Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II
aa r X i v : . [ m a t h . P R ] S e p The Annals of Probability (cid:13)
Institute of Mathematical Statistics, 2016
VISCOSITY SOLUTIONS OF FULLY NONLINEAR PARABOLICPATH DEPENDENT PDES: PART II
By Ibrahim Ekren, Nizar Touzi and Jianfeng Zhang ETH Zurich, Ecole Polytechnique Paris and University of SouthernCalifornia
In our previous paper [Ekren, Touzi and Zhang (2015)], we in-troduced a notion of viscosity solutions for fully nonlinear path-dependent PDEs, extending the semilinear case of Ekren et al. [
Ann.Probab. (2014) 204–236], which satisfies a partial comparison re-sult under standard Lipshitz-type assumptions. The main result ofthis paper provides a full, well-posedness result under an additionalassumption, formulated on some partial differential equation, definedlocally by freezing the path. Namely, assuming further that suchpath-frozen standard PDEs satisfy the comparison principle and thePerron approach for existence, we prove that the nonlinear path-dependent PDE has a unique viscosity solution. Uniqueness is impliedby a comparison result.
1. Introduction.
This paper is the continuation of our accompanyingpapers [7, 8]. The main objective of this series of three papers is the following,fully nonlinear parabolic path-dependent partial differential equation: {− ∂ t u − G ( · , u, ∂ ω u, ∂ ωω u ) } ( t, ω ) = 0 , ( t, ω ) ∈ [0 , T ) × Ω . (1.1)Here Ω consists of continuous paths ω on [0 , T ] starting from the origin, G is aprogressively measurable generator and the path derivatives ∂ t u, ∂ ω u, ∂ ωω u are defined through a functional Itˆo formula, initiated by Dupire [5]; seealso Cont and Fournie [3]. Such equations were first proposed by Peng [16], Received May 2013; revised September 2014. Supported by the Chair
Financial Risks of the
Risk Foundation sponsored by Soci´et´eG´en´erale, and the Chair
Finance and Sustainable Development sponsored by EDF andCalyon. Supported in part by NSF Grant DMS-10-08873.
AMS 2000 subject classifications.
Key words and phrases.
Path dependent PDEs, nonlinear expectation, viscosity solu-tions, comparison principle, Perron’s approach.
This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in
The Annals of Probability ,2016, Vol. 44, No. 4, 2507–2553. This reprint differs from the original inpagination and typographic detail. 1
I. EKREN, N. TOUZI AND J. ZHANG and they provide a convenient language for many problems arising in non-Markovian, or say path dependent framework, with typical examples, in-cluding martingales, backward stochastic differential equations, second-orderBSDEs and backward stochastic PDEs. In particular, the value functions ofstochastic controls and stochastic differential games with both drift and dif-fusion controls can be characterized as the solution of the corresponding pathdependent PDEs. This extends the classical results in Markovian frameworkto non-Markovian ones. We refer to [8] and [17] for these connections.A path dependent PDE can rarely have a classical solution. We thus turnto the notion of viscosity solutions, which had great success in the finitedimensional case. There have been numerous publications on viscosity solu-tions of PDEs, both in theory and in applications, and we refer to the classi-cal references [4] and [9]. In our infinite dimensional case, the major difficultyis that the underlying state space Ω is not locally compact, and thus manytools from the standard PDE viscosity theory do not apply to the presentcontext. In our earlier paper [6], which studies semilinear path-dependentPDEs, we replace the pointwise extremality in the standard definition ofviscosity solution in PDE literature with the corresponding extremality inthe context of an optimal stopping problem under a nonlinear expectation E . More precisely, we introduce a set of smooth test processes ϕ , which aretangent from above or from below, to the processes of interest u in the senseof the following nonlinear optimal stopping problems:sup τ E [( ϕ − u ) τ ] , inf τ E [( ϕ − u ) τ ](1.2) where E := sup P ∈P E P , E := inf P ∈P E P . Here τ ranges over a convenient set of stopping times, and P is an appropri-ate set of probability measures. The replacement of the pointwise tangencyby the tangency in the sense of the last optimal stopping problem is the keyingredient needed to bypass the local compactness of the underlying spacein the standard viscosity solution theory (or the Hilbert structure, whichallows us to access local compactness by finite realization approximation ofthe space). Indeed, the Snell envelope characterization of the solution of theoptimal stopping problem allows us to find a “point of tangency.” Interest-ingly, the structure of the underlying space does not play any role, and thestandard first and second-order conditions of maximality in the standardoptimization theory has the following beautiful counterpart in the optimalstopping problem: the supermartingale property (negative drift; notice thatdrift is related to the second derivative) of the Snell envelope and the mar-tingale property (zero drift) up the optimal stopping time (first hitting ofthe obstacle/reward process). ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II In [6], we proved existence and uniqueness of viscosity solutions for semi-linear path-dependent PDEs. In particular, the unique viscosity solution isconsistent with the solution to the corresponding backward SDE.In [6], all probability measures in the class P are equivalent, and conse-quently P is dominated by one measure. In our fully nonlinear context, theclass P becomes nondominated, consisting of mutually singular measuresinduced by certain linearization of the nonlinear generator G . This causesanother major difficulty of the project: the dominated convergence theo-rem fails under E P . To overcome this, we need some strong regularity forthe involved processes, and thus we require some rather sophisticated esti-mates. In particular, the corresponding optimal stopping problem becomesvery technical and is established in a separate paper [7]. We remark thatthe weak compactness of the class P plays a very important role in thesearguments.In [8], we introduced the appropriate class P for fully nonlinear path de-pendent PDEs (1.1) and the corresponding notion of viscosity solutions. Weinvestigated the connection between our new notion and many other equa-tions in the existing literature of stochastic analysis, for example, backwardSDEs, second-order BSDEs and backward SPDEs. Moreover, we provedsome basic properties of viscosity solutions, including the partial comparisonprinciple; that is, for a viscosity subsolution u and a classical supersolution u , if u T ≤ u T , then u ( t, ω ) ≤ u ( t, ω ) for all ( t, ω ) ∈ [0 , T ] × Ω.In this paper we prove our main result, the comparison principle of vis-cosity solutions. That is, for a viscosity subsolution u and a viscosity su-persolution u , if u T ≤ u T , then u ( t, ω ) ≤ u ( t, ω ) for all ( t, ω ) ∈ [0 , T ] × Ω.Again, due to the lack of local compactness and now also due to our newdefinition of viscosity solutions, the standard approach in PDE literature,namely the doubling variable technique combined with Ishii’s lemma, doesnot seem to work here. Our strategy is as follows: We start from the abovepartial comparison established in [8], but we slightly weaken the smoothrequirement of the classical (semi-)solutions. Let u denote the infimum ofthe classical supersolution and u , the supremum of classical subrsolutions,satisfying appropriate terminal conditions. Then the partial comparison im-plies u ≤ u and u ≤ u . Thus the comparison will be a direct consequenceof the following claim: u = u. (1.3)Then clearly our focus is (1.3). We first remark that due to the failureof the dominated convergence theorem under our new E P , the approachin [6] does not work here. In this paper, we shall follow the alternativeapproach proposed in [8], Section 7, which is also devoted to semilinearpath-dependent PDEs. However, as explained in [8], Remark 7.7, there are I. EKREN, N. TOUZI AND J. ZHANG several major difficulties in the fully nonlinear context, and novel ideas areneeded.Note that (1.3) is more or less equivalent to constructing some classi-cal supersolution u ε and classical subsolution u ε , for any ε >
0, such thatlim ε → [ u ε − u ε ] = 0. Our main tool is the following local path-frozen PDE:for any ( t, ω ) ∈ [0 , T ) × Ω, − ∂ t v ( s, x ) − g t,ω ( s, v, Dv, D v ) = 0 , (1.4) s ∈ [ t, t + ε ] , x ∈ R d such that | x | ≤ ε, where g t,ω ( s, y, z, γ ) := G ( s, ω ·∧ t , y, z, γ ) . Here D and D denote the gradient and Hessian of v with respect to x ,respectively, and we emphasize that g t,ω is a deterministic function, andthus (1.4) is a standard PDE. We shall assume that the above PDE has aunique viscosity solution (in standard sense), which can be approximatedby classical subsolutions and classical supersolutions. One sufficient condi-tion is that after certain smooth mollification of g t,ω , the above local PDEwith smooth boundary condition has a classical solution. We then use thisclassical solution to construct the desired u ε and u ε .We remark that this approach is very much like Perron’s approach instandard PDE viscosity theory. However, there are two major differences:First, in the standard Perron approach, u and u are the extremality of vis-cosity semi-solutions, while here they are the extremality of classical semi-solutions. This requires the smoothness of the above u ε and u ε and thusmakes their construction harder. More importantly, the standard Perronapproach assumes the comparison principle and uses it to obtain the ex-istence of viscosity solutions, while we use (1.3) to prove the comparisonprinciple. Thus the required techniques are quite different.Once we have the comparison principle, then following the idea of thestandard Perron approach, we see u = u is indeed the unique viscosity solu-tion of the path-dependent PDE, so we have both existence and uniqueness.Our result covers quite general classes of path-dependent PDEs, includingthose not accessible in the existing literature of stochastic analysis. One par-ticular application is the existence of the game value for a path-dependentzero sum stochastic differential game, due to our well-posedness result of thepath-dependent Bellman–Isaacs equation; see Pham and Zhang [17]. We alsorefer to Henry-Labordere, Tan and Touzi [12] and Zhang and Zhuo [18] forapplications of our results to numerical methods for path-dependent PDEs.We also note that there is potentially an alternative way to prove the com-parison principle. Roughly speaking, given a viscosity subsolution u and aviscosity supersolution u , if we could find certain smooth approximations u i,ε , close to u i , such that u ,ε is a classical subsolution and u ,ε is a classi-cal supersolution, then it follows from partial comparison (actually classical ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II comparison) that u ,ε ≤ u ,ε , which leads to the desired comparison immedi-ately by passing ε to 0. Indeed, in PDE literature the convex/concave con-volution plays this role. However, in the path-dependent setting, we did notsucceed in finding appropriate approximations u i,ε which satisfy the desiredsemi-solution property. In our current approach, instead of approximatingthe (semi-)solution directly, we approximate the path-frozen PDE by molli-fying its generator g t,ω . The advantage of our approach is that provided themollified path-frozen PDE has a classical solution, it will be straightforwardto check that the constructed u ε and u ε are classical semi-solutions.The price of our approach, however, is that we need classical solutionsof fully nonlinear PDEs. Partially for this purpose, in the present paper weassume that G is uniformly nondegenerate, which is undesirable in viscositytheory, and for path-dependent Bellman–Isaacs equations, we can only dealwith the lower dimensional ( d = 1 or 2) problems. We shall investigate theseimportant problems and explore further possible direct approximations of u i as mentioned above, in our future research.The rest of the paper is organized as follows. Section 2 introduces thegeneral framework and recalls the definition of viscosity solutions introducedin our accompanying paper [8]. Section 3 collects all assumptions neededthroughout the paper. The main results are stated in Section 4, where we alsooutline strategy of proof. In particular, the existence and comparison resultsfollow from the partial comparison principle, the consistency of the Perronapproach and the viscosity property of the postulated solution of the PPDE.These crucial results are proved in Sections 5, 6 and 7, respectively. Finally,Section 8 provides some sufficient conditions for our main assumption, underwhich our well-posedness result is established, together with some concludingremarks.
2. Preliminaries.
In this section, we recall the setup and the notation of[8].2.1.
The canonical spaces.
Let Ω := { ω ∈ C ([0 , T ] , R d ) : ω = } be theset 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 componentsare all equal to 0. Let S d denote the set of d × d symmetric matrices, and x · x ′ := d X i =1 x i x ′ i for any x, x ′ ∈ R d ,γ : γ ′ := tr[ γγ ′ ] for any γ, γ ′ ∈ S d . I. EKREN, N. TOUZI AND J. ZHANG
We define a semi-norm 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 ∞ (( t, ω ) , ( t ′ , ω ′ )) := | t − t ′ | + k ω ·∧ t − ω ′·∧ t ′ k T . (2.1)Then (Ω , k · k T ) is a Banach space, and (Λ , d ∞ ) is a complete pseudomet-ric space. We shall denote by L ( F T ) and L (Λ) the collection of all F T -measurable random variables and F -progressively measurable processes, re-spectively. Let C (Λ) [resp., UC(Λ)] be the subset of L (Λ) whose elementsare continuous (resp., uniformly continuous) in ( t, ω ) under d ∞ . The corre-sponding subsets of bounded processes are denoted by C b (Λ) and UC b (Λ).Finally, L (Λ , R d ) denote the space of R d -valued processes with entries in L (Λ), and we define similar notation for the spaces C , C b , UC and UC b .We next introduce the shifted spaces. Let 0 ≤ s ≤ t ≤ T . − Let Ω t := { ω ∈ C ([ t, T ] , R d ) : ω t = } be the shifted canonical space; B t the shifted canonical process on Ω t ; F t the shifted filtration generated by B t , P t the Wiener measure on Ω t , and Λ t := [ t, T ] × Ω t . − Define k · k st on Ω s and d s ∞ on Λ s in the spirit of (2.1), and the sets L (Λ t )etc. in an obvious way. − For ω ∈ Ω s and ω ′ ∈ Ω t , define the concatenation path ω ⊗ t ω ′ ∈ Ω s by( ω ⊗ t ω ′ )( r ) := ω r [ s,t ) ( r ) + ( ω t + ω ′ r ) [ t,T ] ( r ) for all r ∈ [ s, T ] . − Let ξ ∈ L ( F sT ) and X ∈ L (Λ s ). For ( t, ω ) ∈ Λ s , define ξ t,ω ∈ L ( F tT ) and X t,ω ∈ L (Λ t ) by ξ t,ω ( ω ′ ) := ξ ( ω ⊗ t ω ′ ) , X t,ω ( ω ′ ) := X ( ω ⊗ t ω ′ ) , for all ω ′ ∈ Ω t . It is clear that for any ( t, ω ) ∈ Λ and any u ∈ C (Λ), we have u t,ω ∈ C (Λ t ).The other spaces introduced before enjoy the same property.We denote by T the set of F -stopping times, and by H ⊂ T , the subsetof those hitting times h of the form h := inf { t : B t / ∈ O } ∧ t , (2.2)for some 0 < t ≤ T , and some open and convex set O ⊂ R d containing .The set H will be important for our optimal stopping problem, which iscrucial for the comparison and the stability results. 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. (2.3)Define T t and H t in the same spirit. For any τ ∈ T (resp., h ∈ H ) and any( t, ω ) ∈ Λ such that t < τ ( ω ) [resp., t < h ( ω )], it is clear that τ t,ω ∈ T t (resp., h t,ω ∈ H t ).Finally, the following types of regularity will be important in our frame-work: ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II Definition 2.1.
Let u ∈ L (Λ).(i) We say u is right continuous in ( t, ω ) under d ∞ if for any ( t, ω ) ∈ Λand any ε >
0, there exists δ > s, ˜ ω ) ∈ Λ t satisfying d ∞ (( s, ˜ ω ) , ( t, )) ≤ δ , we have | u t,ω ( s, ˜ ω ) − u ( t, ω ) | ≤ ε .(ii) We say u ∈ U if u is bounded from above, right continuous in ( t, ω )under d ∞ and there exists a modulus of continuity function ρ such that forany ( t, ω ) , ( t ′ , ω ′ ) ∈ Λ, u ( t, ω ) − u ( t ′ , ω ′ ) ≤ ρ ( d ∞ (( t, ω ) , ( t ′ , ω ′ ))) whenever t ≤ t ′ . (2.4)(iii) We say u ∈ U if − u ∈ U .The progressive measurability of u implies that u ( t, ω ) = u ( t, ω ·∧ t ), andit is clear that U ∩ U = UC b (Λ). We also recall from [7] Remark 3.2 thatcondition (2.4) implies that u has left-limits and positive jumps.2.2. Capacity and nonlinear expectation.
