{'fraction_non_alphanumeric': 0.08095297805642633, 'fraction_numerical': 0.018557993730407524, 'mean_word_length': 3.3906628495926006, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 15, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 27, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Viscosity solutions of path-dependent PDEs with randomized time

We introduce a new definition of viscosity solution to path-dependent partial differential equations, which is a slight modification of the definition introduced in [8]. With the new definition, we prove the two important results till now missing in the literature, namely, a general stability result and a comparison result for semicontinuous sub-/super-solutions. As an application, we prove the existence of viscosity solutions using the Perron method. Moreover, we connect viscosity solutions of path-dependent PDEs with viscosity solutions of partial differential equations on Hilbert spaces.


Introduction
This paper studies viscosity solutions of the fully nonlinear path-dependent partial differential equation −∂ t u(t, ω) − G t, ω, u(t, ω), ∂ ω u(t, ω), ∂ 2 ωω u(t, ω) = 0 on [0, T) × Ω. (1.1) Here, T > 0 is a given terminal time and ω ∈ Ω is a continuous path from [0, T] to R m starting from the origin. The path derivatives ∂ t , ∂ ω , ∂ 2 ωω were first introduced in the work of Dupire [6]. See also [2] for the related Itô calculus. Such equations arise naturally in many applications. For example, the dynamic programming equation associated with a stochastic control problem of non-Markov diffusions (see [10]) and the one associated with a stochastic differential game with non-Markov dynamics (see [22]) both fall in the class of equation (1.1). The notion of nonlinear path-dependent partial differential equations was first proposed by Peng [19]. We also refer to Peng and Wang [20] for a study on classical solutions of semilinear equations.
The notion of viscosity solutions studied in this paper is a slight modification over the one introduced in Ekren et al. ([8]) in the semilinear context and further extended to the fully nonlinear case in [10,11]. Following the lines of the classical Crandall and Lions notion of viscosity solutions ( [5]), supersolutions and subsolutions are defined through tangent test functions. However, while Crandall and Lions consider pointwise tangent functions, the tangency conditions in the path-dependent setting is in the sense of the expectation with respect to an appropriate class of probability measures P . We refer to [27] for an overview, and to [3,7,14,16,23,25,26,28] for some of the generalizations.
Regardless of the successful development mentioned above, there were some difficulties for the theory of viscosity solutions coming from the definition adopted in [8]. First, a good stability result was missing. In [10] the authors proved a stability result in the sense that if a sequence of uniformly continuous solutions u n uniformly converges to a uniformly continuous function u, then u is also a solution. The assumptions of the uniform continuity and the uniform convergence are often too strong for applications, for example, for the Perron method to prove the existence of viscosity solution. Secondly, also a comparison result for semicontinuous viscosity sub-/supersolutions was missing. The aim of the present paper is to fill these two theoretical gaps. We do this by slightly modifying the definition in [8]. This modification is sufficient to let us overcome the technical difficulties for proving stability and comparison results, but does not compromise the other results till now obtained in the literature, such as existence.
In the previous definition, adopted in [8], the test function ϕ is urged to be tangent to the (sub)solution u at a point (say 0) in the sense that (1.2) where T is the set of all stopping times taking values in [0, T], and P is a family of probability measures on the path space Ω on which B is the canonical process. In the arguments for proving the stability and the comparison, we often need to solve the optimal stopping problem on the right hand side of (1.2), which is not a simple task and requires the uniform continuity of u (see [9]). This turns out to be one of the main difficulties in improving the stability and the comparison results. In the present paper, in order to overcome this fundamental difficulty, we randomize the optimal stopping problem, and the new definition reads where P is a family of probability measures on the time-path product space [0, T] × Ω on which (T, B) is the canonical process. It turns out that the randomized problem can be solved for less regular functions u, and is more stable. With this change of definition, we manage to prove a general stability result for semicontinuous viscosity solutions (Theorem 4.5). The recipe of our proof is composed of the classical argument for stability in [4] and the measurable selection theorem. Moreover, by using the stability result, we are also able to prove the existence of viscosity solution through Perron's method (Theorem 5.3). Further, we prove a comparison result for semicontinuous sub-/super-solutions under some strong assumptions (Theorem 6.9), by approximating semicontinuous solutions with Lipschitz continuous solutions to approximating equations.
Another major contribution of this paper is to connect the path-dependent partial differential equation (1.1) with the partial differential equation where H is a Hilbert space into which the path space Ω can embed, A is an unbounded operator, and D x 0 , D 2 x 0 x 0 are the first-and second-order differentials with respect to a finite dimensional subspace of H. Both equations (1.1) and (1.3) can be used to characterize the value function of non-Markov stochastic control problem. In this paper, we prove that a viscosity solution to the path-dependent equation (1.1), under some regularity assumption, is also a viscosity solution to the corresponding equation (1.3) (Theorem 7.6). The theory of viscosity solutions for partial differential equations on Hilbert spaces (we mainly refer to [12]) is designed of a large class of equations not rescrited to those of delay type, as (1.3). We notice that, till now, when applied to PDEs of the form (1.3), such a theory can deliver a comparison result under more general assumption on the nonlinearity function G, but only for more regular sub-/super-solutions. Moreover, the theory of path-dependent equations as here developed can treat solutions (semi)continuous in the L ∞ -norm, which cannot be settled in a Hilbert space framework.
