ., u , ∂ ω ϕ, ∂ ωω ϕ ) (cid:9) ( t ∗ , ω ∗ )= c − (cid:8) ∂ t u + G ( ., u , ∂ ω u , ∂ ωω u ) (cid:9) ( t ∗ , ω ∗ ) ≥ c − (cid:8) ∂ t u + G ( ., u , ∂ ω u , ∂ ωω u ) (cid:9) ( t ∗ , ω ∗ ) , where the last inequality follows from (5.4). Since c >
0, this is in contradiction with thesupersolution property of u .As a direct consequence of the above partial comparison, we have Proposition 5.4
Let Assumption 3.1 hold true. If PPDE (3.1) has a classical solution u ∈ C , (Λ) ∩ U C b (Λ) , then u is the unique viscosity solution of PPDE (3.1) with terminalcondition u ( T, · ) . In our accompanying paper [12], we shall prove the uniqueness of viscosity solutionswithout assuming the existence of classical solutions.27
Viscosity Solutions of Backward Stochastic PDEs
In this section, we show that our PPDEs includes Backward SPDEs as a special case. Weremark that such BSPDEs arise naturally in many applications, see e.g. [19] and [21].Consider the following BSPDE with F -progressively measurable solution ( u, q ): u ( t, ω, x ) = ξ ( ω, x ) + Z Tt F ( s, ω, x, u, Du, D u, q, Dq ) ds − Z Tt q ( s, ω, x ) · dB s , P -a.s. (6.1)where x ∈ R d ′ , and D, D denote the gradient and Hessian with respect to the x − variable.Assume u ∈ C , (Λ × R d ′ ), namely ∂ t u, ∂ ω u, Du, ∂ ωω u, D∂ ω u, D u exist and are continuous,where the derivatives in x are in standard sense and the smoothness in ( t, ω ) is in the senseof Definition 2.8. Fix x and apply funtional Itˆo’s formula, we have du ( t, ω, x ) = (cid:0) ∂ t u + 12 tr ( ∂ ωω u ) (cid:1) ( t, ω, x ) dt + ∂ ω u ( t, ω, x ) · dB t , P -a.s.Comparing this with (6.1) we obtain q ( t, ω, x ) = ∂ ω u ( t, ω, x ) , (cid:0) ∂ t u + 12 tr ( ∂ ωω u ) (cid:1) ( t, ω, x ) + F ( t, ω, x, u, Du, D u, q, Dq ) = 0 . This leads to a mixed PPDE: b L u ( t, ω, x ) = 0 , u ( T, ω, x ) = ξ ( ω, x ) , x ∈ R d ′ , (6.2)where, for ϕ ∈ C , (Λ × R d ′ ), b L ϕ := − ∂ t ϕ −
12 tr ( ∂ ωω ϕ ) − F ( ., ϕ, Dϕ, D ϕ, ∂ ω ϕ, D∂ ω ϕ ) . (6.3)To incorporate the mixed PPDE (6.2) into our framework, we enlarge the space of pathsto ˆΩ := Ω × { ω ′ ∈ C ([0 , T ] , R d ′ ) : ω ′ = } . Denote ˆΛ := [0 , T ] × ˆΩ, andˆ G x ( t, ˆ ω, y, z, γ ) := 12 γ + F (cid:0) t, ω, x + ω ′ t , y, z , γ , z , γ (cid:1) , ˆ ξ x (ˆ ω ) := ξ ( ω, x + ω ′ T ) , for all x ∈ R d ′ and ( t, ˆ ω, y, z, γ ) ∈ ˆΛ × R × R d + d ′ × S d + d ′ . Note that ˆ G x ( t, . ) and ˆ ξ x dependon ˆ ω = ( ω, ω ′ ) only through the pair ( ω, ω ′ t ) and ( ω, ω ′ T ), respectively. Definition 6.1
We say u is a viscosity solution (resp. supersolution, subsolution) ofBSPDE (6.1) if, for any fixed x , the process ˆ u x ( t, ˆ ω ) := u ( t, ω, x + ω ′ t ) , t ∈ [0 , T ] , ˆ ω =( ω, ω ′ ) ∈ ˆΩ , is a viscosity solution (resp. supersolution, subsolution) of the PPDE: − ∂ t ˆ u x ( t, ˆ ω ) − ˆ G x (cid:0) t, ˆ ω, ˆ u x , ∂ ˆ ω ˆ u x , ∂ ˆ ω ˆ ω ˆ u x (cid:1) = 0 , on ˆΛ , and ˆ u x ( T, ˆ ω ) = ˆ ξ x (ˆ ω ) . emark 6.2 When ξ and F do not depend on ω , one can easily see that q = 0 and u = u ( t, x ) is deterministic. Then BSPDE (6.1) reduces to a standard PDE. In this case,our Definition 6.1 is not the same as the standard viscosity solution of PDEs, but in thesense of Remark 3.12 (i). Remark 6.3