For every constant
L >
0, wedenote by P L the collection of all continuous semimartingale measures 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. Aprobability measure P ∈ P L means that there exists an extension Q of P on˜Ω such that B = A + M A is absolutely continuous, M is a martingale , | α P | ≤ L,
12 tr(( β P ) ) ≤ L where α P t := dA t dt , β P t := r d h M i t dt (2.5) Q -a.s.Similarly, for any t ∈ [0 , T ), we may define P tL on Ω t and P t ∞ := S L> P tL .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 all F tT -measurable r.v. ξ withsup P ∈P tL E P [ | ξ | ] < ∞ . The following nonlinear expectation will play a cru-cial 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 ](2.7) for all ξ ∈ L ( F tT , P tL ) . I. EKREN, N. TOUZI AND J. ZHANG
Definition 2.2.
Let X ∈ L (Λ) satisfy X t ∈ L ( F t , P L ) for all 0 ≤ t ≤ T . We say that X is an E L -supermartingale (resp., submartingale, martin-gale) if, for any ( t, ω ) ∈ Λ and any τ ∈ T t , E Lt [ X t,ωτ ] ≤ (resp., ≥ , =) X t ( ω ).We now state the Snell envelope characterization of optimal stoppingunder the above nonlinear expectation operators. Given a bounded process X ∈ L (Λ), consider the nonlinear optimal stopping problem S Lt [ X ]( ω ) := sup τ ∈T t E Lt [ X t,ωτ ] and S Lt [ X ]( ω ) := inf τ ∈T t E Lt [ X t,ωτ ] , (2.8) ( t, ω ) ∈ Λ . By definition, we have S L [ X ] ≥ X and S LT [ X ] = X T . Theorem 2.3 ([7]).
Let X ∈ U be bounded, h ∈ H and set b X t := X t { t< h } + X h − { t ≥ h } . Define Y := S L [ b X ] 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 -martingaleon [0 , τ ∗ ] . Consequently, τ ∗ is an optimal stopping time. We remark that the nonlinear Snell envelope Y is continuous in [0 , h ) andhas left limit at h . However, in general Y may have a negative left jump at h . 2.3. The path derivatives.
We define the path derivatives via the func-tional Itˆo formula, initiated by Dupire [5].
Definition 2.4.
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 u dt + ∂ ω u · dB t + ∂ ωω u : d h B i t , ≤ t ≤ T, P -a.s.(2.9)We remark that the above ∂ t u , ∂ ω u and ∂ ωω u , if they exist, are unique,and thus are called the time derivative, first-order and second-order spacederivatives of u , respectively. In particular, it holds that ∂ t u ( t, ω ) = lim h ↓ h [ u ( t + h, ω ·∧ t ) − u ( t, ω )] . (2.10)We refer to [8], Remark 2.9, and [2], Remarks 2.3, 2.4, for various discus-sions on these path derivatives, especially on their comparison with Dupire’s ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II original definition. See also Remark 4.5 below. 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 ∂ ω ( u t,ω ) = ( ∂ ω u ) t,ω , ∂ ωω ( u t,ω ) = ( ∂ ωω u ) t,ω .For technical reasons, we shall extend the space C , (Λ) slightly as follows. Definition 2.5.
Let t ∈ [0 , T ], u : Λ t → R . We say u ∈ C , (Λ t ) if thereexist an increasing sequence of { h i , i ≥ } ⊂ T t , a partition { E ij , j ≥ } ⊂ F t h i of Ω t for each i , a constant n i ≥ i , and ϕ ijk ∈ UC b (Λ), ψ ijk ∈ C , (Λ) ∩ UC b (Λ) for each ( i, j ) and 1 ≤ k ≤ n i , such that, denoting h := t , E := Ω t :(i) for each i and ω , h h i ,ωi +1 ∈ H h i ( ω ) whenever h i ( ω ) < T , the set { i : h i ( ω )
By setting h := T , n := 1, ϕ := 1 and ψ := u , we see that C , (Λ t ) ⊂ C , (Λ t ).2.4. Fully nonlinear path dependent PDEs.
Following the accompanyingpaper [8], we continue our study of the following fully nonlinear parabolicpath-dependent partial differential equation (PPDE, for short): L u ( t, ω ) := {− ∂ t u − G ( · , u, ∂ ω u, ∂ ωω u ) } ( t, ω ) = 0 , ( t, ω ) ∈ Λ , (2.14)where the generator G : Λ × R × R d × S d → R satisfies the conditions reportedin Section 3.For any u ∈ L (Λ), ( t, ω ) ∈ [0 , T ) × Ω and
L >
0, define A L u ( t, ω ) := { ϕ ∈ C , (Λ t ) : ( ϕ − u t,ω ) t = 0 = S Lt [( ϕ − u t,ω ) ·∧ h ]for some h ∈ H t } , (2.15) A L u ( t, ω ) := { ϕ ∈ C , (Λ t ) : ( ϕ − u t,ω ) t = 0 = S Lt [( ϕ − u t,ω ) ·∧ h ]for some h ∈ H t } , where S L and S L are the nonlinear Snell envelopes defined in (2.8). Definition 2.6. (i) Let
L >
0. We say u ∈ U (resp., U ) is a viscosity L -subsolution (resp., L -supersolution) of PPDE (2.14) if, for any ( t, ω ) ∈ [0 , T ) × Ω and any ϕ ∈ A L u ( t, ω ) [resp., ϕ ∈ A L u ( t, ω )], {− ∂ t ϕ − G t,ω ( · , ϕ, ∂ ω ϕ, ∂ ωω ϕ ) } ( t, ) ≤ (resp., ≥ ) 0 . (ii) We say u ∈ U (resp., U ) is a viscosity subsolution (resp., supersolution)of PPDE (2.14) if u is viscosity L -subsolution (resp., L -supersolution) ofPPDE (2.14) for some L > u ∈ UC b (Λ) is a viscosity solution of PPDE (2.14) if it is botha viscosity subsolution and a viscosity supersolution.As pointed out in [8] Remark 3.11(i), without loss of generality in (2.15),we may always set h = h tε for some small ε > h tε := inf { s > t : | B ts | ≥ ε } ∧ ( t + ε ) . (2.16)
3. Assumptions.
This section collects all of our assumptions on the non-linearity G , the terminal condition ξ and the underlying path-frozen PDE. ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II Assumptions on the nonlinearity and terminal conditions.
We firstneed the conditions on the nonlinearity G as assumed in [8]. 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 uniformly Lipschitz continuous in ( y, z, γ ), with a Lipschitz con-stant L ;(iii) for any ( y, z, γ ), G ( · , y, z, γ ) is right continuous in ( t, ω ) under d ∞ ;(iv) G is elliptic, that is, nondecreasing in γ .Our main well-posedness result requires the following strengthening of(iii) and (iv) above: Assumption 3.2. (i) G is uniformly continuous in ( t, ω ) under d ∞ witha modulus of continuity function ρ .(ii) For each ω , G is uniformly elliptic. That is, there exits a constant c > G ( · , γ ) − G ( · , γ ) ≥ c tr( γ − γ ) for any γ ≥ γ .Condition (i) is needed for our uniform approximation of G below; inparticular it is used (only) in the proof of Lemma 6.4. We should point outthough, for the semi-linear PPDE and the path-dependent HJB consideredin [8], Section 4, this condition is violated when σ depends on ( t, ω ). However,this is a technical condition due to our current approach for uniqueness.Condition (ii) is used to ensure the existence of the viscosity solution for thepath-frozen PDE (3.3) below. See also Example 4.7.Our first condition on the terminal condition ξ is the following: Assumption 3.3. ξ ∈ L ( F T ) is bounded and uniformly continuous in ω under k · k T , with the same modulus of continuity function ρ as in As-sumption 3.2(i). Remark 3.4.
The continuity of a random variable in terms of ω seemsless natural in stochastic analysis literature. However, since by nature we arein the weak formulation setting, such continuity is in fact natural in manyapplications. This is emphasized in the two following examples: − Let V := E P [ g ( X σ · )], for some bounded function g : Ω −→ R , and somebounded progressively measurable process σ , with dX σt = σ t dB t , P -a.s.In the weak formulation, we define P σ as the probability measure inducedby the process X σ , and we re-write V := E P σ [ g ( B · )]. Thus the uniformcontinuity requirement reduces to that of the function g . I. EKREN, N. TOUZI AND J. ZHANG − Similarly, the stochastic control problem in strong formulation V :=sup σ ≤ σ ≤ σ E P [ g ( X σ · )] for some constants 0 ≤ σ ≤ σ , may be expressed inthe weak formulation as V = sup σ ≤ σ ≤ σ E P σ [ g ( B · )], thus reducing the uni-form continuity requirement of the terminal data to that of the function g .Our next assumption is a purely technical condition needed in our proofof uniqueness. To be precise, it will be used only in the proof of Lemma 6.3below to ensure the function θ εn constructed there is continuous in its pa-rameter π n . When we have a representation for the viscosity solution, forexample, in the semilinear case in [8], Section 7, we may construct the θ εn explicitly and thus avoid the following assumption:For all ε > n ≥ ≤ T < T ≤ T , denoteΠ εn ( T , T ) := { π n = ( t i , x i ) ≤ i ≤ n : T < t < · · · < t n < T , (3.1) | x i | ≤ ε for all 1 ≤ i ≤ n } . For all π n ∈ Π εn ( T , T ), we denote by ω π n ∈ Ω T the linear interpolation of( T , ), ( t i , P ij =1 x j ) ≤ i ≤ n , and ( T, P nj =1 x j ). Assumption 3.5.
There exist 0 = T < · · · < T N = T such that for each i = 0 , . . . , N −
1, for any ε small, any n and any ω ∈ Ω, ˜ ω ∈ Ω T i +1 , the func-tions π n ξ ( ω ⊗ T i ω π n ⊗ T i +1 ˜ ω ) and π n G ( t, ω ⊗ T i ω π n ⊗ T i +1 ˜ ω, y, z, γ )are uniformly continuous in Π εn ( T i , T i +1 ), uniformly on t ≥ T i +1 , ( y, z, γ ) ∈ R × R d × S d and ˜ ω ∈ Ω T i +1 .We note that the uniform continuity of ξ and G implies that the abovemappings are continuous in π n ∈ Π εn ( T i , T i +1 ), but not necessarily uniformlycontinuous. In particular, they may not have limits on the boundary ofΠ εn ( T i , T i +1 ), namely when t i = t i +1 but x i = x i +1 . We conclude this subsec-tion with a sufficient condition for Assumption 3.5, where for ω ∈ Ω, we usethe notation ω t := max s ≤ t ω s and ω t := min s ≤ t ω s , defined componentwise. Lemma 3.6.
Let ξ ( ω ) = g ( ω T , . . . , ω T N , ω T , . . . , ω T N , ω T , . . . , ω T N , ω ) forsome T < T < · · · < T N = T , and some bounded uniformly continuousfunction ( θ, ω ) ∈ R dN × Ω g ( θ, ω ) ∈ R . Assume further that for all θ , i and ω ∈ Ω , there exists a modulus of continuity function ρ and p > (whichmay depend on the above parameters), such that | g ( θ, ω ⊗ T i ω ⊗ T i +1 ˜ ω ) − g ( θ, ω ⊗ T i ω ⊗ T i +1 ˜ ω ) | ≤ ρ (cid:18)Z T i +1 T i | ω t − ω t | p dt (cid:19) , for all ω , ω ∈ Ω T i , ˜ ω ∈ Ω T i +1 . Then ξ satisfies Assumptions 3.3 and 3.5. ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II Proof.
Clearly ξ satisfies Assumption 3.3. For ω ∈ Ω, ˜ ω ∈ Ω T i +1 and π n , ˜ π n ∈ Π εn ( T i , T i +1 ), denote ˆ ω π n := ω ⊗ T i ω π n ⊗ T i +1 ˜ ω and ˆ ω ˜ π n := ω ⊗ T i ω ˜ π n ⊗ T i +1 ˜ ω . Then | ξ (ˆ ω ˜ π n ) − ξ (ˆ ω ˜ π n ) | ≤ ρ n X k =1 | θ k − ˜ θ k | ! + ρ (cid:18)Z T i +1 T i | ω π n t − ω ˜ π n t | p dt (cid:19) , where ρ is the modulus of continuity function of g with respect to ( θ, ω ).Then one can easily check that the π n ξ (ˆ ω π n ) is uniformly continuousin Π εn ( T i , T i +1 ). (cid:3) Path-frozen PDEs.
Our main tool for proving the comparison prin-ciple for viscosity solutions, or, more precisely, for constructing the u ε and u ε , mentioned in the Introduction, so as to prove (1.3), is some path-frozenPDE. Define the following deterministic function on [ t, ∞ ) × R × R d × S d : g t,ω ( s, y, z, γ ) := G ( s ∧ T, ω ·∧ t , y, z, γ ) , ( t, ω ) ∈ Λ . For any ε > η ≥
0, we denote T η := (1 + η ) T , ε η := (1 + η ) ε and O ε := { x ∈ R d : | x | < ε } , O ε := { x ∈ R d : | x | ≤ ε } ,∂O ε := { x ∈ R d : | x | = ε } , (3.2) Q ε,ηt := [ t, T η ) × O ε η , Q ε,ηt := [ t, T η ] × O ε η ,∂Q ε,ηt := ([ t, T η ] × ∂O ε η ) ∪ ( { T η } × O ε η ) , and we further simplify the notation for η = 0 as Q εt := Q ε, t , Q εt := Q ε, t , ∂Q εt := ∂Q ε, t . Our additional assumption is formulated on the following localized and path-frozen PDE defined for every ( t, ω ) ∈ Λ:(E) t,ωε,η L t,ω v := − ∂ t v − g t,ω ( s, v, Dv, D v ) = 0 on Q ε,ηt . (3.3)Notice that for fixed ( t, ω ), this is a standard deterministic partial differentialequation. Lemma 3.7.
Under Assumptions 3.1 and 3.2 (ii) , PDE (3.3) satisfiesthe comparison principle for bounded viscosity solutions (in standard sense,as in [4]). Moreover, for any h ∈ C ( ∂Q ε,ηt ) , PDE (3.3) with the boundarycondition h has a (unique) bounded viscosity solution v . Proof.
The comparison principle follows from standard theory; see, forexample, [4]. Moreover, as we will see later, the v and v defined in (4.11) areviscosity supersolution and subsolution, respectively, of the PDE (3.3) and I. EKREN, N. TOUZI AND J. ZHANG satisfy v = v = h on ∂Q ε,ηt . Then the existence follows from the standardPerron approach in the spirit of [4], Theorem 4.1. (cid:3) We will use the following additional assumption:
Assumption 3.8.
For any ε > , η ≥
0, ( t, ω ) ∈ Λ and h ∈ C ( ∂Q ε,ηt ), wehave v = v = v , where v is the unique viscosity solution of PDE (3.3) withboundary condition h , and v ( s, x ) := inf { w ( s, x ) : w classical supersolution of (E) t,ωε,η and w ≥ h on ∂Q ε,ηt } , (3.4) v ( s, x ) := sup { w ( s, x ) : w classical subsolution of (E) t,ωε,η and w ≤ h on ∂Q ε,ηt } . We first note that the above sets of w are not empty. Indeed, one can checkstraightforwardly that for any δ > λ δ := C + L k h k ∞ δ + L , w ( t, x ) := k h k ∞ + δe λ δ ( T η − t ) , w ( t, x ) := −k h k ∞ − δe λ δ ( T η − t ) (3.5)satisfy the requirement for v ( s, x ) and v ( s, x ), respectively. We also observethat our definition (3.4) of v and v is different from the corresponding def-inition in the standard Perron approach [13], in which the w is a viscositysupersolution or subsolution. It is also different from the recent develop-ment of Bayraktar and Sirbu [1], in which the w is a so called stochasticsupersolution or subsolution. Loosely speaking, our Assumption 3.8 requiresthat the viscosity solution of (E) t,ωε,η can be approximated by a sequence ofclassical supersolutions and a sequence of classical subsolutions. We shalldiscuss further this issue in Section 8 below. In particular, we will providesome sufficient conditions for Assumption 3.8 to hold.