The rest of the paper is organized as follows. Section 2 introduces the main notations. Section 3 presents the modified definition of the viscosity solutions to the path-dependent partial differential equations. In Section 4 we prove the stability result (Theorem 4.5), and using it in Section 5 we prove the existence of viscosity solution with Perron's method (Theorem 5.3). In Section 6 we show the comparison result for semicontinuous solutions (Theorem 6.9). In Section 7 we clarify the connection between the path-dependent equation and the correponding equation on the Hilbert space (Theorem 7.6). Finally, we complete some proofs in Appendix.

Notations
Canonical Space. Let m > 0 be a natural number, T > 0 be a real number. Define We denote by | · | the Euclidean norm in R m and by | · | ∞ the uniform norm on Ω. In this paper, we study equations set on the spacetime space: For technical reasons, we also often work on the enlarged canonical space: where S m is the space of symmetric m × m real matrices endowed with the supremum norm and Hereafter, we will reserve the letter ω for a generic element of Ω, the letter θ for the generic couple θ = (t, ω) ∈ Θ, and the letter ϑ for the generic element ϑ = (θ, a, µ, q) ∈ Θ. We introduce on Θ the pseudo-metric d ∞ defined by Remark 2.1. In this paper, without being otherwise stated, the (semi-)continuity on the canonical spaces is under | · | ∞ . The (semi-)continuity related to other pseudo-metrics will be explicitly expressed.
Remark 2.2. Note the following facts: For any function ξ : Ω → R taking values in some set R, and for any θ = (t, ω) ∈ Θ, we denote Similarly, given a function u : Θ → R, we denote, for θ ∈ Θ, Similarly, we can also shift functions defined on the enlarged canonical space Θ.
Probability Space. We denote by P the set of probability measures on ( Θ, G T ). Unless otherwise specified, the set P is always endowed with the topology of the weak convergence. We recall that Θ is a Polish space with respect to the product topology, hence P is a Polish space too. We denote by T, B, A, M, Q the first, second, third, fourth, and fifth projection of Θ, respectively We stress the fact that T is a G-stopping time and B, A, M, Q are all G-adapted processes. In order to simplify the notations, we also denote the quintuple (T, B, A, M, Q) by X,
Using the canonical processes, we have for all P ∈ P L that T ∈ [0, T], B = A+M P-a.s.
M is a P-martingale
Recall that T is a G-stopping time, so B T∧· , A T∧· , M T∧· are G-adapted and M T∧· is a P-martingale.
We introduce the sublinear and superlinear expectation operators associated with P L :

Definition of P L -viscosity solution
In this paper, we consider the fully nonlinear parabolic path-dependent PDE (PPDE): In [27,26], it is showed that one can define viscosity solutions for PPDEs via jets. In this manuscript, we start directly from the definition via jets. For α ∈ R, β ∈ R m , γ ∈ S m , let where 〈·, ·〉 denotes the standard scalar product on R m . A further tool for the type of localization that we will implement in the definition of viscosity solution if the function H δ . For any δ ∈ (0, T], define the function H δ : Θ → [0, T] by Definition 3.1 (P L -viscosity sub-/supersolution). Let G : Θ × R × R m × S m → R be a function. A d ∞ -upper semicontinuous (resp. d ∞ -lower semicontinuous) function u : Θ → R, locally bounded from above (resp. locally bounded from below) is a P L -viscosity subsolution (resp. P L -viscosity supersolution) of (3.1) if A locally bounded continuous function u is a P L -viscosity solution of (3.1) if it is both a P Lviscosity sub-and supersolution of (3.1).
Note that the space Θ ′ misses the time dimension, compared to the canonical space Θ in the present paper. Define the nonlinear expectation E ′ L := sup P∈P ′ L E P and the subjet where T is the set of all F B -stopping times. Similarly we can define the superjet J As we see, in our new definition of subjet the function (u − ϕ) θ H δ ∧· reaches its maximum at 0 in the sense of E L instead of max τ∈T E ′ L . By doing so, the maximization over the stopping times τ ∈ T is replaced by the maximization of the laws applied on the canonical variable T. Indeed, the new nonlinear expectation E L is a randomized optimal stopping operator. In general, one need fewer assumptions to ensure the existence of In particular, the optimal probability P * for the first optimization exists if f is d ∞ -u.s.c. and bounded from above (see Section A), while in [9] the authors proved for the second optimization the optimal stopping time τ * exists if f is bounded d ∞ -uniformly continuous. This change allows the viscosity solution under the new definition to have better properties. For example, it allows us to prove a stronger stability of solutions (see Section 4) and a comparison result for semicontinuous solutions (see Section 6). Finally, the change of definition does not threat the already proven results in the pathdependant PDE literature, namely, comparison [11,26,28], convergence of numerical schemes [25], etc. In fact, the arguments for these results will stay in the same lines, while necessary modifications need to be made concerning the optimization problems in (3.4).