In the same manner we may also transform the following (forward) StochasticPDE into a (forward) PPDE: u ( t, ω, x ) = u ( x ) + Z t F ( s, ω, x, u, u x , u xx ) ds + Z t σ ( s, ω, x, u, u x ) dB s . (6.4)Due to its forward nature, the definition of viscosity solutions will be quite different. How-ever, the approach which will be specified in next section and in [12] still works in this case.See Buckdahn, Ma and Zhang [3]. In [10], we proved the comparison principle for semilinear PPDE (4.2), in the case σ = I d .One important argument was the Bank-Baum approximation in [1], which unfortunatelydoes not seem to be extendable to the fully nonlinear case. In this section we provide analternative proof of the comparison principle for semilinear PPDE (4.2). This approachworks in fully nonlinear case as well, but with much more involved technicalities, see ouraccompanying paper [12]. It has also been applied by Henry-Labordere, Tan, and Touzi[16] to study a new type of numerical methods for BSDEs.In order to focus on the main idea and simplify the presentation, we restrict to the case σ = I d . That is, we shall consider the following PPDE: L u ( t, ω ) := − ∂ t u − I d : ∂ ωω u − F ( t, ω, u, ∂ ω u ) = 0 . (7.1)We first give an alternative definition for viscosity solutions of semilinear PPDE (7.1). Weremark that the key point in (3.5) and Definition 3.7 is that the class P L covers all theprobability measures induced by the linearization of the generator G . In the semilinearcase, since the diffusion term σ is already fixed, we shall only consider the drift uncertaintyinduced by the linearization of generator F , as we did in [10]. To be precise, define M t,αT := exp (cid:16) Z Tt α s · dB ts − Z Tt | α s | ds (cid:17) ; P tL := n P ( · ) := Z · M t,αT d P t : α ∈ L (Λ t , R d ) such that | α | ≤ L o ; E Lt [ ξ ] := sup P ∈P tL E P [ ξ ] , E Lt [ ξ ] := inf P ∈P tL E P [ ξ ];29nd A L u ( t, ω ) := n ϕ ∈ C , (Λ t ) : ( ϕ − u t,ω ) t = 0 = inf τ ∈T t E Lt (cid:2) ( ϕ − u t,ω ) ·∧ h (cid:3) for some h ∈ H t o , A L u ( t, ω ) := n ϕ ∈ C , (Λ t ) : ( ϕ − u t,ω ) t = 0 = sup τ ∈T t E Lt (cid:2) ( ϕ − u t,ω ) ·∧ h (cid:3) for some h ∈ H t o . Definition 7.1 (i)
Let
L > . We say u ∈ U (resp. U ) is a viscosity L -subsolution(resp. L -supersolution) of semilinear PPDE (7.1) if, for any ( t, ω ) ∈ [0 , T ) × Ω and any ϕ ∈ A L u ( t, ω ) (resp. ϕ ∈ A L u ( t, ω ) ): (cid:8) − ∂ t ϕ − I d : ∂ ωω ϕ − F t,ω ( ., ϕ, ∂ ω ϕ ) (cid:9) ( t, ) ≤ ( resp. ≥ ) 0 . (ii) u ∈ U (resp. U ) is a viscosity subsolution (resp. supersolution) of PPDE (7.1) if u isviscosity L -subsolution (resp. L -supersolution) of PPDE (7.1) for some L > . (iii) u ∈ U C b (Λ) is viscosity solution of PPDE (7.1) if it is viscosity sub- and supersolution. Under Assumption 4.3, following almost the same arguments and after obvious modifi-cations when necessary, one can easily check that Theorems 3.16, 5.1, and Proposition 4.4still hold. Moreover, we may improve the partial comparison principle of Proposition 5.3as follows. First, we extend the space C , (Λ): Definition 7.2