4. Main results.
The following theorem is the main result of this paper:
Theorem 4.1.
Let Assumptions 3.1, 3.2, 3.3, 3.5 and 3.8 hold true: (i)
Let u ∈ U be a viscosity subsolution and u ∈ U a viscosity superso-lution of PPDE (2.14) with u ( T, · ) ≤ ξ ≤ u ( T, · ) . Then u ≤ u on Λ . (ii) PPDE (2.14) with terminal condition ξ has a unique viscosity solu-tion u ∈ UC b (Λ) . ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II Strategy of the proof.
There are two key ingredients for the proofof this main result. The first is the following partial comparison, provedin Section 5, which extends the corresponding result in Proposition 5.3 of[8] to the set C , (Λ). The reason for extending C , (Λ) to C , (Λ) is thattypically we can construct the approximations u ε and u ε , mentioned in theIntroduction, only in the space C , (Λ), and not in C , (Λ). Proposition 4.2.
Assume Assumption 3.1 holds true. Let u ∈ U be aviscosity supersolution of PPDE (2.14) and u ∈ C , (Λ) bounded from abovesatisfying L u ( t, ω ) ≤ for all ( t, ω ) ∈ Λ with t < T . If u ( T, · ) ≤ u ( T, · ) ,then u ≤ u on Λ .A similar result holds if we switch the roles of u and u . The second key ingredient follows the spirit of the Perron approach as in[6]. Let u ( t, ω ) := inf { ψ t : ψ ∈ D ξT ( t, ω ) } , (4.1) u ( t, ω ) := sup { ψ t : ψ ∈ D ξT ( t, ω ) } , where D ξT ( t, ω ) := { ψ ∈ C , (Λ t ) : ψ − bounded , ( L ψ ) t,ω ≥ t, T ) × Ω t , ψ T ≥ ξ t,ω } , (4.2) D ξT ( t, ω ) := { ψ ∈ C , (Λ t ) : ψ + bounded , ( L ψ ) t,ω ≤ t, T ) × Ω t , ψ T ≤ ξ t,ω } . By using the functional Itˆo formula (2.13), and following the arguments in[8], Theorem 3.16, we obtain a similar result as the partial comparison ofProposition 4.2, implying that u ≤ u. (4.3)Moreover, these processes satisfy naturally a partial dynamic programmingprinciple which implies the following viscosity properties. Proposition 4.3.
Let Assumptions 3.1, 3.2 and 3.3 hold true. Thenthe processes u and u are bounded, uniformly continuous viscosity supersolutions and subsolutions, respectively, of PPDE (2.14). This result will be proved in Section 7. A crucial step for our proof is toshow the consistency of the Perron approach in the sense that equality holdsin the last inequality, under our additional assumptions. I. EKREN, N. TOUZI AND J. ZHANG
Proposition 4.4.
Under the conditions of Theorem 4.1, with N = 1 inAssumption 3.5, we have u = u . The proof of this proposition is reported in Section 6. Given Propositions4.2, 4.3 and 4.4, Theorem 4.1 follows immediately.
Proof of Theorem 4.1.
We prove the theorem in three steps:
Step
1. We first consider the case N = 1 in Assumption 3.5. By Proposi-tion 4.2, we have u ≤ u and u ≤ u . Then Proposition 4.4 implies u ≤ u immediately, which implies (i) and the uniqueness of the viscosity solution.Finally, by Propositions 4.4 and 4.3, u := u = u is a viscosity solution of(2.14). Step
2. For general N , it follows from step 1 that the comparison, ex-istence and uniqueness of the viscosity solution holds on [ T N − , T N ]. Let u denote the unique viscosity solution on [ T N − , T N ] with terminal condi-tion ξ , constructed by the Perron approach. Now consider PPDE (2.14) on[ T N − , T N − ] with terminal condition u ( T N − , · ). We shall prove below that u ( T N − , · ) satisfies the requirement of step 1. Then we may extend the com-parison, existence and uniqueness of the viscosity solution to the interval[ T N − , T N ]. By repeating the arguments backwardly, we complete the proofof Theorem 4.1. Step
3. It remains to verify Assumptions 3.3 and 3.5 with N = 1 for u ( T N − , · ) on [ T N − , T N − ]. First, by Proposition 4.3 it is clear that u ( T N − , · )is bounded. Given ω ∈ Ω, note that PPDE (2.14) on [ T N − , T N ] can beviewed as a PPDE with generator G T N − ,ω and terminal condition ξ T N − ,ω .Then, following the arguments in Lemma 7.3(i) below, one can easily showthat u ( T N − , ω ) is uniformly continuous in ω , and it follows from Assump-tion 3.5 that u ( T N − , ω ⊗ T N − ω π n ) is uniformly continuous in π n ∈ Π εn ( T N − , T N − ). (cid:3) Heuristic analysis on Proposition 4.4.
While highly technical, Propo-sition 4.2 follows along the same lines as the partial comparison of [8], Propo-sition 5.3. Proposition 4.3 has a corresponding result in PDE literature, andis proved in the spirit of the stability result of [8], Theorem 5.1. In thissubsection, we provide some heuristic discussions on Proposition 4.4, focus-ing on the case u = u , and the rigorous arguments will be carried out inSection 6 below.We shall follow [8], Section 7, where Proposition 4.4 is proved in a muchsimpler, semi-linear setting. The idea is to construct u ε ∈ D ξT (0 ,
0) and u ε ∈ D ξT (0 ,
0) such that lim ε → [ u ε − u ε ] = 0. To be precise, modulus sometechnical properties, the approximations u ε , u ε should satisfy: • they are piecewise smooth and L u ε ≥ ≥ L u ε ; ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II • they are continuous in t ; • u εT and u εT are close to ξ .To achieve this, we shall discretize the path ω so that we can utilize thepath-frozen PDE (3.3). We note that such discretization of ω will not inducebig errors, thanks to the uniform continuity of the involved processes. Fix ε >
0, and set h := 0, h i +1 := { t ≥ h i : | B t − B t i | = ε } ∧ T. Denote ˆ π n := { ( h i , B h i ) , ≤ i ≤ n } . Let π n = { ( t i , x i ) , ≤ i ≤ n } be a typ-ical value of ˆ π n ( ω ), and ω π n ∈ Ω be the linear interpolation of π n . Themain idea is to construct a sequence of deterministic functions v εn ( π n ; t, x )so that we may construct the desired u ε and u ε from a common process u εt := v εn (ˆ π n ; t, B t − B h n ), h n ≤ t < h n +1 . For this purpose, we require v εn ,and hence u ε , satisfying the following three corresponding properties: • For each π n , the function v εn ( π n ; · ) is in C , ( Q εt n ) and is a classicalsolution of a certain mollified path-frozen PDE, − ∂ t v εn − g π n ε ( t, v εn , Dv εn , D v εn ) = 0 , (4.4)where g π n ε = g t n ,ω πn ε . Consequently, the process u ε is approximately a classi-cal solution of PPDE (2.14) on [ h n , h n +1 ], thanks to the fact that g ˆ π n ( ω ) ε ( t, · )is a good approximation of G ( t, ω, · ). • v εn (ˆ π n ; h n +1 , B h n +1 − B h n ) = v εn +1 (ˆ π n +1 ; h n +1 , ) so that u ε is continuousin t and is more or less in C , (Λ). • v εn ( π n ; T, x ) is constructed from ξ , so that u εT is close to ξ .Now by the uniform continuity of ξ and G , we will see that u ε := u ε + ρ (2 ε ) and u ε := u ε − ρ (2 ε ) satisfy the desired classical semi-solution prop-erty. Clearly u ε − u ε ≤ ρ (2 ε ), implying the result.In [8], Section 7, the functions v εn can be constructed explicitly via ap-proximating backward SDEs. In the present setting, since we do not have arepresentation for the candidate solution, we cannot construct v εn directly.By some limiting procedure, in Lemma 6.3 below, we shall construct cer-tain deterministic functions θ εn which satisfy all the above three properties,except that θ εn is only a viscosity solution of PDE (4.4). Now to constructsmooth v εn from θ εn , we apply Assumption 3.8. In fact, given the viscositysolution θ εn , Assumption 3.8 allows us to construct the classical superso-lution v εn and the classical subsolution v εn , rather than one single smoothfunction v εn , such that v εn ≤ θ εn ≤ v εn , and v εn − v εn is small. This procedureis carried out in Lemma 6.4 below, and the construction is done piece bypiece, forwardly on each random interval [ h n , h n +1 ]. Remark 4.5.
As we see in the above discussion, the processes we willuse to prove the comparison takes the form v (Π n ; t, B t − B h n ), h n ≤ t < h n +1 , I. EKREN, N. TOUZI AND J. ZHANG for some deterministic function v , which is smooth in ( t, x ). Then it suf-fices to apply the standard Itˆo formula on v , rather than the functionalItˆo formula. Indeed, under our assumptions, we can prove rigorously thewell-posedness of viscosity solutions, including existence, stability and com-parison and uniqueness, without using the functional Itˆo formula. In otherwords, technically speaking, we can establish our theory without involvingthe path derivatives. However, we do feel that the path derivatives and thefunctional Itˆo formula are the natural and convenient language in this path-dependent framework. In particular, it is much more natural to talk aboutclassical solutions of PPDEs by using the path derivatives. Moreover, thecurrent proof relies heavily on the discretization of the underlying path ω ,with the help of the path-frozen PDEs. This discretization induces the abovepiecewise Markovian structure. The functional Itˆo formula allows us to ex-plore in future research other approaches without using such discretization.4.3. The bounding equations.
The proof of Proposition 4.4 requires someestimates, which involve the following particular example analyzed in [8].Recall the constants L C , and c from Assumptions 3.1 and 3.2, andconsider the operators g ( z, γ ) := sup | α |≤ L , √ c ≤| β |≤√ L (cid:20) α · z + 12 β : γ (cid:21) ,g ( y, z, γ ) := g ( z, γ ) + L | y | + C , (4.5) g ( z, γ ) := inf | α |≤ L , √ c ≤| β |≤√ L (cid:20) α · z + 12 β : γ (cid:21) ,g ( y, z, γ ) := g ( z, γ ) − L | y | − C , which clearly satisfy Assumptions 3.1 and 3.2, and g ≤ G ≤ g. (4.6)These operators induce the PPDEs L u := − ∂ t u − g ( u, ∂ ω u, ∂ ωω u ) = 0 and(4.7) L u := − ∂ t u − g ( u, ∂ ω u, ∂ ωω u ) = 0 . Let B tL := { b ∈ L (Λ t ) : | b | ≤ L } and P tL ,c := {P tL : | β P | ≥ √ c } , E L ,c t := sup P ∈P tL ,c E P , (4.8) E L ,c t := inf P ∈P tL ,c E P . ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II Following the arguments in our accompanying paper ([8], Proposition 4), wesee that for a bounded, uniformly continuous F T -measurable r.v. ξ , w ( t, ω ) := sup b ∈B tL E L ,c t (cid:20) ξ t,ω e R Tt b r dr + C Z Tt e R st b r dr ds (cid:21) , (4.9) w ( t, ω ) := inf b ∈B tL E L ,c t (cid:20) ξ t,ω e R Tt b r dr − C Z Tt e R st b r dr ds (cid:21) are viscosity solutions of the PPDE L w := 0 and L w := 0, respectively.By Lemma 3.7, the PDE version of (4.7), L v := − ∂ t v − g ( v, Dv, D v ) = 0 and(4.10) L v := − ∂ t v − g ( v, Dv, D v ) = 0 in Q ε,ηt , satisfies the comparison principle. Moreover, we have the following: Lemma 4.6.
Under Assumptions 3.1 and 3.2 (ii) , for any h ∈ C ( ∂Q ε,ηt ) ,the following functions are the unique viscosity solutions of PDEs (4.10) withboundary condition h : v ( t, x ) := sup b ∈B tL E L ,c t (cid:20) e R h t b r dr h ( h , x + B t h ) + C Z h t e R st b r dr ds (cid:21) , (4.11) v ( t, x ) := inf b ∈B tL E L ,c t (cid:20) e R h t b r dr h ( h , x + B t h ) − C Z h t e R st b r dr ds (cid:21) , where h := h t,x := { s > t : ( s, x + B ts ) / ∈ Q ε,ηt } . Proof.
First, by the arguments in [7], one may easily check that v and v are continuous and satisfy dynamic programing principle for t < h , whichimplies the viscosity property immediately. Then it remains to check theboundary conditions. For x ∈ O ε η , since t ≤ H t,x ≤ T and h is uniformlycontinuous with certain modulus of continuity function ρ h , it is clear that | v ( t, x ) − h ( T, x ) |≤ sup b ∈B tL E L t (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) e R h t b r dr h ( h , x + B t h ) + C Z h t e R st b r dr ds − h ( T, x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) = sup b ∈B tL E L t (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) [ e R h t b r dr − h ( h , x + B t h ) + [ h ( h , x + B t h ) − h ( T, x )]+ C Z h t e R st b r dr ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) (4.12) I. EKREN, N. TOUZI AND J. ZHANG ≤ C E L t [ H − t + ρ h ( T − h + | B t h | )] ≤ C E L t [ T − t + ρ h ( T − t + k B t k T )] → , as t ↑ T . Furthermore, let t < T and = x ∈ O ε η . Note that for any a > P ∈ P tL ,c , P ( h t,x − t ≥ a ) ≤ P (cid:18) sup t ≤ s ≤ t + a x | x | · B ts ≤ ε η − | x | (cid:19) ≤ P (cid:18) sup t ≤ s ≤ t + a Z st x | x | · β P r dW P r ≤ ε η − | x | + L a (cid:19) . Let A s := R st x T | x | ( β P r ) x | x | dr and τ s := inf { r ≥ t : A r ≥ s } . Then M s := R τ s t x T | x | β P r dW P r is a P -Brownian motion, and A s ≥ c ( s − t ). Thus P ( h t,x − t ≥ a ) ≤ P (cid:16) sup t ≤ s ≤ t +2 c a M s ≤ ε η − | x | + L a (cid:17) = P ( k B k c a ≤ ε η − | x | + L a )= P ( | B c a | ≤ ε η − | x | + L a )= P (cid:18) | B | ≤ √ c a [ ε η − | x | + L a ] (cid:19) ≤ C √ a [ ε η − | x | + L a ] . Set a := ε η − | x | , and we get P ( h t,x − t ≥ ε η − | x | ) ≤ C q ε η − | x | . Following similar arguments to those in (4.12), one can easily show that forsome modulus of continuity function ρ , | v ( t, x ) − h ( t, ˜ x ) | ≤ ρ ( ε η − | x | ) where ˜ x := | x | ε η x ∈ ∂O ε η . Then, for t < T , x ∈ ∂O ε η , t < T and x ∈ O ε η , noting that | x − ˜ x | ≤ ε η − | x | = | x | − | x | ≤ | x − x | , we have, as ( t, x ) → ( t , x ), | v ( t, x ) − h ( t , x ) | ≤ | v ( t, x ) − h ( t, ˜ x ) | + | h ( t, ˜ x ) − h ( t , x ) |≤ ρ ( ε η − | x | ) + ρ h ( | t − t | + | x − ˜ x | ) ≤ ρ ( | x − x | ) + ρ h ( | t − t | + 2 | x − x | ) → . ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II This implies that v is continuous on Q ε,η . Similarly one can prove the resultfor v . (cid:3) We remark that (4.9) provides representation for viscosity solutions ofPPDEs (4.7), even in the degenerate case c = 0. However, this is not truefor the PDEs (4.10), due to the boundedness of the domain Q ε,ηt , whichinduces the hitting time h and ruins the required regularity, as we will seein next example. Example 4.7.