Stability
For all G-stopping time τ ≤ T, define the family: P L (τ, ϑ) := P ∈ P L : T ≤ T − τ(ϑ), P-a.s. . We list the following important properties concerning the families of probabilities P L and P L (τ, ϑ). The proofs are postponed to Appendix.
Define the nonlinear (conditional) expectations: Proposition 4.3. Let f : Θ → R be d ∞ -u.s.c. bounded from above, and τ be a continuous G-stopping time.
We have In particular we have Moreover, let P * ∈ P L be such that Then we have As explained in Remark 3.2, in this section we will exploit the advantage of the new definition of the jets in order to prove a better stability result. As we will see in the rest of the paper, one may apply many pseudo-metrics on Θ other than d ∞ .
In what follows, we denote by d a pseudo-metric on Θ which is d ∞ -continuous and such that d(θ, θ ′ ) = 0 implies t = t ′ and ω t∧· = ω ′ t∧· . In order to recall that d has these properties, we will often write d ≪ d ∞ . Then, for any (α, β, γ) ∈ J L u(θ), there existθ n ∈ Θ and (α n , β n , γ) ∈ J L u n (θ n ) such that Proof.
For the simplicity of notation, we denote Since P L is compact (Proposition 4.1) and the maps P → R, P → E P [U n ] are upper semicontinuous, there exists P n ∈ P L for each n ∈ N such that By considering a subsequence if necessary, again denoted by P n , we can find P * ∈ P L such that P n → P * . By Skorohod's representation there exists a probability space ( Ω, G , P) on which where d = denotes equality in distribution. Together with (4.3) we obtain The first inequality in the last line is due to Fatou's lemma, the second one is due to the definition of u, whereas the last equality is due to the assumption (α, β, γ) ∈ J L u(θ) and by choosing any δ sufficiently small. Therefore which, together with (α, β, γ) ∈ J L u(θ), provides Since H δ is continuous, from (4.6) and by denoting T * = T(X * ), T n = T(X n ), we have (4.7) Further, denoting B * = B(X * ) and B n = B(X n ), by (4.4),(4.6) and by d ∞ -continuity of d, we have (4.8) Step 2. It follows from (4.3) and Proposition 4.3 that Therefore, for all n ∈ N, and by (4.7), (4.8) and (4.9) we have (4.10) Step 3. It follows from (4.6) that P lim n→∞ T n = 0 = 1.
Together with (4.5) we obtain By the definition of u we know that lim sup n→∞ u θ n n (T n , B n ) ≤ u(θ) P-a.s. Therefore, Together with (4.10), we obtain (4.11) It follows from (4.11) that there exist ϑ ′ and n such that Finally, since the constants δ, ε can be chosen arbitrarily small, the desired result follows.
As direct consequence of Proposition 4.4 we have the following If u n is a P L -viscosity subsolution of then u is a P L -viscosity subsolution of

Existence of P L -viscosity solution: Perron's method
As we see in the classical literature on the viscosity solutions, one can apply the stability and the comparison results for semi-continuous solutions to prove the existence of viscosity solution via the so-called Perron's method. In Section 4, we have established a quite general stability result (Theorem 4.5), and we will leave the discussion on the comparison result to the next sections. In this section, we adapt the Perron's method to the context of P L -viscosity solution, assuming some comparison result holds true.
ii) For any d-u.s.c. viscosity subsolution u and d-l.s.c. viscosity supersolution v of PPDE (3.1) taking values between u * and v, we have u ≤ v on Θ.
Before proving Theorem 5.3, we show some useful lemmas. The following proposition is a direct corollary of the stability result in Proposition 4.4.
Proposition 5.4. Let u be d ∞ -u.s.c., θ ∈ Θ and (α, β, γ) ∈ J L u(θ). Suppose also that u n is a sequence of d ∞ -u.s.c. functions uniformly bounded from above such that Then there existθ n ∈ Θ, (α n , β n , γ) ∈ J L u n (θ n ) such that Proof. Recall the definition of u in Proposition 4.4. We clearly have It the follows that (α, β, γ) ∈ J L u(θ). Then the desired result follows from Proposition 4.4.
Then φ is a locally bounded d ∞ -u.s.c. function and is a P L -viscosity subsolution to By the dynamic programming result (see e.g. Theorem 2.3 in [18]), we have Then it is easy to verify ψ is a P L -viscosity subsolution to and, using (5.3), it is not difficult too see that φ is a P L -viscosity subsolution to (5.2).
Let θ n ∈ Θ and P n ∈ P ′ L such that d ∞ (θ n , θ) → 0 and P n → P. By Skorohod's representation there exists a probability space ( Ω, G , P) on which where the inequality is due to Fatou's lemma and the and P ′ L is compact (see [17]). By Proposition 7.32 of [1], we obtain that φ is d ∞ -l.s.c.