Let t ∈ [0 , T ) , u ∈ L (Λ t ) . We say u ∈ C , (Λ t ) if there exist randomtimes t = h ≤ h · · · ≤ T (not necessarily hitting times) such that, (i) h i < h i +1 whenever h i < T , and for all ω ∈ Ω t , the set { i : h i ( ω ) < T } is finite; (ii) For each i , ω ∈ Ω t , and s ∈ [ h i ( ω ) , h i +1 ( ω )) , there exist h ∈ H s and ˜ u s,ω ∈ C , (Λ s ) such that h < h s,ωi +1 and u s,ω = ˜ u s,ω on [ s, h ] . (iii) u is bounded and continuous in t . Roughly speaking, C , (Λ) consists of processes u which are piecewise C , (Λ), in thesense that u is smooth on [ h i , h i +1 ) mentioned above.For u ∈ C , (Λ t ) and ( s, ω ) ∈ [ t, T ) × Ω t , we may define the derivatives of u at ( s, ω )as the derivatives of ˜ u s,ω at ( s, ), where ˜ u s,ω is defined in Definition 7.2 (ii). Clearly thesederivatives are independent of the choices of ˜ u . We remark that the processes in C , (Λ t ) arein general not continuous in ω . We also note that the space C , (Λ t ) here is slightly differentfrom that in [10], and in [12] we shall modify it slightly further for technical reasons.We first extend the partial comparison principle Proposition 5.3 to the case that either u or u is only in C , (Λ), instead of C , (Λ). Our proof relies heavily on the theory ofReflected BSDEs, for which we refer to El Karoui et al [13], Hamad`ene [15], and Peng and30u [29] for details. We note that Remark 3.11 in our earlier paper [10] on this issue isheuristic. The precise statements are given below. Remark 7.3
Let X ∈ L (Λ) be bounded and c`adl`ag with positive jumps. Fix L >
Y , ˜ Z, ˜ K ) denote the unique F -measurable solution to the following RBSDE: ˜ Y t = X T + Z Tt L | ˜ Z s | ds − Z Tt ˜ Z s · dB s + ˜ K T − ˜ K t ;˜ Y t ≥ X t , [ ˜ Y t − − X t − ] d ˜ K t = 0; P -a.s. (7.2)Then ˜ Y and ˜ K are continuous in t , P -a.s. Moreover, there exists τ ∗ ∈ T such that τ ∗ = inf { t ≥ Y t = X t } , P -a.s. (7.3)We remark that, since we require ˜ Y to be F -measurable, we cannot claim ˜ Y is continuous forall ω . Consequently, the right side of (7.3) may not be an F -stopping time, but a stoppingtime adapted to the P -augmented filtration of F .(ii) Define Y t ( ω ) = sup τ ∈T t E Lt [ X t,ωτ ] , ( t, ω ) ∈ Λ . Then Y = ˜ Y . Moreover, for any τ ∈ T , following standard arguments one may easily showthat, for P -a.e. ω , ( ˜ Y τ,ω , ˜ Z τ,ω , ˜ K τ,ω ) satisfies the following RBSDE on [ τ ( ω ) , T ]: ˜ Y τ,ωt = X τ,ωT + Z Tt L | ˜ Z τ,ωs | ds − Z Tt ˜ Z τ,ωs · dB τ ( ω ) s + ˜ K τ,ωT − ˜ K τ,ωt ;˜ Y τ,ωt ≥ X τ,ωt , [ ˜ Y τ,ωt − − X τ,ωt − ] d ˜ K τ,ωt = 0; P τ ( ω )0 -a.s.Then, Y τ ( ω ) ( ω ) = ˜ Y τ,ωτ ( ω ) = ˜ Y τ ( ω ) ( ω ). That is, Y τ = ˜ Y τ , P -a.s. for all τ ∈ T . In other words, Y and ˜ Y are P -modifications.(iii) However, since X is not required to be continuous in ω , we are not able to provethe desired regularity of Y . In particular, we are not able to prove that Y and ˜ Y are P -indistinguishable. Consequently, we cannot verify rigorously that Y solves RBSDE (7.2).(iv) In Theorem 2.6, although ˆ X is also not continuous in ω , due to its special structurewe proved in [11] that the Snell envelope Y has certain regularity in ω , which is crucial forproving the optimality of τ ∗ in Theorem 2.6.We now establish the partial comparison principle.31 roposition 7.4 Let Assumption 4.3 hold and σ = I d . Let u ∈ U be a viscosity subsolu-tion of PPDE (3.1) and u ∈ C , (Λ) satisfying L u ( t, ω ) ≥ for all ( t, ω ) ∈ [0 , T ) × Ω . If u ( T, · ) ≤ u ( T, · ) , then u ≤ u on Λ .The result also holds if we assume instead that u ∈ C , (Λ) satisfies L u ( t, ω ) ≤ forall ( t, ω ) ∈ [0 , T ) × Ω and u ∈ U is a viscosity supersolution of PPDE (3.1). Proof
As in Proposition 5.3, we shall only prove u ≤ u under the additional condition(5.4). Let u ∈ C , (Λ) with corresponding h i , i ≥