Assume Assumption 3.1 holds, but G is degenerate, andthus c = 0. Let d = 1, and set h ( s, x ) := s on ∂Q ε,ηt . Then the v defined by(4.11) is discontinuous in [0 , T η ) × ∂O ε η ⊂ ∂Q ε,η and thus is not a viscositysolution of the PDE (4.10). Proof.
It is clear that v ( t, x ) = E L t (cid:20) e L ( h − t ) h + C Z h t e L ( s − t ) ds (cid:21) , where the integrand is increasing in h which takes values on [ t, T η ]. Then,by taking the P corresponding to α = β = 0, we have h = T η , P -a.s. and thus v ( t, x ) = e L ( T η − t ) T η + C Z T η t e L ( s − t ) ds, ( t, x ) ∈ Q ε,η . However, we have v ( t, x ) = t on ∂Q ε,η , so v is discontinuous in [0 , T η ) × ∂O ε η . (cid:3) A change of variables formula.
We conclude this section with achange of variables formula, which is interesting in its own right. We havepreviously observed in [8], Remark 3.15, that the classical change of vari-ables formula is not known to hold true for our notion of viscosity solutionsunder Assumption 3.1. We now show that it holds true under the additionalAssumption 3.8.Let u ∈ C , b (Λ) and Φ ∈ C , ([0 , T ] × R ). Assume Φ is strictly increasingin x , and let Ψ denote its inverse function. Note that Ψ is increasing in x and Ψ x >
0. Define˜ u ( t, ω ) := Φ( t, u ( t, ω )) and thus u ( t, ω ) = Ψ( t, ˜ u ( t, ω )) . (4.13)Then direct calculation shows that L u ( t, ω ) = Ψ x ( t, ˜ u ( t, ω )) ˜ L ˜ u ( t, ω ) and(4.14) ˜ L ˜ u := − ∂ t ˜ u − ˜ G ( t, ω, ˜ u, ∂ ω ˜ u, ∂ ωω ˜ u ) , I. EKREN, N. TOUZI AND J. ZHANG where˜ G ( t, ω, y, z, γ ):= Ψ t ( t, y ) + G ( t, ω, Ψ( t, y ) , Ψ x ( t, y ) z, Ψ xx ( t, y ) z + Ψ x ( t, y ) γ )Ψ x ( t, y ) . Then the following result is obvious:
Proposition 4.8.
Under the above assumptions on Ψ , u is classicalsolution (resp., supersolution, subsolution) of L u = 0 if and only if ˜ u :=Φ( t, u ) is a classical solution (resp., supersolution, subsolution) of ˜ L ˜ u = 0 . Moreover, we have the following:
Theorem 4.9.
Assume both ( G, ξ ) and ( ˜ G, Φ( T, ξ )) satisfy the condi-tions of Theorem 4.1. Then u is the viscosity solution of PPDE (2.14) withterminal condition ξ if and only if ˜ u := Φ( t, u ) is the viscosity solution ofPPDE (4.14) with terminal condition ˜ ξ := Φ( T, ξ ) . Proof.
One may easily check that w = Φ( t, u ) , w = Φ( t, u ), where w ( t, ω ) := inf { ψ t : ψ ∈ C , (Λ t ) , ψ − bounded , ˜ L ψ ≥ , ψ T ≥ Φ( T, ξ t,ω ) } ; w ( t, ω ) := sup { ψ t : ψ ∈ C , (Λ t ) , ψ + bounded , ˜ L ψ ≤ , ψ T ≤ Φ( T, ξ t,ω ) } . Then the result follows immediately from Proposition 4.4 and the argumentsin the proof of Theorem 4.1. (cid:3)
We observe that the above operator ˜ G is quadratic in the z -variable, sowe need somewhat stronger conditions to ensure the well-posedness.
5. Partial comparison of viscosity solutions.
In this section, we proveProposition 4.2. The proof is crucially based on the optimal stopping prob-lem reported in Theorem 2.3.We first prove a lemma. Recall the partition { E ij , j ≥ } ⊂ F h i , the con-stant n i and the uniform continuous mappings ϕ ijk and ψ ijk in (2.12) corre-sponding to u ∈ C , (Λ). For δ >
0, let 0 = t < t < · · · < t N = T such that t k +1 − t k ≤ δ for k = 0 , . . . , N −
1, and define t N +1 := T + δ . Lemma 5.1.
For all i, j ≥ , there is a partition ( ˜ E ij,k ) k ≥ ⊂ F h i of E ij and a sequence ( p k ) k ≥ taking values , . . . , N , such that h i ∈ [ t p k , t p k +1 ) on ˜ E ij,k , sup ω,ω ′ ∈ ˜ E ij,k k ω ·∧ h i ( ω ) − ω ′·∧ h i ( ω ′ ) k ≤ δ and min ω ∈ ˜ E ij,k h i ( ω ) = h i (cid:16) ω ij,k (cid:17) =: ˜ t ij,k for some ω ij,k ∈ ˜ E ij,k . ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II Proof.
Since i, j are fixed, we simply denote E := E ij and h := h i . De-note E k := E ∩ { t k ≤ h < t k +1 } , k ≤ n . Then { E k } k ⊂ F h forms a partitionof E . Since Ω is separable, there exists a finer partition { E k,l } k,l ⊂ F h suchthat, for any ω, ω ′ ∈ E k,l , k ω ·∧ h ( ω ) − ω ′·∧ h ( ω ′ ) k ≤ δ .Next, for each E k,l , there is a sequence ω k,l,m ∈ E k,l such that t k,l,m := h ( ω k,l,m ) ↓ inf ω ∈ E k,l h ( ω ). Denote t k,l, := t k +1 . Define E k,l,m := E k,l ∩ { t k,l,m +1 ≤ h < t k,l,m } ∈ F h i , and renumerate them as ( ˜ E k ) k ≥ . Wethen verify directly that ( ˜ E k ) k ≥ defines a partition of E satisfying the re-quired conditions. (cid:3) Proof of Proposition 4.2.
We only prove u ≤ u . The inequality forgeneral t can be proved similarly. Assume u is a viscosity L -supersolutionand u ∈ C , (Λ) with corresponding hitting times h i , i ≥
0. By Proposi-tion 3.14 of [8], we may assume without loss of generality that G ( t, ω, y , z, γ ) − G ( t, ω, y , z, γ ) ≥ y − y for all y ≤ y . (5.1)We now prove the proposition in three steps. Throughout the proof, denoteˆ u := u − u . Since u is bounded from above and u bounded from below, we see thatˆ u + is bounded. Step
1. We first show that for all i ≥ ω ∈ Ω,ˆ u + h i ( ω ) ≤ E L h i ( ω ) [(ˆ u + h i +1 − ) h i ,ω ] . (5.2)Since ( u ) t,ω ∈ C , (Λ t ), clearly it suffices to consider i = 0. Assume on thecontrary that 2 T c := ˆ u +0 ( ) − E L [ˆ u + h − ] > . (5.3)Recall (2.12). Notice that E = Ω and that ϕ k (0 , ) are constants, and wemay assume without loss of generality that n = 1 and u t = ψ ( t, B ) , ≤ t ≤ h , where ψ ∈ C , (Λ) ∩ UC b (Λ) with bounded derivatives. Denote X t := ( ψ t − u t ) + + ct, ≤ t ≤ T. Since u is bounded from below, by the definition of U , one may easily checkthat X is a bounded process in U , and X t := ˆ u + t + ct, ≤ t ≤ h . Define b X := X [0 , h ) + X h − [ h ,T ] ; Y := S L [ b X ] , τ ∗ := inf { t ≥ Y t = b X t } . I. EKREN, N. TOUZI AND J. ZHANG
Applying Theorem 2.3 and by (5.3), we have E L [ b X τ ∗ ] = Y ≥ X = ˆ u +0 ( ) = 2 T c + E L [ˆ u + h − ] ≥ T c + E L [ b X h ] . Then there exists ω ∗ ∈ Ω such that t ∗ := τ ∗ ( ω ∗ ) < h ( ω ∗ ). Next, by the E L -supermartingale property of Y of Theorem 2.3, we haveˆ u + ( t ∗ , ω ∗ ) + ct ∗ = X t ∗ ( ω ∗ ) = Y t ∗ ( ω ∗ ) ≥ E Lt ∗ [ X t ∗ ,ω ∗ h t ∗ ,ω ∗ − ] ≥ E Lt ∗ [ c h t ∗ ,ω ∗ ] > ct ∗ , implying that 0 < ˆ u + ( t ∗ , ω ∗ ) = ˆ u ( t ∗ , ω ∗ ). Since u ∈ U , by (2.3) there exists h ∈ H t ∗ such that h < h t ∗ ,ω ∗ and ˆ u t ∗ ,ω ∗ t > t ∈ [ t ∗ , h ] . (5.4)Then X t ∗ ,ω ∗ t = ϕ t − ( u ) t ∗ ,ω ∗ t for all t ∈ [ t ∗ , h ], where ϕ ( t, ω ) := ψ t ∗ ,ω ∗ ( t, ω ) + ct . Observe that ϕ ∈ C , (Λ t ∗ ). Using again the E L -supermartingale propertyof Y of Theorem 2.3, we see that for all τ ∈ T t ∗ ,( ϕ − ( u ) t ∗ ,ω ∗ ) t ∗ = X t ∗ ( ω ∗ ) = Y t ∗ ( ω ∗ ) ≥ E Lt ∗ [ Y t ∗ ,ω ∗ τ ∧ h ] ≥ E Lt ∗ [ X t ∗ ,ω ∗ τ ∧ h ]= E Lt ∗ [( ϕ − ( u ) t ∗ ,ω ∗ ) τ ∧ h ] . That is, ϕ ∈ A L u ( t ∗ , ω ∗ ), and by the viscosity L -supersolution property of u , 0 ≤ {− ∂ t ϕ − G ( · , u , ∂ ω ϕ, ∂ ωω ϕ ) } ( t ∗ , ω ∗ )= − c − { ∂ t u + G ( · , u , ∂ ω u , ∂ ωω u ) } ( t ∗ , ω ∗ ) ≤ − c − { ∂ t u + G ( · , u , ∂ ω u , ∂ ωω u ) } ( t ∗ , ω ∗ ) , where the last inequality follows from (5.4) and (5.1). Since c >
0, this is incontradiction with the subsolution property of u and thus completes theproof of (5.2). Remark 5.2.
The rest of the proof is only needed in the case where u ∈ C , (Λ) \ C , (Λ). Indeed, if u ∈ C , (Λ), then H = T , and it followsfrom step 1 that ˆ u +0 ≤ E L [ˆ u + T − ] ≤ E L [ˆ u + T ] = 0, and then u ≤ u . In fact, thisis the partial comparison principle proved in [8], Proposition 5.3. Step
2. We continue by using the following result which will be proved instep 3:For i ≥ P ∈ P L and P L ( P , h i ) := { P ′ ∈ P L : P ′ = P on F h i } , we have(5.5)∆ i := ˆ u + h i − − P ess-sup P ′ ∈P L ( P , h i ) E P ′ [ˆ u + h i +1 − |F h i ] ≤ , P -a.s. ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II Then by standard arguments, we have E P [ˆ u + h i − ] ≤ sup P ′ ∈P L ( P , h i ) E P ′ [ˆ u + h i +1 − ] ≤ E L [ˆ u + h i +1 − ] . Since P ∈ P L is arbitrary, this leads to E L [ˆ u + h i − ] ≤ E L [ˆ u + h i +1 − ], and by induc-tion, ˆ u +0 ≤ E L [ˆ u + h i − ], for all i . Notice that ˆ u + is bounded, lim i →∞ C L [ h i 3. It remains to prove (5.5). Clearly it suffices to prove it on each E ij . As in the proof of Lemma 5.1, we omit the dependence on the fixedpair ( i, j ), thus writing E := E ij , n = n i , h := h i , h := h i +1 , ϕ k := ϕ ij,k , ψ k := ψ ij,k , ∆ := ∆ i , and let C denote the common bound of ϕ k , ψ k and ρ ,the common modulus of continuity function of ϕ k , ψ k , 1 ≤ k ≤ n . We alsodenote ˜ E k := ˜ E ij,k , ω k := ω ij,k and ˜ t k := ˜ t ij,k , as defined in Lemma 5.1.Fix an arbitrary P ∈ P L and ε > 0. Since u ∈ U , we have u h − ≥ u h . Then,for each k , it follows from (5.2) thatˆ u + h − ( ω k ) ≤ ˆ u + h ( ω k ) ≤ E P k [(ˆ u + h − ) ˜ t k ,ω k ] + ε for some P k ∈ P ˜ t k L . Define ˆ P ∈ P L ( P , h ) such that for P -a.e. ω ∈ ˜ E k , the ˆ P h ( ω ) ,ω -distribution of B h ( ω ) is equal to the P k -distribution of B ˜ t k , where ˆ P h ( ω ) ,ω denotes the r.c.p.d.Then P -a.s. on ˜ E k , E ˆ P [ˆ u + h − |F h ]( ω )= E ˆ P h ( ω ) ,ω [ˆ u + ( h ( ω ⊗ h ( ω ) B h ( ω ) · ) − , ω ⊗ h ( ω ) B h ( ω ) · )]= E P k [ˆ u + ( h ( ω ⊗ h ( ω ) ˜ B ˜ t k · ) − , ω ⊗ h ( ω ) ˜ B ˜ t k . )] , where ˜ B ˜ t k s := B ˜ t k s − h ( ω )+˜ t k , s ≥ h ( ω ). Recalling that ˆ u + is bounded, P -a.s. thisprovides∆( ω ) ≤ ˆ u + h − ( ω ) − E ˆ P [ˆ u + h − |F h ]( ω ) ≤ ε + X k ≥ ˜ E k ( ω )(ˆ u + h − ( ω ) − ˆ u + h − ( ω k ))+ X k ≥ ˜ E k ( ω ) E P k [(ˆ u + h − ) ˜ t k ,ω k − ˆ u + ( h ( ω ⊗ h ( ω ) ˜ B ˜ t k · ) − , ω ⊗ h ( ω ) ˜ B ˜ t k · )](5.6) ≤ ε + X k ≥ ˜ E k ( ω )(ˆ u h − ( ω ) − ˆ u h − ( ω k )) + I. EKREN, N. TOUZI AND J. ZHANG + X k ≥ ˜ E k ( ω ) E P k [((ˆ u h − ) ˜ t k ,ω k − ˆ u ( h ( ω ⊗ h ( ω ) ˜ B ˜ t k · ) − , ω ⊗ h ( ω ) ˜ B ˜ t k · )) + ∧ C ] . We now estimate the above error for fixed ω ∈ ˜ E k :(1) To estimate the terms of the first sum, we recall that d ∞ (( h ( ω ) , ω ) , (˜ t k , ω k )) ≤ δ on ˜ E k , by Lemma 5.1. Then since u is continuous, it followsfrom (2.12) that on ˜ E k , u h i − ( ω ) − u h i − ( ω j ) = u h i ( ω ) − u h i ( ω j )= n X l =1 [ ϕ l ( h ( ω ) , ω ) − ϕ l (˜ t k , ω k )] ψ l (0 , ) ≤ Cnρ (2 δ ) . Moreover, denoting by ρ the modulus of continuity of − u ∈ U in (2.4), wesee that u h − ( ω k ) − u h − ( ω )= u (˜ t k − , ω k ) − u (˜ t k − , ω ) + u (˜ t k − , ω ) − u ( h ( ω ) − , ω ) ≤ ρ ( δ ) + sup h ( ω ) − δ ≤ t ≤ h ( ω ) [ u ( t − , ω ) − u ( h ( ω ) − , ω )] . By the last two estimates, we see that the first sum in (5.6) X k ≥ ˜ E k ( ω )(ˆ u h − ( ω ) − ˆ u h − ( ω k )) + −→ δ ց . (5.7)(2) Recall from Lemma 5.1 that 0 ≤ h ( ω ) − ˜ t k ≤ δ . Then (2.11) leads to0 ≤ [ h ( ω k ⊗ ˜ t k ˜ B ˜ t k · ) − ˜ t k ] − [ h ( ω ⊗ h ( ω ) ˜ B ˜ t k · ) − h ( ω )] ≤ h ( ω ) − ˜ t k ≤ δ, and therefore, denoting η δ ( ω ) := δ + sup {| ω s − ω t | : 0 ≤ t ≤ T, t ≤ s ≤ ( t + δ ) ∧ T } , d ∞ (( h ( ω k ⊗ ˜ t k ˜ B ˜ t k · ) − ˜ t k , ˜ B ˜ t k ) , ( h ( ω ⊗ h ( ω ) ˜ B ˜ t k · ) − h ( ω ) , ˜ B ˜ t k ))(5.8) ≤ η δ ( ˜ B ˜ t k ) ≤ η δ ( B ˜ t k ) . Then, by using (2.12) again, we see that( u ) ˜ t k ,ω k h ˜ tk,ωk − − u ( h ( ω ⊗ h ( ω ) ˜ B ˜ t k · ) − , ω ⊗ h ( ω ) ˜ B ˜ t k · )= u ( h ( ω k ⊗ ˜ t k B ˜ t k · ) , ω k ⊗ ˜ t k ˜ B ˜ t k · ) − u ( h ( ω ⊗ h ( ω ) ˜ B ˜ t · ) , ω ⊗ h ( ω ) ˜ B ˜ t k · ) ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II = n X l =1 [ ϕ l (˜ t k , ω k ) ψ l ( h ( ω ⊗ h ( ω ) ˜ B ˜ t k · ) − h ( ω ) , ˜ B ˜ t k )(5.9) − ϕ l ( h ( ω ) , ω ) ψ l ( h ( ω k ⊗ ˜ t k ˜ B ˜ t k · ) − ˜ t k , ˜ B ˜ t k )] ≤ Cn [ ρ (2 δ ) + ρ ( η δ ( B ˜ t k ))] . We now similarly estimate the corresponding term with u . Since ˜ t k ≤ h ( ω ),by (2.4) and (5.9) we have u ( h ( ω ⊗ h ( ω ) ˜ B ˜ t k · ) − , ω ⊗ h ( ω ) ˜ B ˜ t k · ) − ( u h − ) ˜ t k ,ω k = ( − u )( h ( ω k ⊗ ˜ t k B ˜ t k · ) − , ω k ⊗ ˜ t k ˜ B ˜ t k · ) − ( − u )( h ( ω ⊗ h ( ω ) ˜ B ˜ t k · ) − ,ω ⊗ h ( ω ) ˜ B ˜ t k · ) ≤ ρ ( d ∞ (( h ( ω ⊗ h ( ω ) ˜ B ˜ t k ) , ω ⊗ h ( ω ) ˜ B ˜ t k ) , ( h ( ω k ⊗ ˜ t k ˜ B ˜ t k ) , ω k ⊗ ˜ t k ˜ B ˜ t j ))) ≤ ρ ( d ∞ (( h ( ω ) , ω ) , (˜ t k , ω k ))+ d ∞ (( h ( ω k ⊗ ˜ t k ˜ B ˜ t k ) − ˜ t k , ˜ B ˜ t k ) , ( h ( ω ⊗ h ( ω ) ˜ B ˜ t k ) − h ( ω ) , ˜ B ˜ t k ))) ≤ ρ (2 δ + η δ ( B ˜ t k )) . Combining the above with (5.9), this implies that the second summation in(5.6) satisfies X k ≥ ˜ E k ( ω ) E P k [((ˆ u h − ) ˜ t k ,ω k − ˆ u ( h ( ω ⊗ h ( ω ) ˜ B ˜ t k · ) − , ω ⊗ h ( ω ) ˜ B ˜ t k · )) + ∧ C ] ≤ X k ≥ E P k [( Cn ( ρ + ρ )(2 δ + η δ ( B ˜ t k ))) ∧ C ] ˜ E k ( ω ) ≤ Cn E L [( ρ + ρ )(2 δ + η δ ( B )) ∧ C ] . One can easily check that lim δ → E L [( ρ + ρ )(2 δ + η δ ( B )) ∧ C ] = 0. Then bysending δ → ε → (cid:3) 6. Consistency of the Perron approach. This section is dedicated to theproof of Proposition 4.4. We follow the strategy outlined in Section 4.2,which is based on the idea in [8], Proposition 7.5. However, as pointed outin [8], Remark 7.7, due to fully nonlinearity, the arguments here are muchmore involved. We shall divide the proof into several lemmas. As in theprevious section, we may assume without loss of generality that G satisfiesthe monotonity (5.1).We start with some estimates for viscosity solutions of PDE (3.3). I. EKREN, N. TOUZI AND J. ZHANG Lemma 6.1. Let Assumptions 3.1 and 3.2 (ii) hold true. Let h i : ∂Q εt → R be continuous and v i be the viscosity solution of the PDE (E) t,ωε, withboundary condition h i , i = 1 , . Then, denoting δv := v − v , δh := h − h ,on Q εt we have δv ( s, x ) ≤ E L ,c s [( δh ) + ( h , x + B s h )] , (6.1) where h := T ∧ inf { r ≥ s : | x + B sr | = ε } . Proof. Let w denote the right-hand side of (6.1). Following the argu-ments in Lemma 4.6, it is clear that w is the unique viscosity solution ofPDE with boundary condition ( δh ) + , − ∂ t w − g ( Dw, D w ) = 0 on O εt . (6.2)Let K be a smooth nonnegative kernel with unit total mass. For all η > 0, wedefine the mollification w η := w ∗ K η of w . Then w η is smooth, and it followsfrom a convexity argument of Krylov [14] that w η is a classical supersolutionof − ∂ t w η − g ( Dw η , D w η ) ≥ O εt , (6.3) w η = ( δh ) + ∗ K η on ∂ O εt . We claim that ˜ w η + v supersolution of the PDE(E) t,ωε, , where ˜ w η := w η + k w η − ( δh ) + k L ∞ ( ∂Q εt ) .(6.4)Then, noting that ˜ w η + v = w η + h + k w η − ( δh ) + k L ∞ ( ∂Q εt ) ≥ h = v on ∂Q εt , we deduce from the comparison result of Lemma 3.7 that ˜ w η + v ≥ v on Q εt . Sending η ց 0, this implies that w + v ≥ v , which is the requiredresult.It remains to prove that ˜ w η + v is a supersolution of the PDE (E) t,ωε, . Let( t , x ) ∈ O εt , φ ∈ C , ( O εt ) be such that 0 = ( φ − ˜ w η − v )( t , x ) = max( φ − ˜ w η − v ). Then it follows from the viscosity supersolution property of v that L t,ω ( φ − ˜ w η )( t , x ) ≥ 0. Hence, at the point ( t , x ), by (5.1) and (6.3),we have L t,ω φ ≥ L t,ω φ − L t,ω ( φ − ˜ w η )= − ∂ t w η − g t,ω ( · , φ, Dφ, D φ )+ g t,ω ( · , φ − ˜ w η , D ( φ − w η ) , D ( φ − w η )) ≥ − ∂ t w η − g t,ω ( · , φ, Dφ, D φ ) + g t,ω ( · , φ, D ( φ − w η ) , D ( φ − w η )) ≥ g ( Dw η , D w η ) − α · Dw η − γ : D w η ≥ , where | α | ≤ L and | γ | ≤ L , thanks to Assumption 3.1. This proves (6.4). (cid:3) ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II Viscosity solutions of a discretized path-frozen PDE. Denote Π εn :=Π εn (0 , T ) in (3.1), and by Π εn its closure. Under Assumption 3.5 (with N = 1),clearly one may extend the mapping π n ∈ Π εn ξ ( ω π n ) continuously to thecompact set Π εn , and we shall still denote it as ξ ( ω π n ) for all π n ∈ Π εn .We first construct some stopping times, in light of Definition 2.5. For π n ∈ Π εn and ( t, x ) ∈ Q εt n , define the sequence h ε,π n ,t,xm := h m as follows: First, h := t , and h := inf { s ≥ t : | x + B ts | = ε } ∧ T, h m +1 := { s > h m : | B ts − B t h m | = ε } ∧ T, m ≥ π mn ( t, x, B t ) := ( π n , ( h , x + B t h ) , ( h , B t h − B t h ) , . . . , ( h m , B t h m − B t h m − )) . It is clear that π mn ( t, x, B t ) ∈ Π εn + m whenever h m < T . Lemma 6.2. { h ε,π n ,t,xm , m ≥ } satisfies the requirements of Definition2.5 (i)–(ii) , with E mj = Ω t in (ii) . Proof. For notational simplicity, we omit the superscripts ε,π n ,t,x . It isclear that h h m ,ωm +1 ∈ H h m ( ω ) whenever h m ( ω ) < T . Next, if h m ( ω ) < T for all m , then | B t h m +1 − B t h m | ( ω ) = ε for all m . This contradicts the fact that ω is(left) continuous at lim m →∞ h m ( ω ), and thus h m ( ω ) = T when m is largeenough. Moreover, for each m , { h m < T } ⊂ {| B t h i +1 − B t h i | = ε, i = 1 , . . . , m − }⊂ ( m − X i =1 | B t h i +1 − B t h i | ≥ ( m − ε ) . Then, for any L > C Lt [ h m < T ] ≤ m − ε E Lt " m − X i =1 | B t h i +1 − B t h i | (6.6) ≤ CL ( m − ε → m → ∞ . Similarly one can show that lim m →∞ C Ls [ h s,ωm < T ] = 0 for any ( s, ω ) ∈ Λ t .Finally, for ω, ˜ ω ∈ Ω and using the notation in Definition 2.5(ii), we have h m +1 ( ω ⊗ h m ( ω ) ˜ ω ) = T ∧ inf { t ≥ h i ( ω ) : | ˜ ω t − h m ( ω ) | = ε } = T ∧ [ h m ( ω ) + ˜ h (˜ ω )] , I. EKREN, N. TOUZI AND J. ZHANG where ˜ h (˜ ω ) := inf { t : | ˜ ω t | = ε } is independent of ω . Then, given h n ( ω ) ≤ h n ( ω ′ ), (2.11) follows immediately. (cid:3) We next prove the existence of the functions θ εn , as mentioned in Sec-tion 4.2, which allows us to construct classical super and subsolutions inLemma 6.4 below. Lemma 6.3. Let Assumptions 3.1, 3.2 (ii) , 3.3 and 3.5 with N = 1 holdtrue. Then there exists a sequence of continuous functions θ εn : ( π n , ( t, x )) ∈ Π εn +1 R , bounded uniformly in ( ε, n ) , such that θ εn ( π n ; · ) is a viscosity solution of (E) t n ,ω πn ε, ; θ εn ( π n ; t, x ) = ξ ( ω π n , ( t,x ) ) if t = T, (6.7) θ εn ( π n ; t, x ) = θ εn +1 ( π n , ( t, x ); t, if | x | = ε. Proof. Step 1. We first prove the lemma in the cases G = g and G = g ,as introduced in (4.5). For any n , denote θ εn,n ( π n ; t n , ) := ξ ( ω π n ) , which is continuous for π n ∈ Π εn , thanks to Assumption 3.5 (with N = 1).For m := n − , . . . , 0, let θ := θ εn,m ( π m ; · ) be the unique viscosity solution ofthe PDE L θ := − ∂ t θ − g ( θ, Dθ, D θ ) = 0 in Q εt m , (6.8) θ ( t, x ) = θ εn,m +1 ( π m , ( t, x ); t, ) on ∂Q εt m . Applying Lemma 6.1 repeatedly and recalling Assumption 3.5 (with N = 1)again, we see that θ εn,m ( π m ; t, x ) are uniformly bounded and continuous inall variables ( π m , t, x ). Now for any π m ∈ Π εm and ( t, x ) ∈ Q εt m , define θ εm ( π m , t, x ) := sup b ∈B tL E L t (cid:20) e R Tt b r dr lim n →∞ ξ ( ω π n − mm ( t,x,B t ) ) + C Z Tt e R st b r dr ds (cid:21) . Then, by (6.6), | θ εm ( π m , t, x ) − θ εn,m ( π m , t, x ) | ≤ C C L t m [ h n − m < T ] ≤ C ( n − m − ε −→ n → ∞ . This implies that θ εm ( π m ; t, x ) are uniformly bounded, uniform in ( ε, m ) andare continuous in all variables ( π m , t, x ). Moreover, by stability of the vis-cosity solutions, we see that θ εm ( π m ; · ) is the viscosity solution of PDE (6.8) in Q εt m ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II with the boundary condition θ εm ( π m ; T, x ) = ξ ( ω π m , ( T,x ) ) , | x | ≤ ε,θ εm ( π m ; t, x ) = θ εm +1 ( π m , ( t, x ); t, , | x | = ε. Similarly we may define from g the following θ εm satisfying the correspondingproperties: θ εm ( π m , t, x ) := inf b ∈B tL E L t (cid:20) e R Tt b r dr lim n →∞ ξ ( ω π n − mm ( t,x,B t ) ) − C Z Tt e R st b r dr ds (cid:21) . Step 2. We now prove the lemma for G . Given the construction of step 1,define θ ε,mm ( π m ; t, x ) := θ εm ( π m ; t, x ) , θ ε,mm ( π m ; t, x ) := θ εm ( π m ; t, x ); m ≥ . For i = m − , . . . , 0, by Lemma 3.7 we may define θ ε,mi and θ ε,mi as the uniqueviscosity solution of the PDE (E) t i ,ω πi ε, with boundary conditions θ ε,mi = θ ε,mi +1 and θ ε,mi = θ ε,mi +1 on ∂Q εt i . Note that for ( t, x ) ∈ ∂Q εt m , θ ε,mm ( π m ; t, x ) = θ ε,m +1 m +1 ( π t,xm ; t, , θ ε,mm ( π m ; t, x ) = θ ε,m +1 m +1 ( π t,xm ; t, . Since g ≤ g t,ω ≤ g , it follows from the comparison result of the PDEs definedby the operators g and g that θ ε,mm ( π m ; · ) ≥ θ ε,m +1 m ( π m ; · ) ≥ θ ε,m +1 m ( π m ; · ) ≥ θ ε,mm ( π m ; · ) in Q εt m . Then, by an immediate backward induction, the comparison result ofLemma 3.7 implies θ ε,mi ( π i ; · ) ≥ θ ε,m +1 i ( π i ; · ) ≥ θ ε,m +1 i ( π i ; · ) ≥ θ ε,mi ( π i ; · )(6.9) in Q εt i , for all i ≤ m. Denote δθ ε,mi := θ ε,mi − θ ε,mi . For any π i and any ( t, x ) ∈ Q εt i , recall the no-tation in (6.5). Applying Lemma 6.1 repeatedly, and following similar butmuch easier arguments as those in Lemma 5.5, we see that | δθ ε,mi ( π i ; t, x ) | ≤ E L t [ | δθ ε,mm ( π m − ii ( t, x, B t ); h m − i , | ] . Note that δθ ε,mi ( π i ; t, x ) = 0 when t = T . Then, by (6.6) again, | δθ ε,mi ( π i ; t, x ) | ≤ C C L t [ h m − i < T ] ≤ C ( m − i − ε → m → ∞ . Together with (6.9), this implies the existence of θ εi such that θ ε,mi ց θ εi , θ ε,mi ր θ εi , as m → ∞ . Clearly θ εi are uniformly bounded and continuous.Finally, it follows from the stability of the viscosity solutions that θ εi satisfies(6.7). (cid:3) I. EKREN, N. TOUZI AND J. ZHANG Approximating classical super and subsolutions of the PPDE. Wenow apply Assumption 3.8 to θ εn to construct smooth approximations of u and u , namely the u ε and u ε mentioned in Section 4.2. Define h εi := h ε, (0 , ) , (0 , ) i , that is, h ε := 0 and h εn +1 := T ∧ inf { t ≥ h εn : | B t − B h εn | = ε } for all n ≥ . Let ˆ π n denote the sequence ( h εi , B h εi ) ≤ i ≤ n , and ω ε := lim n →∞ ω ˆ π n . It is clearthat k ω − ω ε k T ≤ ε and k ω ˆ π n ·∧ h n − ω k h n +1 ≤ ε for all n, ω. (6.10)Recall the common modulus of continuity function ρ of G in Assump-tion 3.2, and let θ εn be given as in Lemma 6.3. We then approximate θ ε bya piecewise smooth processes in C , (Λ). Lemma 6.4. Under the conditions of Theorem 4.1, with N = 1 in As-sumption 3.5, there exists ψ ε ∈ C , (Λ) bounded from below with correspond-ing stopping times h εn such that ψ ε (0 , ) = θ ε (0 , ) + ε + T ρ (2 ε ) , (6.11) ψ ε ( T, ω ) ≥ ξ ( ω ε ) , L ψ ε ≥ on [0 , T ) . Proof. For notational simplicity, in this proof we omit the superscript ε and denote θ n := θ εn , ψ = ψ ε etc. Moreover, we extend the domain of θ n ( π n ; · )to [ t n , ∞ ) × R d , θ n ( π n ; t, x ) := θ ( π n ; t ∧ T, proj O ε ( x )) , where proj O ε is the orthogonal projection on O ε , the closed centered ballwith radius ε . We shall construct ψ on each [ h n , h n +1 ) forwardly, by induc-tion on n . Step 1. First, let η > λ > Q ε,η , and recall the operators L and L at (4.10). Thanks to Lemma 3.7, let v η,λ , v η,λ and v η,λ denote theunique viscosity solutions of PDEs (E) , ε,η , L v = 0 and L v = 0, respectively,with the same boundary condition θ + λ on ∂Q ε,η .By comparison, we have v η,λ ≤ v η,λ ≤ v η,λ . Then, by using the estimatein Lemma 6.1, one can easily show that there exist η ( λ ) and C ( λ ), whichmay depend on L , λ and the regularity of θ , such that, for all η ≤ η ( λ ),0 ≤ v η,λ − θ ≤ C ( λ ) on Q ε,η \ Q ε with C ( λ ) ց , as λ ց . In particular, the above inequalities hold on ∂Q ε . Then, by the comparisonprinciple, Lemmas 3.7 and 6.1, we have0 ≤ v η,λ − θ ≤ C ( λ ) in Q ε,η . ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II Fix λ such that C ( λ ) < ε , and set η := η ( λ ). Then v η ,λ < θ + ε . Onthe other hand, by Assumption 3.8, there exists v ∈ C , ( Q ε,η ) satisfying v (0 , ) ≤ v η ,λ (0 , ) + ε < θ (0 , ) + ε , L , v ≥ Q ε,η , v ≥ v η ,λ on ∂Q ε,η . By the comparison principle and Lemma 3.7, the last inequality on ∂Q ε,η implies that v ≥ v η ,λ ≥ θ on Q ε,η . By modifying v outside of Q ε,η / and by the monotonicity (5.1), withoutloss of generality we may assume v ∈ C , ([0 , T ] , R d ) with bounded deriva-tives such that v (0 , ) = θ (0 , ) + ε , L , v ≥ Q ε , v ≥ θ on ∂Q ε . We now define ψ ( t, ω ) := v ( t, ω t ) + ε ρ (2 ε )( T − t ) , t ∈ [0 , h ] . (6.12)Note that ( t, ω t ) ∈ Q ε for t < h , ( h , ω h ) ∈ ∂Q ε , and θ is bounded. Then ψ (0 , ) = θ (0 , ) + ε + T ρ (2 ε ) , (6.13) v ( h , ω ) ≥ θ ( h , ω h ) = θ (ˆ π ; h , , ψ ≥ − C on [0 , h ] . Moreover, by monotonicity (5.1) again, and by Assumption 3.2 and (6.10), L ψ ( t, ω ) = ρ (2 ε ) − ∂ t v ( t, ω t ) − G ( t, ω, ψ, Dv ( t, ω t ) , D v ( t, ω t )) ≥ ρ (2 ε ) − ∂ t v ( t, ω t ) − G ( t, ω, v ( t, ω t ) , Dv ( t, ω t ) , D v ( t, ω t ))(6.14) ≥ − ∂ t v ( t, ω t ) − g , ( t, v ( t, ω t ) , Dv ( t, ω t ) , D v ( t, ω t ))= L , v ( t, ω t ) ≥ ≤ t < h ( ω ) . Here we use the fact that ∂ ω [ v ( t, ω t )] = ( Dv )( t, ω t ); see [8], Remark 2.9(i). Step 2. Let η , λ , δ be small positive numbers which will be decidedlater. Set s i := (1 − δ ) i T , i ≥ 0. Since O ε is compact, there exists a par-tition D , . . . , D n such that | y − ˜ y | ≤ T δ for any y, ˜ y ∈ D j , j = 1 , . . . , n .For each j , fix a point y j ∈ D j . Now for each ( i, j ), let v η,λij denote theunique viscosity solution of the PDE (E) s i ,ω ( si,yj ) ε,η with the boundary condi-tion v η,λij ( t, x ) = θ ( s i , y j ; t, x ) + λ on ∂Q ε,ηs i . Here ω ( s i ,y j ) denotes the linearinterpolation of (0 , ) , ( s i , y j ) , ( T, y j ). Then, by the same arguments as in I. EKREN, N. TOUZI AND J. ZHANG step 1, there exist η ( λ ) and C ( λ ), which may depend on L , λ and theregularity of θ , but independent of δ and ( i, j ), such that for all η ≤ η ( λ ),0 ≤ v η,λij ( t, x ) − θ ( s i , y j ; t, x ) ≤ C ( λ ) on Q ε,ηs i \ Q εs i and C ( λ ) ց λ ց . Following the arguments in step 1, we may fix λ , η , independently of δ and ( i, j ), and there exists v ij ∈ C , ([ s i , T ] , R d ) with bounded derivativessuch that v ij ( s i , ) = θ ( s i , y j ; s i , ) + ε , L s i ,ω ( si,yj ) v ij ≥ Q εs i ,v ij ≥ θ ( s i , y j ; · ) on ∂Q εs i . Denote E ij := { s i +1 < h ≤ s i } ∩ { B h ∈ D j } ∈ F h . Here we are using ( i, j ) instead of j as the index, and clearly E ij forma partition of Ω. We then define ψ on [ h , h ] in the form of (2.12) with n = 2, ψ t := X i,j (cid:20) v ( h , B h ) + v ij ( s i + t − h , B t − B h ) − v ij ( s i , ) + ε (cid:21) E ij (6.15) + ρ (2 ε )( T − t ) , t ∈ [ h , h ] . We show that ψ satisfies all the requirements on [ h , h ] when δ is smallenough. • First, by (6.15), we have ψ h = X i,j (cid:20) v ( h , B h ) + ε (cid:21) E ij + ρ (2 ε )( T − h )= v ( h , B h ) + ε ρ (2 ε )( T − h ) , which is consistent with (6.12), and thus ψ is continuous at t = h . • We next check, similar to (6.14), that L ψ ( t, ω ) ≥ , h ≤ t < h . (6.16)Note that ( h , B h ) ∈ ∂Q ε and 0 ≤ s i − h ≤ s i − s i +1 = δs i ≤ δT on E ij ,then v ( h , B h ) − v ij ( s i , ) + ε ≥ θ ( h , B h ; h , ) − θ ( s i , y j ; s i , ) + ε ≥ ε − ρ (3 T δ ) on E ij , ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II where ρ is the modulus of continuity function of θ . In particular, ρ (3 T δ ) < ε when δ is small enough. Now on E ij , denoting t := h , x := ω h , ˜ t := s i − h + t , by (5.1), Assumption 3.2(i) and (6.10) again, we have L ψ ( t, ω ) ≥ L ψ ( t, ω ) − L s i ,ω ( si,yj ) v ij (˜ t, x )= ρ (2 ε ) − G ( t, ω, ψ ( t, ω ) , Dv ij (˜ t, x ) , D v ij (˜ t, x ))+ G (˜ t ∧ T, ω ( s i ,y j ) ·∧ s i , v ij (˜ t, x ) , Dv ij (˜ t, x ) , D v ij (˜ t, x ))(6.17) ≥ ε − ρ (3 T δ ) − G ( t, ω ˆ π ·∧ t , v ij (˜ t, x ) , Dv ij (˜ t, x ) , D v ij (˜ t, x ))+ G (˜ t ∧ T, ω ( s i ,y j ) ·∧ s i , v ij (˜ t, x ) , Dv ij (˜ t, x ) , D v ij (˜ t, x )) ≥ ε − ρ (3 T δ ) − ρ ( d ∞ (( t, ω ˆ π ·∧ t ) , (˜ t ∧ T, ω ( s i ,y j ) ·∧ s i ))) . Without loss of generality, assume ε ≤ T . Then d ∞ (( t, ω ˆ π ·∧ t ) , (˜ t ∧ T, ω ( s i ,y j ) ·∧ s i )) ≤ | t − ˜ t | + sup ≤ s ≤ T (cid:12)(cid:12)(cid:12)(cid:12) s ∧ t t x − s ∧ s i s i y j (cid:12)(cid:12)(cid:12)(cid:12) ≤ δT + sup ≤ s ≤ T (cid:12)(cid:12)(cid:12)(cid:12) s ∧ t t x − s ∧ t t y j (cid:12)(cid:12)(cid:12)(cid:12) + sup ≤ s ≤ T (cid:12)(cid:12)(cid:12)(cid:12) s ∧ t t y j − s ∧ s i s i y j (cid:12)(cid:12)(cid:12)(cid:12) ≤ δT + ε sup ≤ s ≤ T (cid:12)(cid:12)(cid:12)(cid:12) s ∧ t t − s ∧ s i s i (cid:12)(cid:12)(cid:12)(cid:12) = 2 δT + ε (cid:20) − t s i (cid:21) ≤ δT + ε (cid:20) − s i +1 s i (cid:21) = 3 δT. Then L ψ ( t, ω ) ≥ ε − [ ρ + ρ ](3 T δ ). By choosing δ small enough, we ob-tain (6.16). • Finally, we emphasize that the bound of v ij and its derivatives dependonly on the properties of θ (and the η which again depends on θ ), butnot on ( i, j ). Then ψ satisfies Definition 2.5(iii) on [ h , h ]. Moreover, since θ is bounded, by comparison we see that ψ ≥ − C on [ h , h ]. Step 3. Repeating the arguments, we may define ψ on [ h n , h n +1 ] forall n . From the construction and recalling Lemma 6.2, we see that ψ ∈ C , (Λ) bounded from below, ψ (0 , ) = θ (0 , ) + ε + T ρ (2 ε ) and L ψ ≥ , T ). Finally, since h n = T when n is large enough, we see that ψ ( T, ω ) = ψ ( h n ( ω ) , ω ) ≥ θ n ( ω ε ) = ξ ( ω ε ). (cid:3) Now we are ready to prove the main result of this section. I. EKREN, N. TOUZI AND J. ZHANG Proof of Proposition 4.4. For any ε > 0, let h εn , n ≥ ψ ε be asin Lemma 6.4, and define ψ ε := ψ ε + ρ (2 ε ). Then clearly ψ ε ∈ C , (Λ), ψ ε is bounded from below, and ψ ε ( T, ω ) − ξ ( ω ) = ψ ε ( T, ω ) + ρ (2 ε ) − ξ ( ω ) ≥ ξ ( ω ε ) − ξ ( ω ) + ρ (2 ε ) ≥ , where the last inequality follows from (6.10). Moreover, for t ∈ [ h n , h n +1 ),by (5.1) again, L ψ ε ( t, ω ) = − ∂ t ψ ε ( t, ω ) − G ( t, ω, ψ ε + ρ (2 ε ) , ∂ ω ψ ε , ∂ ωω ψ ε ) ≥ − ∂ t ψ ε ( t, ω ) − G ( t, ω, ψ ε , ∂ ω ψ ε , ∂ ωω ψ ε ) = L ψ ε ( t, ω ) ≥ . Then by the definition of u we see that u (0 , ) ≤ ψ ε (0 , ) = ψ ε (0 , ) + ρ (2 ε ) ≤ θ ε (0 , ) + ε + ( T + 1) ρ (2 ε ) . Similarly, u (0 , ) ≥ θ ε (0 , ) − ε − ( T + 1) ρ (2 ε ). This implies that u (0 , ) − u (0 , ) ≤ ε + ( T + 1) ρ (2 ε )) . Since ε > u (0 , ) = u (0 , ). Similarly we canshow that u ( t, ω ) = u ( t, ω ) for all ( t, ω ) ∈ Λ. (cid:3) Fo later use, we conclude this section with a complete well-posednessresult for a special PPDE. Corollary 6.5. Let G ( t, ω, y, z, γ ) = g ( y, z, γ ) satisfy Assumptions 3.1and 3.2, and assume that ξ satisfies Assumptions 3.3 and 3.5 with N = 1 .Then u = u and is the unique viscosity solution of the PPDE (2.14). Proof. We first observe that g satisfies Assumption 3.2(i). We shallprove in Proposition 8.2 below that it also satisfies Assumption 3.8. Then itfollows from the last proof that u = u . Moreover, the process w introduced in(4.9) is a viscosity solution of PPDE (2.14) with terminal condition ξ . Thenit follows from the partial comparison of Proposition 4.2 that u ≤ w ≤ u ,hence equality. (cid:3) 7. Viscosity properties of u and u . This section is devoted to the proofof Proposition 4.3. The idea is similar to the corresponding result in the PDEliterature, and in spirit is similar to the stability of the viscosity solutions asin [8], Theorem 5.1. However, we shall first establish the required regularitiesof u and u . Lemma 7.1. Under Assumptions 3.1 and 3.3, the processes u, u arebounded. ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II Proof. We shall only prove the result for u , the proof for u beingsimilar. Fix ( t, ω ), and set ψ ( s, ˜ ω ) := C ( L + 1) e ( L +1)( T − s ) . Then ψ ∈ C , (Λ t ) ⊂ C , (Λ t ), ψ ≥ ψ T ≥ C ( L + 1) ≥ C ≥ ξ t,ω , and wecompute that( L ψ ) t,ωs = ( L + 1) ψ s − G t,ω ( · , ψ s , , ) ≥ ψ s − G t,ω ( · , , , ) ≥ C ( L + 1) − C ≥ . This implies that ψ ∈ D ξT ( t, ω ), and thus u ( t, ω ) ≤ ψ ( t, ).On the other hand, by similar arguments one can show that − ψ is aclassical subsolution of PPDE (2.14) satisfying − ψ T ≤ ξ t,ω . Then by partialcomparison Proposition 4.2, u ( t, ω ) ≥ − ψ ( t, ). Hence | u ( t, ω ) | ≤ ψ ( t, ) ≤ C ( L + 1) e ( L +1) T . (cid:3) We next prove that u and u satisfy a partial dynamic programming prin-ciple. Lemma 7.2. Under Assumptions 3.1 and 3.3, for ≤ t < t ≤ T , wehave u ( t , ω ) ≥ inf { ψ t : ψ ∈ D u t t ( t , ω ) } , u ( t , ω ) ≤ sup { ψ t : ψ ∈ D u t t ( t , ω ) } . Proof. We only prove the result for u . For any arbitrary ψ ∈ D ξT ( t , ω ),notice that ψ t ,ω ′ ∈ C , (Λ t ) and ψ t ( ω ′ ) ≥ u t ,ωt ( ω ′ ) for any ω ′ ∈ Ω t . Then ψ ∈ D u t t ( t , ω ), and the result follows. (cid:3) The next result shows that the functions u, u are uniformly continuous. Weobserve that with this regularity in hand, and following standard techniques,we may prove that the equality holds in Lemma 7.2, so that u, u satisfy adynamic programming principle. However, this is not needed for the presentanalysis. Moreover, the result is true in degenerate case c = 0 as well. Lemma 7.3. Under Assumptions 3.1, 3.2 (ii) and 3.3, we have u, u ∈ UC b (Λ) . Proof. We only prove the result for u .