Proof. We only show the result for subsolution. The one for the supersolution follows from the same argument. First, notice that d ≪ d ∞ implies that u * is d ∞ -u.s.c. Now let θ ∈ Θ and (α, β, γ) ∈ J u * (θ). By the very definition of u * , it is clear that the assumptions of Proposition 5.4 are fulfilled (with the functions u n , u appearing in the statement replaced by u, u * , respectively). Hence there existθ n ∈ Θ, (α n , β n , γ) ∈ J u(θ n ) such that Since u is a P L -viscosity subsolution to (3.1), we have . Now take any φ ∈ D, and define, for θ ′ ∈ Θ, It is easy to verify that Φ is d ∞ -u.s.c. and that, by the viscosity subsolution property of φ and φ, Φ is a P L -viscosity subsolution to (3.1). Then, it follows from Lemma 5.8 that the d-u.s.c. envelop Φ * is also a P L -viscosity subsolution to (3.1). Finally note that which is a contradiction to the definition of u. Therefore, we have u(θ) = u(0).
Proof of Theorem 5.3. Thanks to Lemma 5.9, we only need to check the property of viscosity solution at the point θ = (0, 0).
Step 1. We first prove that u is d-u.s.c. and that it is a P L -viscosity subsolution. Let (α, β, γ) ∈ J u * (0, 0), where u * is the d-u.s.c. envelop of u. By the definition of u and u * , there exists a sequence of u n ∈ D and θ n ∈ Θ such that It follows from Proposition 5.4 that there existθ n ∈ Θ and (α n , β n , γ) ∈ J u n (θ n ) such that Since u n are all P L -viscosity subsolution to (3.1), we have If we let n → ∞, we obtain −α − G (0, 0), u * (0, 0), β, γ ≤ 0. Therefore, u * is a P L -viscosity subsolution. By definition of u, we have u * ≤ u. On the other hand, by definition of u * , we also have u * ≥ u. So finally we conclude that u = u * is d-u.s.c. and a P L -viscosity subsolution.
Step 2. Let u * be the d ∞ -l.s.c. envelop of the function u. We now prove u * is a P L -viscosity Therefore, for δ small enough we have For the simplicity of notation, we denote for all θ ∈ Θ By recalling the definitions of E ′ L and E L , we have Step 1 we know that u is d-hence d ∞ -u.s.c. Then, by Lemma 5.6, it follows that φ is a locally bounded d ∞ -u.s.c. function and a P L -viscosity subsolution to . By the Lipschitz continuity assumptions on G and Lemma 5.5, we then have that φ is a P L -viscosity subsolution to Recall that u, φ are both d ∞ -u.s.c. and that H δ is continuous, and observe that u Further, since u is a P L -viscosity subsolution to (3.1) and φ is a P L -viscosity subsolution to (5.8) on {t < H δ (ω)}, it is easy to verify that U is a P L -viscosity subsolution to (3.1). Then it follows from Lemma 5.8 that the d-u.s.c. envelop of U, namely U * , is a P L -viscosity subsolution to (3.1). However, by definition of u * there is a sequence θ n ∈ Θ such that d ∞ θ n , (0, 0) → 0 and u * (0, 0) = lim n→∞ u(θ n ), and thus lim inf The second last inequality is due to the d ∞ -l.s.c. of the function φ. Therefore, there is θ n such that U * (θ n ) > u(θ n ), which is in contradiction with the definition of u.
Step 4. Note that u * ≤ u * ≤ v, in particular, u * (T, ·) = ξ. By Step 3 and Assumption 5.2, we have u ≤ u * . Together with the definition of u * , we have that u = u * is d-l.s.c. and a P L -viscosity supersolution to (3.1). Then, recalling what proved in Step 1, we conclude that u is a d-continuous P L -viscosity solution to (3.1).

Comparison result for ← − d p -semicontinuous solutions
We have seen that for Perron's method the comparison result for semicontinuous solutions is crucial. However, up to now there is no such result for the fully nonlinear path-dependent PDE in the literature. The main difficulty, as explained in Remark 3.2, is due to the optimal stopping problem presented in (3.4). As an advantage of our modified definition of P L -viscosity solutions, we are able to show such a comparison result by combining the comparison result for uniformly continuous solutions proved in [28] and the convolution in backward pseudo-metric developed in [24].
In the present section we will deal with the pseudo-metric ← − d p on Θ defined, for 1 ≤ p < ∞, by where we extend ω, ω ′ on the negative real line by ω s = ω ′ s = 0 for s < 0. Notice that Remark 6.1. Let us we recall the pseudo-metric d p (1 ≤ p < ∞) on Θ defined in [28]: where we set ω s : The comparison result in this section will hold under the following strong assumption on the nonlinearity G. (ii) For every (θ, β) ∈ Θ × R m , Note that if Assumption 6.2 holds true then Assumption 5.1 holds true with d = ← − d p .

Proof.
A similar comparison result, in which u, v, G are d p -uniformly continuous, is proved under the old definition of viscosity solution in [28]. As mentioned in Remark 3.2, the change of the definition does not add trouble for proving the existing comparison result. Further, we can indeed apply the same argument as in [28] to prove the desired comparison result, where the d p -uniform continuity is replaced by ← − d p -uniform continuity. Since the whole argument is too long, we refer the reader to [28] for the technical details.
In the rest of this section, we show a comparison result for ← − d p -semicontinuous solutions. The main idea is to approximate the semicontinuous solutions by uniform continuous functions with the following convolution. For a bounded ← − d p -u.s.c. function u, we define u n (θ) := sup Then u n is a bounded ← − d p -Lipschitz function and u n → u pointwise as n → ∞.