0. Assume to the contrary that c := T [ u − u ] >
0, and denote X t := ( u t − u t ) + + ct. By Definition 7.2 (iii), X satisfies the requirements in Remark 7.3. Let ( ˜ Y , ˜ Z, ˜ K ) and τ ∗ be defined as in Remark 7.3 (i). Then one can easily see that ˜ K t = 0 for t < τ ∗ , and thus2 cT = X ≤ ˜ Y = E L [ ˜ Y τ ∗ ] = E L [ X τ ∗ ] . This implies that E L [ X T ] = cT < cT ≤ E L [ X τ ∗ ] , and thus P [ τ ∗ < T ] >
0. On the other hand, apply Remark 7.3 (ii) to τ ∗ , then there exists ω ∗ ∈ Ω such that t ∗ := τ ∗ ( ω ∗ ) < T and Y t ∗ ( ω ∗ ) = ˜ Y t ∗ ( ω ∗ ). Thus X t ∗ ( ω ∗ ) = ˜ Y t ∗ ( ω ∗ ) = Y t ∗ ( ω ∗ ) = sup τ ∈T t ∗ E Lt ∗ [ X t ∗ ,ω ∗ τ ] . In particular, this implies that( u − u ) + ( t ∗ , ω ∗ ) + ct ∗ = X t ∗ ( ω ∗ ) ≥ E Lt ∗ [ X t ∗ ,ω ∗ T ] = cT, and thus ( u − u )( t ∗ , ω ∗ ) >
0. Assume without loss of generality that t ∗ ∈ [ h i ( ω ∗ ) , h i +1 ( ω ∗ )),and we may choose the h ∈ H t ∗ in Definition 7.2 (ii) small enough so that ( u − u ) t ∗ ,ω ∗ > t ∗ , h ). Now following the arguments in Proposition 5.3, in particular by replacing the( u ) t ∗ ,ω ∗ there with the ˜ u t ∗ ,ω ∗ ∈ C , (Λ t ∗ ) in Definition 7.2 (ii), we can easily obtain thedesired contradiction.We now turn to comparison and uniqueness. First, define u ( t, ω ) := inf (cid:8) ψ ( t, ) : ψ ∈ D ( t, ω ) (cid:9) , u ( t, ω ) := sup (cid:8) ψ ( t, ) : ψ ∈ D ( t, ω ) (cid:9) , (7.4)32here, for the L in (7.1) and denoting by L t,ω the corresponding operator on the shiftedspace with coefficient F t,ω , D ( t, ω ) := n ψ ∈ C , (Λ t ) : L t,ω ψ ≥ t, T ) × Ω t and ψ T ≥ ξ t,ω o , D ( t, ω ) := n ψ ∈ C , (Λ t ) : L t,ω ψ ≤ t, T ) × Ω t and ψ T ≤ ξ t,ω o . (7.5)Following the arguments in the consistency Theorem 3.16, one can easily show that u ≤ u. (7.6)A crucial step for our proof is to show that equality holds in the last inequality. Proposition 7.5
Let Assumption 4.3 hold with σ = I d , and F is uniformly continuous in ( t, ω ) under d ∞ . Then have u = u . We then have the following wellposedness result.
Theorem 7.6
Assume all the conditions in Proposition 7.5 hold true. (i)
Let u ∈ U be a viscosity subsolution and u ∈ U a viscosity supersolution of semilinearPPDE (7.1), in the sense of Definition 7.1, with u T ≤ ξ ≤ u T . Then u ≤ u on Λ . (ii) The semilinear PPDE (7.1) with terminal condition ξ has a unique viscosity solution u ∈ U C b (Λ) , in the sense of Definition 7.1. Proof
First by the partial comparison principle Proposition 7.4, we have u ≤ u and u ≤ u . Then Proposition 7.5 implies u ≤ u immediately, which implies further theuniqueness of viscosity solution. Finally by Proposition 4.4 we have the existence. Proof of Proposition 7.5.
Without loss of generality, we shall only prove u (0 , ) ≤ u (0 , ).In light of Proposition 3.14, we may also assume without loss of generality that F is nonincreasing in y . (7.7)For any ε >