(i) We first prove that u is uniformly continuous in ω , uniformly in t . For t ∈ [0 , T ] and ω , ω ∈ Ω, denote δ := k ω − ω k t . For ψ ∈ D ξT ( t, ω ), define ψ ( s, ˜ ω ) := ψ ( s, ˜ ω ) + ψ ( s ) where ψ ( s ) := e ( L +1)( T − s ) [ ρ ( δ ) + δ ] . I. EKREN, N. TOUZI AND J. ZHANG Notice that e − ( L +1) s = e − ( L +1) h i e − ( L +1)( s − h i ) , and one can easily checkthat ψ ∈ C , (Λ t ) with the same h i as those of ψ . Moreover, ψ is boundedfrom below, and ψ T = ψ T + ψ T ≥ ξ t,ω T + ρ ( δ ) ≥ ξ t,ω ;( L ψ ) t,ω s ≥ ( L ψ ) t,ω s − ( L ψ ) t,ω s = ( L + 1) ψ s − G t,ω ( s, · , ψ , ∂ ω ψ , ∂ ωω ψ )+ G t,ω ( s, · , ψ , ∂ ω ψ , ∂ ωω ψ ) ≥ ( L + 1) ψ s − ρ ( δ ) − L ψ s = ψ s − ρ ( δ ) ≥ δ > . Then ψ ∈ D ξT ( t, ω ), and therefore u ( t, ω ) ≤ ψ ( t, ), implying that u ( t, ω ) − ψ ( t, ) ≤ ψ ( t, ) − ψ ( t, ) = e ( L +1)( T − t ) [ ρ ( δ ) + δ ] ≤ C [ ρ ( δ ) + δ ] . Since ψ ∈ D ξT ( t, ω ) is arbitrary, we obtain u ( t, ω ) − u ( t, ω ) ≤ C [ ρ ( δ ) + δ ].By symmetry, this shows the required uniform continuity of u in ω , uniformlyin t .(ii) We now prove that − u satisfies (2.4). Fix t < t ≤ T , and considerthe process w ( t, ω ) := inf b ∈B tL E L ,c t (cid:20) e R t t b r dr u ( t , ω ⊗ t B t ) − C Z t t e R st b r dr ds (cid:21) , (7.1) ( t, ω ) ∈ [0 , t ] × Ω . By (4.9), w is a viscosity solution of the PPDE L w := − ∂ t w − g ( w, ∂ ω w, ∂ ωω w ) = 0 , (7.2) t ∈ [0 , t ) , ω ∈ Ω , w ( t , ω ) = u ( t , ω ) . Recalling (4.6) and applying partial comparison principle Proposition 4.2on PPDE (7.2), we see that ψ t ≥ w ( t , ω ) for any ψ ∈ D u t t ( t , ω ). Then u ( t , ω ) ≥ w ( t , ω ), and thus u ( t , ω ) − u ( t , ω ) ≤ u ( t , ω ) − w ( t , ω )= sup b ∈B t L E L ,c t (cid:20) u ( t , ω ) − e R t t b r dr u ( t , ω ⊗ t B t ) + C Z t t e R st b r dr ds (cid:21) . ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II Then it follows from (i) and Lemma 7.1 that u ( t , ω ) − u ( t , ω ) ≤ C ( t − t ) + C E L ,c t [ | u ( t , ω ) − u ( t , ω ⊗ t B t ) | ] ≤ C ( t − t ) + C E L ,c t [ ρ ( d ∞ (( t , ω ) , ( t , ω )) + k B t k t )] , where ρ is the modulus of continuity of u ( t , · ). Now it is straightforward tocheck that − u satisfies (2.4).(iii) We finally prove that u satisfies (2.4). This, together with Lemma 7.1and (ii), implies that u ∈ UC b (Λ). For t < t , ω ∈ Ω and ψ ∈ D ξT ( t , ω ),define ξ t (˜ ω ) := ψ ( t , ) + e L ( T − t ) ρ ( d ∞ (( t , ω ) , ( t , ω )) + k ˜ ω k t ) , ˜ ω ∈ Ω t and w ( t, ˜ ω ) := sup b ∈B tL E L ,c t (cid:20) e R t t b r dr ξ t ( t , ˜ ω ⊗ t B t ) + C Z t t e R st b r dr ds (cid:21) , (7.3) ( t, ˜ ω ) ∈ [ t , t ] × Ω t . By Lemma 7.1, we may assume without loss of generality that | ψ ( t , ) | ≤ C . Then | w ( t , ) − ψ ( t , ) |≤ C ( t − t ) + C E L ,c t [ ρ ( d ∞ (( t , ω ) , ( t , ω )) + k B t k t )](7.4) ≤ Cρ ( d ∞ (( t , ω ) , ( t , ω ))) , for some modulus of continuity ρ .By (4.9), the process w is a viscosity solution of the PPDE L w := − ∂ t w − g ( w, ∂ ω w, ∂ ωω w ) = 0 , (7.5) ( t, ˜ ω ) ∈ [ t , t ) × Ω t and w ( t , · ) = ξ t . Notice that ξ t satisfies the conditions of Corollary 6.5, and therefore w =( w ), where ( w ) is defined for PPDE (7.5) in the spirit of (4.1). Then for any ε > 0, there exists ψ ∈ C , (Λ t ) bounded from below such that ψ ( t , ) ≤ w ( t , ) + ε,ψ ( t , ˜ ω ) ≥ w ( t , ˜ ω ) and(7.6) − ∂ t ψ − g ( ψ , ∂ ω ψ , ∂ ωω ψ ) ≥ . I. EKREN, N. TOUZI AND J. ZHANG Therefore, for t ∈ [ t , t ), by (4.5) and (4.6), we have L ψ = − ∂ t ψ − G ( · , ψ , ∂ ω ψ , ∂ ωω ψ )(7.7) ≥ g ( ψ , ∂ ω ψ , ∂ ωω ψ ) − G ( · , ψ , ∂ ω ψ , ∂ ωω ψ ) ≥ . Now define ψ on Λ t by ψ ( t, ˜ ω ) := ψ ( t, ˜ ω ) [ t ,t ) ( t )(7.8) + [ ψ ( t, ˜ ω t ) + ( ψ ( t , ˜ ω ) − ψ ( t , )) e L ( t − t ) ] [ t ,T ] ( t ) , where ˜ ω t s := ˜ ω s − ˜ ω t for ˜ ω ∈ Ω t and s ∈ [ t , T ]. Since ψ , ψ and − ψ ( t , )are bounded from below, then so is ψ . We shall prove in (iv) below that ψ ∈ C , (Λ t ). Then it follows from (7.5) and (7.6) that ψ ( t , ˜ ω ) ≥ w ( t , ˜ ω ) ≥ ψ ( t , ), and thus ψ ( t, ˜ ω ) ≥ ψ ( t, ˜ ω t ) for t ≥ t . Then, for t ∈ [ t , T ], L ψ = − ∂ t ψ + L ( ψ − ψ ( t, ˜ ω t )) − G ( · , ψ , ∂ ω ψ , ∂ ωω ψ ) ≥ L ( ψ − ψ ( t, ˜ ω t )) + G ( · , ψ , ∂ ω ψ , ∂ ωω ψ )(7.9) − G ( · , ψ , ∂ ω ψ , ∂ ωω ψ ) ≥ . Moreover, by (7.8), (7.6) and (7.5), ψ ( T, ˜ ω ) ≥ ψ ( T, ˜ ω t ) + ( w ( t , ˜ ω ) − ψ ( t , )) e L ( t − T ) ≥ ξ t ,ω (˜ ω t ) + ρ ( d ∞ (( t , ω ) , ( t , ω )) + k ˜ ω k t ) ≥ ξ t ,ω (˜ ω ) . This, together with (7.7) and (7.9), implies that ψ ∈ D ξT ( t , ω ). Then itfollows from (7.6) and (7.4) that u ( t , ω ) ≤ ψ ( t , ) = ψ ( t , ) ≤ w ( t , ) + ε ≤ ψ ( t , ) + Cρ ( d ∞ (( t , ω ) , ( t , ω ))) + ε. Since ψ ∈ D ξT ( t , ω ) and ε > ψ ∈ C , (Λ t ). Let h i , E ,ij correspond to ψ and h i , E ,ij correspond to ψ in Definition 2.5. Define a random index I := inf { i : h i ≥ t } . Set h i := h i for i < I and h i ( ω ) := h i − I ( ω t ) for i ≥ I . Moreover, set E ,i j − := E ,ij ∩ { I > i } and E ,i j := E ,i − Ij ∩ { I ≤ i } , j ≥ h i +1 = h i +1 ∧ t whenever h i < t , it is clear that h i are F -stopping times and ( h ) h i ( ω ) ,ωi +1 ∈ H h i ( ω ) whenever h i ( ω ) < T . From theconstruction of E ,ij one can easily see that { E ,ij , j ≥ } ⊂ F h i and form a ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II partition of Ω t . Moreover, since on each E ,ij , either h i = h i or h i = h i − I ,Definitions 2.5(ii)–(iv) are obvious.It remains to prove { i : h i < T } is finite and lim i →∞ C Lt [( h i ) t,ω < T ] = 0(7.10) for any ( t, ω ) ∈ Λ t . Notice that, denoting by [ i ] the largest integer below i , { h i < T } = (cid:26) h i < T, I > (cid:20) i (cid:21)(cid:27) ∪ (cid:26) h i < T, I ≤ (cid:20) i (cid:21)(cid:27) ⊂ { h i/ < t } ∪ { ω ∈ Ω t : h i/ ( ω t ) < T } . Then { i : h i ( ω ) < T } is finite for all ω . Furthermore, for any L > P ∈ P t L , P [ h i < T ] ≤ P [ h i/ < t ] + P [ { ω ∈ Ω t : h i/ ( ω t ) < T } ] ≤ C Lt [ h i/ < T ] + E P [ P t ,ω [ h i/ < T ]] ≤ C Lt [ h i/ < T ] + C Lt [ h i/ < T ] , and thuslim i →∞ C Lt [ h i < T ] ≤ lim i →∞ [ C Lt [ h i/ < T ] + C Lt [ h i/ < T ]] = 0 . Similarly one can show (7.10) for any ( t, ω ) ∈ Λ t . (cid:3) Proof of Proposition 4.3. In view of Lemmas 7.1 and 7.3, it remainsto prove that u and u are the viscosity L -supersolution and subsolution,respectively, of PPDE (2.14). Without loss of generality, we may assumethat the generator G satisfies (5.1), and we prove only that u is a viscosity L -supersolution at (0 , ).Assume to the contrary that there exists ϕ ∈ A L u (0 , ) such that − c := L ϕ (0 , ) < 0. Following the proof of the partial dynamic programming prin-ciple of Lemma 7.2, we observe that for any ψ ∈ D ξT (0 , ) and any ( t, ω ) ∈ Λ,it is clear that ψ t,ω ∈ D ξT ( t, ω ) and then ψ ( t, ω ) ≥ u ( t, ω ). By the definitionof u in (4.1), there exist ψ n ∈ C , (Λ) such that δ n := ψ n (0 , ) − u (0 , ) ↓ n → ∞ , (7.11) ( L ψ n ) s ≥ ψ ns ≥ u s , s ∈ [0 , T ] . I. EKREN, N. TOUZI AND J. ZHANG Let h be the hitting time required in A L u (0 , ), and since ϕ ∈ C , (Λ) and u ∈ UC b (Λ) ⊂ U , without loss of generality, we may assume L ϕ ( t, ω ) ≤ − c | ϕ t − ϕ | + u t − u ≤ c L , (7.12) for all t ≤ h . We emphasize that the above h is independent of n . Now let { h ni , i ≥ } correspond to ψ n ∈ C , (Λ). Since ϕ ∈ A L u (0 , ), this implies for all P ∈ P L and n, i that 0 ≥ E P [( ϕ − u ) h ∧ h ni ] ≥ E P [( ϕ − ψ n ) h ∧ h ni ] . (7.13)Recall the processes α P , β P in the definition of P ∈ P L [see (2.5)], and denote G P φ := α P · ∂ ω φ + ( β P ) : ∂ ωω φ . Then, applying functional Itˆo formula in(7.13) and recalling that ψ n is a semi-martingale on [0 , h ni ], it follows from(7.11) that δ n ≥ E P [ ψ n − ψ n h ∧ h ni + ϕ h ∧ h ni − ϕ ]= E P (cid:20)Z h ∧ h ni ( ∂ t + G P )( ϕ − ψ n ) ds (cid:21) ≥ E P (cid:20)Z h ∧ h ni (cid:18) c − G ( · , ϕ, ∂ ω ϕ, ∂ ωω ϕ ) + G ( · , ψ n , ∂ ω ψ n , ∂ ωω ψ n )+ G P ( ϕ − ψ n ) (cid:19) ds (cid:21) ≥ E P (cid:20)Z h ∧ h ni (cid:18) c − G ( · , ϕ, ∂ ω ϕ, ∂ ωω ϕ ) + G ( · , u, ∂ ω ψ n , ∂ ωω ψ n )+ G P ( ϕ − ψ n ) (cid:19) ds (cid:21) , where the last inequality follows from (5.1) and the fact that u ≤ ψ n by(7.11). Since ϕ = u , by (7.12) and (5.1), we get δ n ≥ E P (cid:20)Z h ∧ h ni (cid:18) c − G ( · , u , ∂ ω ϕ, ∂ ωω ϕ ) + G ( · , u , ∂ ω ψ n , ∂ ωω ψ n )+ G P ( ϕ − ψ n ) (cid:19) ds (cid:21) . Now let η > n , define τ n := 0, and τ nj +1 := h ∧ inf { t ≥ τ nj : ρ ( d ∞ (( t, ω ) , ( τ nj , ω ))) + | ∂ ω ϕ ( t, ω ) − ∂ ω ϕ ( τ nj , ω ) | + | ∂ ωω ϕ ( t, ω ) − ∂ ωω ϕ ( τ nj , ω ) | + | ∂ ω ψ n ( t, ω ) − ∂ ω ψ n ( τ nj , ω ) | + | ∂ ωω ψ n ( t, ω ) − ∂ ωω ψ n ( τ nj , ω ) | ≥ η } . ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II Recalling Definitions 2.5(iii)–(iv), we see the uniform regularity of ψ n on[0 , h ni ] for each i . Then, together with the smoothness of G and ϕ , one caneasily check that τ nj ↑ h as j → ∞ . Thus δ n ≥ (cid:20) c − Cη (cid:21) E P [ h ∧ h ni ]+ X j ≥ E P [( τ nj +1 ∧ h ni − τ nj ∧ h ni ) × ( G ( · , u , ∂ ω ψ n , ∂ ωω ψ n ) − G ( · , u , ∂ ω ϕ, ∂ ωω ϕ )+ G P ( ϕ − ψ n )) τ nj ]= (cid:20) c − Cη (cid:21) E P [ h ∧ h ni ]+ X j ≥ E P (cid:20) ( τ nj +1 ∧ h ni − τ nj ∧ h ni ) × (cid:18) α τ nj · ∂ ω ( ψ n − ϕ ) + 12 β τ ni : ∂ ωω ( ψ n − ϕ ) + G P ( ϕ − ψ n ) τ nj (cid:19)(cid:21) for some appropriate α τ nj , β τ nj . Now choose P n ∈ P L such that α P n t = α τ nj , β P n t = β τ nj for all τ nj ≤ t < τ nj +1 . Then δ n ≥ [ c − Cη ] E P n [ h ∧ h ni ]. Set η := c C ,send i → ∞ and recall from Definition 2.5 that lim i →∞ C L ( h ni < T ) = 0.This leads to δ n ≥ c E P n [ h ] ≥ E L [ h ], and by sending n → ∞ , we obtain E L [ h ] = 0. However, since h ∈ H , by [8], Lemma 2.4, we have E L [ h ] > (cid:3) 8. On Assumptions 3.8 and 3.2(i). Sufficient conditions for Assumption 3.8. In this subsection we dis-cuss the validity of our Assumption 3.8 which is clearly related to the clas-sical Perron approach, the key argument for the existence in the theory ofviscosity solutions, as shown by Ishii [13]. However, our definition of v and v involves classical supersolutions and subsolutions, while the classical defini-tion in [13] involves viscosity solutions. We remark that Fleming and Vermes[10, 11] have some studies in this respect. The main issue here is to approx-imate viscosity solutions by classical supersolutions or subsolutions. This isa difficult problem which requires some restrictions on the nonlinearity. Inthis section, we provide some sufficient conditions, and we hope to addressthis issue in a more systematic way in future research. I. EKREN, N. TOUZI AND J. ZHANG For ease of presentation, we first simplify the notation in Assumption 3.8.Let O := { x ∈ R d : | x | < } , O := { x ∈ R d : | x | ≤ } ,∂O := { x ∈ R d : | x | = 1 } ;(8.1) Q := [0 , T ) × O, Q := [0 , T ] × O,∂Q := ([0 , T ] × ∂O ) ∪ ( { T } × O ) . We shall consider the following (deterministic) PDE on Q : L v := − ∂ t v − g ( s, x, v, Dv, D v ) = 0 in Q and(8.2) v = h on ∂Q. We remark that in (3.3) the generator g is independent of x . Assumption 8.1. (i) g and h are continuous in ( t, x );(ii) g is uniformly Lipschitz continuous in ( y, z, γ ) and uniformly ellipticin γ .As in Lemma 3.7, under the above assumption, we see that PDE (8.2)has a unique viscosity solution v , and the comparison principle holds in thesense of viscosity solutions within the class of bounded functions. Define v ( t, x ) := inf { w ( t, x ) : w classical supersolution of PDE (8.2) } ,v ( t, x ) := sup { w ( t, x ) : w classical subsolution of PDE (8.2) } . By the comparison principle we have v ≤ v ≤ v .Denote S d + := { γ ∈ S d : γ ≥ } . The following proposition is the main resultof this section: Proposition 8.2. Under Assumption 8.1, we have v = v if g is eitherconvex in γ or the dimension d ≤ . Proof. For the case d ≤ g is convex in γ . As in (5.1), weassume without loss of generality that g ( · , y , · ) − g ( · , y , · ) ≤ y − y for all y ≥ y . (8.3)For any α > 0, we define O α := { x ∈ R d : | x | < α } , Q δ := [0 , (1 + α ) T ) × O α , and similar to (8.1), define their closures and boundaries. Let µ, η be ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II smooth mollifiers on Q and Q × R × R d × S d , and define for any α ′ > h α ( t, x ) := ( h ∗ µ α ) (cid:18) t α , x (1 + α ) (cid:19) , ( t, x ) ∈ Q α ,g ( t, x, y, z, γ ) := min ( t ′ ,x ′ ) ∈ Q { g ( t ′ , x ′ , y, z, γ ) + 2 ρ ( | t − t ′ | + | x − x ′ | ) } ,g α ′ := ( g ∗ η α ′ ) , ( t, x, y, z, γ ) ∈ Q × R × R d × S d . By the uniform continuity of g , we have c ( α ′ ) := k g − g α ′ k ∞ → α ′ ց g α ′ := g α ′ − c ( α ′ ) and g α ′ := g α ′ + c ( α ′ ) . By our assumptions on g and h , it follows from Theorem 14.15 of Lieber-man [15] that there exist v α,α ′ , v α,α ′ ∈ C , ( Q α ) ∩ C ( Q α ) solutions of theequations( E α,α ′ ) : − ∂ t v − g α ′ ( · , v, Dv, D v ) = 0 in Q α and v = h α on ∂Q α , ( E α,α ′ ) : − ∂ t v − g α ′ ( · , v, Dv, D v ) = 0 in Q α and v = h α on ∂Q α , respectively. In particular, their restriction to Q are in C , ( Q ). By thecomparison principle, v α,α ′ ≤ v α,α ′ . Moreover, it follows from (8.3) that g α ′ ( · , y + 2 c ( α ′ ) , · ) ≤ g α ′ ( · , y, · ) − c ( α ′ ) = g α ′ ( · , y, · ) . This shows that v α,α ′ + 2 c ( α ′ ) is a classical supersolution of ( E α,α ′ ), andtherefore v α,α ′ + 2 c ( α ′ ) ≥ v α,α ′ ≥ v α,α ′ . Additionally, notice that the solutions v α,α ′ , v α,α ′ are bounded uniformlyin α, α ′ for α, α ′ small enough. The generators g α ′ , g α ′ have the same uni-form ellipticity constants as g , and they verify the hypothesis of Theo-rem 14.13 of Liebermann [15] uniformly in α ′ . Therefore v α,α ′ , v α,α ′ areLipschitz continuous with the same Lipshitz constant for all α, α ′ . Then,denoting h α,α ′ := v α,α ′ | ∂Q and h α,α ′ := v α,α ′ | ∂Q , this implies that c ( α, α ′ ) := max {k h α,α ′ − h k ∞ , k h α,α ′ − h k ∞ } −→ α → , uniformly in α ′ . Now for fixed ε > 0, choose α , α ′ > c ( α , α ′ ) < ε/ α ′ > c ( α ′ ) ≤ ε/ 4. Then w α ,α ′ := v α ,α ′ + c ( α , α ′ ) and w α ,α ′ := v α ,α ′ − c ( α , α ′ ) are respectively the classical supersolution and subsolution of (8.2)on Q . Thus w α ,α ′ ≤ v and w α ,α ′ ≥ v . Therefore, v − v ≤ w α ,α ′ − w α ,α ′ = v α ,α ′ − v α ,α ′ + 2 c ( α , α ′ ) ≤ c ( α ′ ) + 2 c ( α , α ′ ) ≤ ε. Then it follows from the arbitrariness of ε that v = v . (cid:3) I. EKREN, N. TOUZI AND J. ZHANG A weaker version of Assumption 3.2 (i). We remark that, while seem-ingly reasonable, the uniform continuity of G in ( t, ω ) is violated even forsemilinear PPDEs when the diffusion coefficient σ depends on ( t, ω ). In thissubsection we weaken the uniform regularity in Assumption 3.2 slightly soas to fit into the framework of Pham and Zhang [17], which deals withpath-dependent Bellman–Isaacs equations associated to stochastic differen-tial games. Assumption 8.3. There exist a modulus of continuity functions ρ , ˜ ρ such that, for any ( t, ω ) , (˜ t, ˜ ω ) ∈ Λ and any ( y, z, γ ), | G ( t, ω, y, z, γ ) − G (˜ t, ˜ ω, y, z, γ ) |≤ ˜ ρ ( | t − ˜ t | )[ | z | + | γ | ] + ρ ( d ∞ (( t, ω ) , (˜ t, ˜ ω ))) . Recall the parameters ε, δ, η and the functions v ij introduced in the proofof Lemma 6.4. Notice that Assumption 3.2 is used only in the proof ofLemma 6.4, more precisely in (6.14) and (6.17). We also note that the smoothfunctions v ij are typically constructed as the classical solution to some PDE,as in Section 8 and in [17], and thus satisfy certain estimates. Assume thefollowing:There exists a constant C η > 0, which may depend on η (and ε ), but is independent of δ , such that | Dv ij ( t, x ) | ≤ C η , | D v ij ( t, x ) | ≤ C η for all ( t, x ) ∈ Q ε . (8.4)We claim that Lemma 6.4, hence our main result, Theorem 4.1, still holdstrue if we replace Assumption 3.2 by (8.4) and Assumption 8.3.Indeed, in (6.14), note that G ( t, ω, ( v , Dv , D v )( t, ω t )) − g , ( t, ( v , Dv , D v )( t, ω t ))= G ( t, ω, ( v , Dv , D v )( t, ω t )) − G ( t, , ( v , Dv , D v )( t, ω t )) ≤ ρ ( ε ) , thanks to Assumption 8.3. Thus we still have (6.14).To see (6.17) under our new assumption, we first note that as in (6.17)and by (5.1), L ψ ( t, ω ) ≥ ρ (2 ε ) + ε − ρ (3 T δ ) − G ( t, ω, v ij (˜ t, x ) , Dv ij (˜ t, x ) , D v ij (˜ t, x ))+ G (˜ t ∧ T, ω ( s i ,y j ) ·∧ s i , v ij (˜ t, x ) , Dv ij (˜ t, x ) , D v ij (˜ t, x )) . Now by Assumption 8.3 and (8.4) we have, at (˜ t, x ) ∈ Q ε , G ( t, ω, v ij , Dv ij , D v ij ) − G (˜ t ∧ T, ω ( s i ,y j ) ·∧ s i , v ij , Dv ij , D v ij )= G ( t, ω, v ij , Dv ij , D v ij ) − G ( t, ω ˆ π ·∧ t , v ij , Dv ij , D v ij ) ISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES II + G ( t, ω ˆ π ·∧ t , v ij , Dv ij , D v ij ) − G (˜ t ∧ T, ω ( s i ,y j ) ·∧ s i , v ij , Dv ij , D v ij ) ≤ ρ ( k ω − ω ˆ π ·∧ t k t ) + ˜ ρ ( | t − ˜ t ∧ T | )[ | Dv ij | + | D v ij | ]+ ρ ( d ∞ (( t, ω ˆ π ·∧ t ) , (˜ t ∧ T, ω ( s i ,y j ) ·∧ s i ))) ≤ ρ (2 ε ) + C η ˜ ρ ( T δ ) + ρ ( d ∞ (( t, ω ˆ π ·∧ t ) , (˜ t ∧ T, ω ( s i ,y j ) ·∧ s i ))) . Thus L ψ ( t, ω ) ≥ ε − ρ (3 T δ ) − C η ˜ ρ ( T δ ) − ρ ( d ∞ (( t, ω ˆ π ·∧ t ) , (˜ t ∧ T, ω ( s i ,y j ) ·∧ s i ))) . Substituting this inequality to (6.17), we see that the rest of the proof ofLemma 6.4 remains the same.8.3. Concluding remarks. We now summarize the conditions under whichwe have the complete wellposedness result. Theorem 8.4. Assume the following hold true: • Assumptions 3.1 and 3.2 (ii) ; • Assumptions 3.3 and 3.5 or, more specifically, the sufficient conditions ofLemma 3.6; • G is either convex in γ or the dimension d ≤ ; • Assumption 3.2 (i) , or more generally, Assumption 8.3 and (8.4).Then the results of Theorem 4.1 hold true. We conclude with some final remarks on our assumptions. We first notethat the highly technical requirements of the space C , (Λ) are needed onlyin the proofs, and are not part of our assumptions. Assumptions 3.1 and3.3 are more or less standard, and are in fact the conditions used in [8].In particular, due to the failure of the dominated convergence theorem un-der E P L , the regularity of the involved processes become crucial, and someassumptions on regularity of data are more or less necessary.Assumption 3.5 on the additional structure of ξ is purely technical, dueto our current approach. Indeed, in situations where we have a represen-tation for the viscosity solution, for example, in the semilinear case, as in[8], Section 7, this assumption is not needed. We believe this assumptioncan also be removed if we consider path-dependent HJB equations wherethe function θ εn in Lemma 6.3 can be constructed directly via second-orderBSDEs.The uniform continuity of G in ( t, ω ) in Assumption 3.2(i) excludes thedependence of the diffusion coefficient σ on ( t, ω ) for stochastic control orstochastic differential game problems (see [8], Section 4 and [17]) and thus I. EKREN, N. TOUZI AND J. ZHANG is not desirable. This is due to our approach of approximating PPDEs bypath-frozen PDEs. This assumption may not be needed if we do not use thisapproximation.The uniform nondegeneracy of G in Assumption 3.2(ii) is of course serious,as in PDE literature.Finally, Assumption 3.8 is crucial in our current approach. For path-dependent HJB equations, namely when G is convex in γ , we have, more orless, complete results in the uniformly nondegenerate case. However, in thepresent paper we verify this assumption by the existence of classical solu-tions of the mollified path-frozen PDE. Unfortunately, for Bellman–Isaacsequations, we are able to obtain classical solutions only when d ≤ 2; see[17]. It will be very interesting to explore more PDE estimates to see if wecan verify Assumption 3.8 directly without getting into classical solutionsof high-dimensional Bellman–Isaacs equations.We note that the essential point of our whole argument is to find approxi-mations u ε , u ε ∈ C , (Λ) such that L u ε ≥ ≥ L u ε . Assumptions 3.2, 3.5 and3.8 all serve this purpose. There is potentially an alternative way to provethe comparison principle directly. Let u be a viscosity subsolution and u aviscosity supersolution such that u T ≤ u T . Instead of mollifying the PDE toobtain classical solutions, we may try to mollify u i directly so that the corre-sponding u i,ε will be automatically smooth (in some appropriate sense). Infact, in the PDE literature, the convex/concave convolution exactly servesthis purpose. However, in this case, the main challenge is that we need tocheck that u ,ε is a classical subsolution, and u ,ε a classical supersolution,which, if true, will imply the comparison immediately. It will be interestingto explore this approach as well in future research.REFERENCES [1] Bayraktar, E. and Sˆırbu, M. (2013). Stochastic Perron’s method for Hamilton–Jacobi–Bellman equations. SIAM J. Control Optim. Buckdahn, R. , Ma, J. and Zhang, J. (2015). Pathwise Taylor expansions for ran-dom fields on multiple dimensional paths. Stochastic Process. Appl. Cont, R. and Fourni´e, D.-A. (2013). Functional Itˆo calculus and stochastic integralrepresentation of martingales. Ann. Probab. Crandall, M. 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(2014). Monotone schemes for fully nonlinear parabolic pathdependent PDEs. Journal of Financial Engineering I. EkrenDepartment of MathematicsETH ZurichHG E 66.1 Ramistrasse 101 8092 ZurichSwitzerlandE-mail: [email protected] N. TouziCMAPEcole Polytechnique ParisRoute de Saclay 91128Palaiseau CedexFranceE-mail: [email protected]