Assumption 6.4. For a function u : Θ → R, there is a constant C 0 such that where we adopt the convention 0 0 = 0. Example 6.5. Let u, v be two ← − d p -Lipschitz functions with Lipschitz constant C and assume that u(T, ·) = v(T, ·) and u ≤ v on Θ. Then all the functions u, such that u ≤ u ≤ v, satisfy Assumption 6.4. Lemma 6.6. Let u be a bounded ← − d p -u.s.c. function satisfing Assumption 6.4. Then we have u n (T, ·) = u(T, ·) with u n defined in (6.1) for n big enough.
The main advantage of the convolution in (6.1) is that u n inherits the viscosity subsolution property of u. We first prove a lemma, which is an adaptation of the important Lemma 3.3 of [28] to the new definition of P L -viscosity solution. Note that in Lemma 3.3 of [28] the result holds true only for uniformly continuous functions, while the following lemma is proved for u.s.c. functions. This improvement is due to our new definition.
Proof. Let U := u − ϕ α,β,γ and define the value function V : Θ → R by where ϑ = (t, ω, a, µ, q) ∈ Θ is such that θ = (t, ω). Notice that the definition of V (θ) does not depend on the representative ϑ. By compactness of P L and by upper semicontinuity of the map we can find P * ∈ P L such that On the other hand, by the assumption of the lemma, we have Therefore, By taking into account (6.4) and (6.5), we conclude that there exists ϑ * ∈ Θ such that t * < H δ (ω * ) and U(θ * ) = V (θ * ), which is equivalent to (6.3).
If G satisfies Assumption 6.2, we define the modulus of continuity ρ G by Proposition 6.8. Let u be a ← − d p -u.s.c. P L -viscosity subsolution to (3.1), bounded by a constant C > 0, and let u n be defined as in (6.1). Assume that G satisfies Assumption 6.2. Then, for n big enough, u n is a P L -viscosity subsolution to the following equation: Proof. Let (α, β, γ) ∈ J L u n (θ). Then for any ε > 0 we have for a suitably δ > 0 arbitrarily small. By definition of u n and by (6.6), we can find θ * ∈ Θ such that Further, we have It is important to note that Now we apply Lemma 6.7 and obtain that there exists θ such that and thus, by the subsolution property of u, where ρ(ω t∧· , ·) is the modulus of continuity of the path ω t∧· . By using the definition of ρ G together with (6.7) and (6.8), we get Finally, we let δ, ε tend to 0 and obtain −α − G(θ, β, γ) ≤ ρ G 2C+1 n . Now we are ready to prove the main result of this section. Theorem 6.9. Let G satisfy Assumption 6.2. Let u (resp. v) be a bounded ← − d p -u.s.c. (resp. l.s.c.) P L -viscosity subsolution (resp. supersolution) to PPDE (3.1). In addition, assume that u, v satisfy Assumption 6.4. Then, if u(T, ·) ≤ v(T, ·), we have u ≤ v on Θ.
Proof. As a direct consequence of Proposition 6.8, we know that, for n sufficiently large, is a bounded ← − d p -uniformly continuous P L -viscosity subsolution to PPDE (3.1) and u n (T, ·) = u n (T, ·). Further, by Lemma 6.6, we have u n (T, ·) = u(T, ·) and thus u n (T, ·) = u(T, ·) for n big enough. We can similarly define v n , so that v n → v pointwise and that v n is a bounded ← − d puniformly continuous P L -viscosity supersolution to (3.1) and v n (T, ·) = v(T, ·) for n big enough.

Representation of PPDEs as PDEs in infinite dimension and comparison under weak-continuity
The aim of this section is to start with a P L -viscosity (sub-/super-)solution u defined on the space Θ, then to associate to it a function u defined on a product space [0, T] × H where H is a suitably chosen Hilbert space, and finally show that u is a viscosity (sub-/super-)solution of a PDE on [0, T] × H. As a corollary of such relationship, we can exploit the comparison theorem available for viscosity solutions in Hilbert spaces to obtain uniqueness for P L -viscosity solutions. We start by introducing the Hilbert space H on which we will set our new PDE associated with the original PPDE (3.1). Then we will address the problem of associating a function on the original space Θ with a function on the product space [0, T] × H. In order to perform this change of variable in such a way to end up with a function regular enough to exploit the comparison theorem for viscosity solutions in Hilbert spaces, we need to introduce a pseudo-metric d B on Θ, weaker than ← − d p , and an associated norm | · | B on H, weaker than the original norm | · | H . Once provided these preliminaries, we can introduce the PDE on [0, T] × H associated with the original PPDE on Θ, recall the notion of viscosity solution in Hilbert spaces, and prove the main theorem of this section (Theorem 7.6), thanks to which we can use [12,Theorem 3.50] to get uniqueness of P L -viscosity solutions (Corollary 7.7).
For the theory of viscosity solutions in Hilbert spaces we always refer to [12]. For the basic notions of stochastic calculus in Hilbert spaces that we need, we refer to [21].