0, we denote O ε := { x ∈ R d : | x | < ε } , O ε := { x ∈ R d : | x | ≤ ε } , ∂O ε := { x ∈ R d : | x | = ε } ; Q εt := [ t, T ) × O ε , Q εt := [ t, T ] × O ε , ∂Q εt := (cid:0) [ t, T ] × ∂O ε (cid:1) ∪ (cid:0) { T } × O ε (cid:1) . Moreover, we introduce the following space of discrete sequences:Π εn := n π n = ( t i , x i ) ≤ i ≤ n : t = 0 , x = , t i < t i +1 ∧ T and | x i | ≤ ε, for all i o . π n ∈ Π εn , and any ( t, x ) ∈ Q εt n , define h t,x,ε := T ∧ inf { s ≥ t : | B ts + x | = ε } , h t,x,εi +1 := T ∧ inf (cid:8) s ≥ h t,x,εi : | B ts − B t h t,x,εi | = ε (cid:9) , i ≥ . We denote by ˆ B ε,π n ,t,x ( ω ) the linear interpolation ( t i , P ij =0 x j ) ≤ i ≤ n and ( h t,x,εi ( ω ) , P nj =0 x j + x + B t h t,x,εi ( ω )) i ≥ , namely, abbreviating h i := h t,x,εi ,ˆ B ε,π n ,t,xs = i X j =0 x j + s − t i t i +1 − t i x i +1 , ≤ i ≤ n − , s ∈ [ t i , t i +1 ];ˆ B ε,π n ,t,xs = n X j =0 x j + x + B t h i + s − t i t i +1 − t i ( B t h i +1 − B t h i ) , i ≥ , s ∈ [ h i , h i +1 ] . Define θ εn (cid:0) π n ; ( t, x ) (cid:1) := Y ε,π n ,t,xt where, denoting h t,x,ε := t and omitting the superscripts ε,π n ,t,x , Y s = ξ ( ˆ B ) + Z Ts F (cid:16) r, X i ≥ ˆ B ·∧ h t,x,εi [ h t,x,εi , h t,x,εi +1 ) ( r ) , Y r , Z r (cid:17) dr − Z Ts Z r · dB r , P t -a.s.We remark that F (cid:16) r, X i ≥ ˆ B ·∧ h t,x,εi [ h t,x,εi , h t,x,εi +1 ) ( r ) , Y r , Z r (cid:17) = X i ≥ F (cid:16) r, ˆ B ·∧ h t,x,εi , Y r , Z r (cid:17) [ h t,x,εi , h t,x,εi +1 ) ( r )is well defined and F -adapted. One can easily prove that the deterministic function θ εn := θ εn ( π n ; · ) ∈ C , ( Q εt n ) and that θ εn is continuous on the boundary (cid:0) ( t n , T ) × ∂O ε (cid:1) ∪ (cid:0) { T }× O ε (cid:1) .Indeed, it satisfies the following standard PDE in Q εt n : − ∂ t θ εn − I d : D θ εn − F ( s, ω π n , ( T, ) , θ εn , Dθ εn ) = 0 in Q εt n , (7.8)with boundary conditions θ εn ( T, x ) = ξ ( ω π n , ( T,x ) ) , | x | ≤ ε ; θ εn ( t, x ) = θ εn +1 ( π n , ( t, x ); t, ) , t ∈ ( t n , T ) , x ∈ ∂O ε . where ω π n , ( T,x ) denotes the linear interpolation of ( t i , P ij =0 x j ) ≤ i ≤ n , ( T, P nj =0 x j + x ), andis deterministic.We now let h εi := h , ,εi , and ˆ B ε the linear interpolation of { ( h εi , B h εi ) i ≥ } . Define ψ ε ( t, ω ) := ∞ X n =0 θ εn (cid:0) (0 , ) , ( h εi , B h εi − B h εi − ) ≤ i ≤ n ; t, B t − B h εn (cid:1) [ h εn , h εn +1 ) ( t ) . h εi satisfies Definition 7.2 (i) and ψ ε satisfies Definition 7.2 (iii).Moreover, for each n ≥ ω ∈ Ω, and h εn ( ω ) ≤ t < h εn +1 ( ω ), we have ( t, ω t − ω h εn ( ω ) ) ∈ Q ε h εn ( ω ) . Set ( t , x ) := (0 , ), ( t j , x j ) := ( h εj ( ω ) , ω h εj ( ω ) − ω h εj − ( ω ) ), j = 1 , · · · , n , andlet δ > t, ω t − ω t n ) ∈ Q ε,δt n := [ t n , T − δ ) × O ε − δ . Onemay modify θ εn outside of Q ε,δt n to obtain ˜ θ εn (( t j , x j ) ≤ j ≤ n ; · ) ∈ C , ([ t n , T ] × R d ). Nowby setting h := inf { s ≥ t : ( s, B ts + ω t − ω t n ) / ∈ Q ε,δt n } < ( h εi +1 ) t,ω and ˜( ψ ε ) t,ω ( s, B t ) :=˜ θ εn (( t j , x j ) ≤ j ≤ n ; s, B ts + ω t − ω t n ), it is clear that ψ ε satisfies Definition 7.2 (ii). That is, ψ ε ∈ C , (Λ) with corresponding hitting times h εn .One may easily check further that ψ ε ( T, ω ) = ξ ( ˆ B ε ), and − ∂ t ψ ε − I d : ∂ ωω ψ ε − F (cid:16) s, X i ≥ ˆ B ε ·∧ h εi [ h εi , h εi +1 ) , ψ ε , ∂ ω ψ ε (cid:17) = 0 . (7.9)Notice that k ˆ B ε − B k T ≤ ε . Then k ξ ( ˆ B ε ) − ξ ( B ) k ≤ ρ (2 ε ) , (cid:12)(cid:12)(cid:12) F (cid:16) s, ∞ X i =0 ˆ B ε ·∧ h εi [ h εi , h εi +1 ) , y, z (cid:17) − F ( s, B, y, z ) (cid:12)(cid:12)(cid:12) ≤ ρ (2 ε ) . (7.10)Set ψ ε := ψ ε + ρ (2 ε )[1 + T − t ] , ψ ε := ψ ε − ρ (2 ε )[1 + T − t ] . Then ψ ε ≥ ψ ε , ψ ε ∈ C , (Λ) , ψ ε ( T, ω ) ≥ ψ ε ( T, ω ) + ρ (2 ε ) = ξ ( ˆ B ε ) + ρ (2 ε ) ≥ ξ ( B ) , and, by (7.7), (7.10), and (7.9) − ∂ t ψ ε − I d : ∂ ωω ψ ε − F ( s, B · , ψ ε , ∂ ω ψ ε ) ≥ − ∂ t ψ ε + ρ (2 ε ) − I d : ∂ ωω ψ ε − F ( s, B · , ψ ε , ∂ ω ψ ε ) ≥ − ∂ t ψ ε − I d : ∂ ωω ψ ε − F (cid:16) s, X i ≥ ˆ B ε ·∧ h εi [ h εi , h εi +1 ) , ψ ε , ∂ ω ψ ε (cid:17) = 0 . That is, ψ ε ∈ D (0 , ). Then u (0 , ) ≤ ψ ε (0 , ). Similarly, one can prove u (0 , ) ≥ ψ ε (0 , ).Thus u (0 , ) − u (0 , ) ≤ ψ ε (0 , ) − ψ ε (0 , ) = 2 ρ (2 ε )(1 + T ) . Send ε →