We start by introducing the Hilbert spaces where H ′ is endowed with its standard scalar product 〈·, ·〉 L 2 induced by the L 2 -norm and H is endowed with scalar product and norm given by We next consider the C 0 -semigroup S on H defined by The infinitesimal generator A of S is given by To express the regularity assumptions for the comparison results (Theorem 7.6 and Corollary 7.7), we will consider the following pseudo-metric d B on Θ: Similarly, for u : It is clear that the functions G * H , G * H , u * H , u * H are well-defined, because for all (t, x) ∈ [0, T] × H we can find a sequence (t, ω n ) ∈ Θ such that (ω n (t), ω n (· + t)1 [−t,0] ) converges to (x 0 , x 1 1 [−t,0] ) in the norm | · | H , hence in | · | B .
We will now recall the definition of viscosity solution for PDEs on Hilbert spaces as provided by [12,Ch. 3], and it is in order to fit such a framework that we write (7.8) by emphasizing the maximal dissipative operator A − I. LetĜ : [0, T] × H × R × R m × S m → R be a function. Consider the following parabolic equation x 0 x 0 v) = 0 on (0, T) × H, (7.8) where  (ii) h ∈ C 1,2 ((0, T) × R, R) and is such that, for every t ∈ (0, T), h(t, ·) is even and h(t, ·) is nondecreasing on R + .
The first main result of this section is the following Theorem 7.6. Let u be a P L -viscosity subsolution (resp. supersolution) of (3.1). Then u * H (resp. u * H ) is a H-viscosity subsolution (resp. supersolution) of Theorem 7.6 allows to exploit the comparison for H-viscosity solutions ([12, Theorem 3.50]) to get as corollary a comparison result for P L -viscosity solutions (Corollary 7.7). As a byproduct, we also obtain a sufficient condition for the Perron-type result we proved in Section 5. Corollary 7.7. Let G satisfy Assumptions 5.1, with d = d B . Let u be a bounded P L -viscosity subsolution of (3.1) and let v be a bounded P L -viscosity supersolution of (3.1). Suppose that lim r,η→0 Then u ≤ v on Θ. Moreover, Assumption 5.2 is fulfilled with u := u, v := v, d := d B .
Before proving Theorem 7.6 and Corollary 7.7, we need a preliminary discussion useful to relate the two different settings of P L -viscosity solutions and of viscosity solutions on H as defined by Definition 7.4. (x 0 , x 1 )). Notice that, for any (s, x) ∈ H, the measurability of U entails the measurability of (7.13) where, for any function ω ∈ Ω and (s, x) ∈ [0, T] × H, Let (t, x) ∈ [0, T] × H and let U : [0, T] × H → R be a measurable function, locally bounded from above (resp. locally bounded from below). We define the jet J H Now let u n be a sequence of d ∞ -u.s.c. functions uniformly bounded from above and let us define a function u H : [0, T] × H → R (resp. u H ) by  H (t, x)), there exist θ n ∈ Θ, (α n , β n , γ) ∈ J L u n (θ n ) such that t n → t, (ω n (t n ), ω n (· + t n )1 [−t n ,0] ) and u n (θ n ), α n , β n → u H (t, x), α, β (resp. u n (θ n ), α n , β n → u H (t, x), α, β ).
Proof. The desired result is very similar to Proposition 4.4 and can be proved with the same argument.
Proposition 7.9. Let u be a P L -viscosity subsolutions to (3.1). Then x). By the very definition of u * H , we can apply Proposition 7.8 to get sequences θ n ∈ Θ and (α n , β n ) ∈ R × R m such that (α n , β n , γ) ∈ J L u(θ n ) and H (t, x), α, β). (7.18) Since u is a P L -viscosity solution to (3.1), we have, for all n, −α n − G(t n , ω n , u n (θ n ), β n , γ) ≤ 0. By taking the limit n → ∞, by using (7.18), and by the very definition of G * H , we obtain which concludes the proof.
In what follows, for t ∈ [0, T] and P ∈ P L , we denote by G t the translated filtration {G 0∨(s−t) } s∈[0,T] and by G P,t is predictable, and hence also the map is predictable. We then obtain the predictability of the left-hand side derivative Now let y 1 ∈ L 2 ((−∞, 0), R m ). By the very definition of Z t,x s , we have, P-a.s.,  By collecting (7.21), (7.22), (7.23), (7.24) and recalling that y 0 , y 1 were arbitrarily chosen, we obtain (7.20).
Notice that the H-valued process Z t,x defined by (7.19) is independent on P ∈ P L and everywhere continuous, whereas equation (7.20) depends on the chosen P through the chosen version of the stochastic integral.
Proof of Theorem 7.6. We write the proof only for the subsolution case, since the other case is symmetric.
By the very definition, u * H is |·|+|·| B -u.s.c., hence by compactness of (A−I) −1 (Proposition A.5) it is weakly sequentially u.s.c. Now let ψ(s, y) = ϕ(s, y) + h(s, |y| H ) be a test function and let (t, x) ∈ (0, T) × H be a maximum for Notice that A −1/2 is maximal dissipative. Let A n := nA(n− A) −1 , n ≥ 1, be the Yosida approximation of A. Then there exists n ≥ 1 such that |S n t | L(H) ≤ e t , t ≥ 0, n ≥ n, where S n denotes the continuous semigroup generated by A n .