0, we obtain u (0 , ) ≤ u (0 , ). This, together with (7.6), implies that u (0 , ) = u (0 , ).We shall remark that the regularity of θ εn is quite subtle. In fact, in general θ εn may bediscontinuous on { t n } × ∂O ε . However, we do not need the continuity at those points inabove proof. 35 emark 7.7 The above proof of Proposition 7.5 takes advantage of the following threefacts in the semi-linear case, which do not hold anymore in the fully nonlinear case in [12].(i) The proof of partial comparison principle Proposition 7.4 uses the RBSDE theory,which applies implicitly the dominated convergence theorem. In the fully nonlinear case, inorder to avoid the application of the dominated convergence theorem, we shall modify thespace C , (Λ) slightly.(ii) The functions θ εn can be defined via BSDEs. In the fully nonlinear case, in particularwhen there is no representation formula, we shall prove the existence of θ εn satisfying (7.8)in an abstract way in [12].(iii) The functions θ εn are already in C , ( Q εt n ). In the fully nonlinear case, this istypically not true, and then we shall approximate θ εn by smooth functions. In this section we study the following first order PPDE: {− ∂ t u − G ( ., u, ∂ ω u ) } ( t, ω ) = 0 , ( t, ω ) ∈ Λ , (8.1)where G : Λ × R × R d → R verifies the following counterpart of Assumption 3.1: Assumption 8.1 (i) G ( · , y, z ) ∈ L (Λ) for any fixed ( y, z ) , and | G ( · , , ) | ≤ C . (ii) G is uniformly Lipschitz continuous in ( y, z ) with a Lipschitz constant L , and locallyuniformly continuous in ( t, ω ) under d ∞ , namely for any ( t, ω ) , there exists ρ t,ω such that sup y,z | G (˜ t, ˜ ω, y, z ) − G ( t, ω, y, z ) | ≤ ρ t,ω ( d ∞ (( t, ω ) , (˜ t, ˜ ω ))) . We note that here we require G to be locally uniformly continuous in ( t, ω ), which is strongerthan Assumption 3.1 (iv), but weaker than the uniform continuity required in Theorem 7.6or in [12]. The uniform regularity is used mainly for the proof of comparison principle.However, in this case we will employ some compactness arguments, and then continuityimplies locally uniform continuity.This PPDE was studied by Lukoyanov [18] by using the compactness of the set Ω tL defined below. In this section we explain briefly how to reduce our general formulation tothis case so as to adapt to Lukoyanov’s arguments. However, we emphasize again that thistype of compactness argument encounters a fundamental difficulty in the second order case,see Remark 8.2 below. We shall establish the wellposedness of second order equations inour accompanying paper [12] by using the optimal stopping result Theorem 2.6.36s is well known, a first order HJB equation corresponds to deterministic control. Sim-ilar to Section 7, in this case we may restrict our probability measures to degenerate oneswith β = , see Remark 3.10. We then define for any L > P tL := { P α : α : [ t, T ] → R d , | α | ≤ L } where dB ts = α s ds, P α -a.s. (8.2)and the corresponding nonlinear expectations E Lt , E Lt , and nonlinear optimal stopping prob-lems S Lt , S Lt , etc. in an obvious way. DenoteΩ tL := { ω ∈ Ω t : ω is Lipschitz continuous with Lipschitz constant L } , (8.3)and Λ tL := [ t, T ] × Ω tL , P t ∞ := ∪ L> P tL , Ω t ∞ := ∪ L> Ω tL , Λ t ∞ := ∪ L> Λ tL . As in [18], onecan easily check that P (Ω tL ) = 1 for all P ∈ P tL , Ω tL is compact and Ω t ∞ ⊂ Ω is dense under k · k T ;for s < t, ω ∈ Ω sL , ˜ ω ∈ Ω tL , we have ω ⊗ t ˜ ω ∈ Ω sL . (8.4) Remark 8.2