Now we fix P ∈ P L . For n ≥ 1, define the process Z t,x  Let Z be defined as in (7.19). Notice that τ(Z t,x ) is a G t -stopping time and τ(Z t,x n ) is a G P,t -stopping time. For n ≥ 1, define on ( Θ, G P,t , P) the continuous processes Y t,x and Y t,x and  Now, since A n generates a continuous group in L(H), the process Y t,x n is the strong solution to the following linear SDE in integral form on ( Θ, G P,t , P): n )∧s−t P-a.s., ∀s ∈ [t, T).

(7.32)
By the assumptions on the test function ψ = ϕ + h, by using (7.32), and by recalling that ) and take the expectation, to get, for n ≥ n, n,r ) + 〈A * n ∇ x ϕ(r, Z t,x n,r ), Z t,x n,r 〉 dr t 〈∇ x h(r, Z t,x n,r ), Z t,x n,r 〉dr Tr Q r−t D 2 x 0 x 0 ψ(r, Z t,x n,r ) dr , where we have used the fact that h is radial and that 〈A n y, y〉 ≤ |y| 2 H ∀y ∈ H, n ≥ n.
We will now take in consideration each term appearing in the formula above and pass to the limit as n → ∞ to get a useful inequality for E ψ τ(Z . Using (7.20), (7.25), and the standard machinery based on the factorization formula for stochastic convolutions with C 0 -semigroups (see [21, and recall that the quadratic variation of M is bounded in our case) one can see that This entails immediately the two following facts for any p ∈ [1, 2). By the assumptions on ψ and by (7.36) we can then pass to the limit in (7.33) and obtain, x 0 x 0 ψ(r, Z t,x r ) dr . (7.37) Since P was arbitrary, (7.37) holds for all P ∈ P L . Now, x 0 x 0 ψ are uniformly continuous on bounded sets, recalling the definition of τ = τ ρ,ε in (7.26), and noticing that ε ↓ 0 implies τ ε (Z t,x ) ↓ t P-a.s., one can easily see that Then, for any arbitrarily small real number ζ > 0, there exists ε such that, for all P ∈ P L and all G t -stopping time ρ ≥ t, , and by collecting (7.37), (7.38a), (7.38b), (7.38c), (7.38d), we obtain, for all P ∈ P L and all ρ ≥ t, by the very definition of Z t,x in (7.19), we see that we can choose δ * such that If we now define ρ : where we have used the equality (recall (7.13) for the definition of U s,y 0 ,y 1 ). By (7.40) we have that (α, β, γ) ∈ J H L u * H (t, x), hence, by Proposition 7.9, we obtain −α − G * H (t, x, u * H (t, x), α, β, γ) ≤ 0, or, equivalently, We can now conclude by letting ζ tends to 0.
Proof of Corollary 7.7. First notice that, by Theorem 7.6, u * H and v * H are sub-and supersolutions of x 0 x 0 u) = 0 whereĜ := G * H = G * H is well-defined due to Assumption 5.1(i). We want to show that [12, Theorem 3.50] applies to u * H , v * H , from which the corollary follows immediately. But it is easy to see that Assumptions 5.1 imply that Hypotheses 3.44-3.49 in [12] are verified with F(t, x, r, β, γ) := −〈x, β〉 −Ĝ(t, x, r, β, γ), whereĜ := G * H = G * H . In particular, the nuclearity condition Hypothesis 3.47 is automatically satisfied in our case because the second order term inĜ is finite dimensional, and Hypothesis 3.48 comes from the ellipticity condition (Assumption 5.1(iii)), from the local uniform continuity (Assumption 5.1(i)), and from the Lipschitz continuity of G (Assumption 5.1(iv)). Finally, condition (3.72) in [12, p. 206], with u, v replaced by our u * H , v * H , follows by (7.12).

A Appendix
We first address some properties of P L . Let γ ∈ (0, 1], ℓ > 0. Define Lemma A.2. For all γ ∈ 0, 1 2 , and ε > 0, there exists ℓ > 0 such that Proof. Notice that, if a ∈ A( Θ L ) and q ∈ Q( Θ L ), where Θ L is defined in (2.1), then |a s −a s ′ | ≤ L|s−s ′ | and |q s − q s ′ | ≤ L|s − s ′ |, for all s, s ′ ∈ [0, T]. Moreover, since by definition P( Θ L ) = 1 for all P ∈ P L , we only need to show that there exists ℓ > 0 such that L is compact (Remark A.1), and by applying Prokhorov theorem taking into account Lemma A.2, we obtain that P L is tight. To conclude the proof, it remains to show that P L is closed. Let {P n } n∈N ⊂ P L be a sequence converging to P in P . We need to show that P ∈ P L , i.e., M is a P-martingale, 〈M〉 = Q P-a.s.
Step 1. We show that P( Θ L ) = 1. We first notice that Θ L is closed. So by weak convergence of {P n } n∈N to P, we have P( Θ L ) ≥ lim sup n→∞ P n ( Θ L ) = 1.