All the above properties are important in Lukoyanov’s approach for firstorder PPDEs, especially for proving the comparison principle. In the second order case, forexample for the semilinear PPDEs considered in Section 7, since P t (Ω t ∞ ) = 0, the set Ω t ∞ is not appropriate. One may consider to enlarge the space: for 0 < α < L >
0, letΩ tα,L := { ω ∈ Ω t : ω is H¨older- α continuous with H¨older constant L } , Ω tα, ∞ := ∪ L> Ω tα,L . Then for α < we have P t (Ω tα, ∞ ) = 1 , Ω tα,L is compact and Ω tα, ∞ ⊂ Ω is dense under k · k T However, the last property in (8.4) fails in this case:for s < t, ω ∈ Ω sα,L , ˜ ω ∈ Ω tα,L , in general ω ⊗ t ˜ ω / ∈ Ω sα,L . This is the main reason why we were unable to extend this approach to second order case.Notice that PPDE (8.1) involves only derivatives ∂ t u and ∂ ω u , we thus introduce the fol-lowing definitions. Definition 8.3
We say a process u ∈ C (Λ t ) is in C , (Λ t ) if there exist ∂ t u ∈ C (Λ t ) and ∂ ω u ∈ C (Λ t , R d ) such that, du s = ∂ t u s ds + ∂ ω u s · dB ts , t ≤ s ≤ T, P -a.s. for all P ∈ P t ∞ . (8.5)37t is obvious that ∂ t u and ∂ ω u , if they exist, are unique on Λ t ∞ . Then, since Ω t ∞ ⊂ Ω isdense and ∂ t u , ∂ ω u are continuous, we see that they are unique in Λ t .For all u ∈ L (Λ), ( t, ω ) ∈ Λ with t < T , and
L >
0, define A L u ( t, ω ) := n ϕ ∈ C , (Λ t ) : ( ϕ − u t,ω ) t = 0 = S Lt (cid:2) ( ϕ − u t,ω ) ·∧ h (cid:3) for some h ∈ H t o , A L u ( t, ω ) := n ϕ ∈ C , (Λ t ) : ( ϕ − u t,ω ) t = 0 = S Lt (cid:2) ( ϕ − u t,ω ) ·∧ h (cid:3) for some h ∈ H t o . (8.6)We then define viscosity solutions exactly as in Definition 3.7. We may easily check thatall the results in this paper, when reduced to first order PPDEs, still hold under this newdefinition. In particular, the examples in Section 4.1 are still valid, and the value functionof the deterministic control problem is a viscosity solution to the corresponding first orderpath dependent HJB equation.We remark that our Definition 8.3 of derivatives is equivalent to Lukoyanov’s notionof derivatives, which is defined via Taylor expansion. Moreover, instead of using nonlinearexpectation as in (8.6), Lukoyanov uses test functions ϕ such that ϕ − u attains pathwiselocal maximum (or minimum) at ( t, ω ). So, modulus some minor technical difference, inspirit a viscosity solution (resp. subsolution, supersolution) in our sense is equivalent toa viscosity solution (resp. subsolution, supersolution) in Lukoyanov’s sense. Indeed, ourfollowing comparison principle and uniqueness result for first order PPDEs follows fromalmost the same arguments as that of [18]. We nevertheless sketch a proof for completeness. Theorem 8.4
Let Assumption 8.1 hold true. Let u ∈ U (resp. u ∈ U ) be a viscositysubsolution (resp. supersolution) of PPDE (8.1) , in the sense of Definition 3.7 and modifiedin the context of this section. If u ( T, · ) ≤ u ( T, · ) on Ω , then u ≤ u on Λ . Proof
Let u (resp. u ) be a viscosity L -subsolution (resp. L -supersolution) for some L ≥ L . Assume by contradiction that c := u − u >
0. For ε >
0, defineΦ ε ( t, ω ; ˜ t, ˜ ω ) := u ( t, ω ) − u (˜ t, ˜ ω ) − c T [2 T − t − ˜ t ] − ε Ψ ε ( t, ω ; ˜ t, ˜ ω )where Ψ ε ( t, ω ; ˜ t, ˜ ω ) := | t − ˜ t | + | ω t − ˜ ω ˜ t | + R T | ω t ∧ r − ˜ ω ˜ t ∧ r | dr. Then c ε := sup ( t,ω ;˜ t, ˜ ω ) ∈ (Λ L ) Φ ε ( t, ω ; ˜ t, ˜ ω ) ≥ Φ ε (0 , ; 0 , ) = c > . By compactness of thespace, there exists ( t ε , ω ε ; ˜ t ε , ˜ ω ε ) ∈ (Λ L ) such that Φ ε ( t ε , ω ε ; ˜ t ε , ˜ ω ε ) = c ε . Note that u isbounded from above and u is bounded from below, then one can easily see thatΨ ε ( t ε , ω ε ; ˜ t ε , ˜ ω ε ) ≤ Cε, which implies lim ε → d ∞ ( t ε , ω ε ; ˜ t ε , ˜ ω ε ) = 0 . t ε ≤ ˜ t ε and since Φ ε ( t ε , ω ε ; ˜ t ε , ˜ ω ε ) ≥ Φ ε (˜ t ε , ˜ ω ε ; ˜ t ε , ˜ ω ε ), it follows from (2.2) that0 ≤ ε Ψ ε ( t ε , ω ε ; ˜ t ε , ˜ ω ε ) ≤ u ( t ε , ω ε ) − u (˜ t ε , ˜ ω ε ) − c T [˜ t ε − t ε ] → . Now, if ˜ t ε = T , then u (˜ t ε , ˜ ω ε ) ≥ u (˜ t ε , ˜ ω ε ) and thus c ≤ Φ ε (˜ t ε , ˜ ω ε ; ˜ t ε , ˜ ω ε ) ≤ u ( t ε , ω ε ) − u (˜ t ε , ˜ ω ) − c T [˜ t ε − t ε ] − ε Ψ ε ( t, ω ; ˜ t, ˜ ω ) → . This is a contradiction. Thus we have t ε ≤ ˜ t ε < T (or ˜ t ε ≤ t ε < T ) when ε is small enough.We now define test functions: ϕ ( t, ω ) := u (˜ t ε , ˜ ω ε ) + c T [2 T − t − ˜ t ε ] + 1 ε Ψ ε ( t, ω ; ˜ t ε , ˜ ω ε ) − c ε ,ϕ (˜ t, ˜ ω ) := u ( t ε , ω ε ) − c T [2 T − t ε − ˜ t ] − ε Ψ ε ( t ε , ω ε ; ˜ t, ˜ ω ) + c ε . It is straightforward to check that ϕ ∈ A L u ( t ε , ω ε ), ϕ ∈ A L u (˜ t ε , ˜ ω ε ), and ∂ t ϕ ( t ε , ω ε ) = − c T + 2 ε [ t ε − ˜ t ε ] , ∂ ω ϕ ( t ε , ω ε ) = 2 ε h [ ω εt ε − ˜ ω ε ˜ t ε ] + Z Tt ε [ ω εt ε − ˜ ω ε ˜ t ε ∧ r ] dr i ; ∂ t ϕ (˜ t ε , ˜ ω ε ) = c T − ε [˜ t ε − t ε ] , ∂ ω ϕ (˜ t ε , ˜ ω ε ) = − ε h [˜ ω ε ˜ t ε − ω εt ε ] + Z T ˜ t ε [˜ ω ε ˜ t ε ] − ω εt ε ∧ r ] dr i . Note that 0 < Φ ε (˜ t ε , ˜ ω ε ; ˜ t ε , ˜ ω ε ) ≤ u ( t ε , ω ε ) − u (˜ t ε , ˜ ω ). As standard, we may assume withoutloss of generality that G is decreasing in y . Then it follows from the viscosity property of u , u that c T = ∂ t ϕ (˜ t ε , ˜ ω ε ) − ∂ t ϕ ( t ε , ω ε ) ≤ G ( · , u , ∂ ω ϕ )(˜ t ε , ˜ ω ε ) − G ( · , u , ∂ ω ϕ )( t ε , ω ε ) ≤ G (cid:16) ˜ t ε , ˜ ω ε , u ( t ε , ω ε ) , ∂ ω ϕ (˜ t ε , ˜ ω ε ) (cid:17) − G (cid:16) t ε , ω ε , u ( t ε , ω ε ) , ∂ ω ϕ ( t ε , ω ε ) (cid:17) ≤ sup y,z | G (˜ t ε , ˜ ω ε , y, z ) − G ( t ε , ω ε , y, z ) | + L | ∂ ω ϕ (˜ t ε , ˜ ω ε ) − ∂ ω ϕ ( t ε , ω ε ) | . By the compactness of Ω L , { ( t ε , ω ε ) , ε > } has a limit point ( t ∗ , ω ∗ ) ∈ Λ L , and we mayassume without loss of generality that lim ε → d ∞ (( t ε , ω ε ) , ( t ∗ , ω ∗ )) = 0. Then it follows from thelocally uniform continuity of G that lim ε → sup y,z | G (˜ t ε , ˜ ω ε , y, z ) − G ( t ε , ω ε , y, z ) | = 0. Moreover, | ∂ ω ϕ (˜ t ε , ˜ ω ε ) − ∂ ω ϕ ( t ε , ω ε ) | = 2 ε (cid:12)(cid:12)(cid:12) Z t ε ∨ ˜ t ε t ε ∧ ˜ t ε [ ω ετ ε ∧ r − ˜ ω ε ˜ t ε ∧ r ] dr (cid:12)(cid:12)(cid:12) ≤ ε h | t ε − ˜ t ε || ω ετ ε − ˜ ω ε ˜ t ε | + Z t ε ∨ ˜ t ε t ε ∧ ˜ t ε [ | ω εt ε ∧ r − ω εt ε | + | ˜ ω ε ˜ t ε ∧ r − ˜ ω ε ˜ t ε | ] dr i ≤ Cε h | t ε − ˜ t ε | + | ω ετ ε − ˜ ω ε ˜ t ε | i → . This implies c T ≤
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