Step 2. We show that M is a P-martingale. First, notice that the process | M| ∞ is continuous and that (for some constants c 1 > 0) Hence, by weak convergence, we have Let now τ k (ϑ) := inf{s ∈ [0, T] : s + |µ s∧· | ∞ ≥ k} ∧ t. Notice that τ k is continuous on Θ and that it is an G-stopping time. Hence M τ k is continuous and bounded on Θ. Since M is a P n -martingale for all n, we can write, for all t, t ′ ∈ [0, T], t ′ < t, ϕ bounded continuous G t ′ -measurable functions, using the weak convergence of {P n } n , This shows that M τ k ∧· is a P-martingale. Due to the fact that τ k ր t pointwise as k → ∞, we obtain, by passing to the limit in (A.4) after recalling (A.3), and hence we conclude that M is a P-martingale.
Step 3. We show that 〈M〉 = Q P-a.s. To this aim, it is sufficient to show that where M (i) denotes the ith component of the vector M and Q (i, j) denotes the (i, j) entry of the matrix Q. After noting that that (for some c 2 > 0) we can proceed exactly as in Step 2.
In the following discussion, we aim to prove the dynamic programming result stated in Proposition 4.3.

Proof of Lemma 4.2.
We need to show that P * ∈ P L and T ≤ T − τ(ϑ * ), P * -a.s. The former is due to the result of Proposition 4.1, so it remains to prove the latter. Since P n → P * , by Skorohod representation there exists a probability space ( Ω, G , P) on which where d = denotes equality in distribution. Since P n ∈ P L (τ, ϑ n ) and τ is continuous, it follows from Fatou's lemma that which concludes the proof. Proposition A.3. Let f : Θ → R be d ∞ -u.s.c. and bounded from above, and τ be a continuous G-stopping time. Define Then, there is a G T∧τ -measurable kernel ν : Θ → P such that ν(ϑ) ∈ P L (τ, ϑ) for all ϑ ∈ Θ and Proof. First note that the equality in (A.6) follows from Remark 2.3. Consider the set D ⊂ Θ× P L defined by D := (ϑ, P) : ϑ ∈ Θ, P ∈ P L (τ, ϑ) . For all P ∈ P L (τ, ϑ) define f : D → R by f (ϑ, P) := E P f t∧τ(ϑ),ω .
We claim that f is jointly u.s.c., where on D we consider the topology induced by Θ × P L , with Θ endowed with its product topology. Then let |ϑ n − ϑ * | ∞ → 0 and P n → P * . By Skorohod's representation, we have a probability space ( Ω, G , P) in which (A.5) holds true. It follows that lim sup n→∞ f (ϑ n , P n ) = lim sup n→∞ E P f t n ∧ τ(ϑ n ) + T n , ω n ⊗ t n ∧τ(ϑ n ) B n ≤ E P lim sup n→∞ f t n ∧ τ(ϑ n ) + T n , ω n ⊗ t n ∧τ(ϑ n ) B n ≤ E P f t * ∧ τ(ϑ * ) + T * , ω * ⊗ t * ∧τ(ϑ * ) B * = f (ϑ * , P * ), where the first inequality is due to Fatou's lemma and the second one is due to the d ∞ -u.s.c. of f . Next note that f (ϑ, P).
Proof. The first statement follows from the definition (A.8) if f is the indicator function 1 A . The general case follows by a standard approximation procedure. Now the fact P ⊗ τ ν ∈ P L follows easily by considering that P ∈ P L , that ν takes values in P L , and by using (A.8).
Finally, we prove the dynamic programming result.
Proof of Proposition 4.3. By definition of nonlinear expectation and by the fact that any P L (τ, ϑ) contains the dirac measure in ϑ = 0, we immediately have the inequality Regarding the converse inequality, we first note that, by Proposition A.3, there is a G t∧τ -measurable kernel ν : Θ → P such that E τ L f t∧τ,ω (ϑ) = E ν(ϑ) f t∧τ,ω . Therefore which shows the desired inequality. Now (4.2) follows by setting τ = T in (4.1).
To prove the last part of the proposition, let P * ∈ P L be an optimal probability measure such that E L [ f ] = E P * [ f ]. It follows from the definition of nonlinear expectation and (4.2) that Therefore, E P Proof of Proposistion 7.2. The fact that (7.5) is continuous in case f ∈ W 1,2 ((0, T), R) follows by Lemma 7.1 and by the discussion in [29,Remark 2.6], or directly by integration by part and Hölder's inequality. We now prove the converse. We assume m = 1; the proof for m > 1 goes along the same lines. Let f ∈ L 2 ((0, T), R) be such that (7.5) is d B -continuous. We want to show that f ∈ W 1,2 ((0, T), R). Let V denote the subspace of R × L 2 (R − , R) whose elements are the pairs x = (x 0 , x 1 ) where x 1 is continuous on [−T, 0] and x 0 = x 1 (0). By Lemma 7.1 we have that the linear functional is continuous. We can then extend it to a continuous linear functional Λ on the Hilbert space completion (H B , | · | B ) of (V , | · | B ), which coincides with the completion of (H, | · | B ) because V is | · | H -dense in H and | · | B is weaker than | · | H .
By the Riesz representation theorem it then follows that there exists z ∈ H B such that Since y ∈ D(A * ) = R × W 1,2 (R − ) and since (A.14) holds in particular for all x ∈ V , we conclude f (−·) ∈ W 1,2 ((−T, 0), R).