Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions

We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We develop first a local theory of classical solutions and define then viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative one using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. When the diffusion coefficient is semi-linear (but the drift can be fully nonlinear), we establish a complete theory, including global existence and comparison principle. Our methodology relies heavily on the method of characteristics.


Introduction
We study the fully nonlinear second-order SPDE du (t, x, ω) = f (t, x, ω, u, ∂ x u, ∂ 2 x x u) dt + g (t, x, ω, u, ∂ x with initial condition u(0, x, ω) = u 0 (x), where (t, x) ∈ [0, ∞) × R, B is a standard Brownian motion defined on a probability space ( , F, P), f and g are F B -progressively measurable random fields, and • denotes the Stratonovic integration. Our investigation will build on several aspects of the theories of pathwise solutions to SPDEs studied in the past two decades. These include: the theory of stochastic viscosity solutions, initiated by Lions and Souganidis (1998a;1998b;2000a;2000b) and also studied by Buckdahn and Ma (2001a;2001b;2002); path-dependent PDEs (PPDEs) studied by Buckdahn et al. (2015), based on the notion of path derivatives in the spirit of Dupire (2019); and the aspect of rough PDEs studied by Keller and Zhang (2016), in terms of the rough path theory (initiated by Lyons (1998)) and using the connection between Gubinelli's derivatives for "controlled rough paths" (2004) and Dupire's path derivatives. The main purpose of this paper is to integrate all these notions into a unified framework, in which we shall investigate the most general well-posedness results for fully nonlinear SPDEs of the type (1.1).

A brief history
SPDE (1.1), especially when both f and g are linear or semilinear, has been studied extensively in the literature. We refer to the well-known reference Rozovskii (1990) for a fairly complete theory on linear SPDEs and to Krylov (1999) for an L p -theory of linear and some semilinear cases. When SPDE (1.1) is fully nonlinear, as often encountered in applications such as stochastic control theory and many other fields (cf. the lecture notes of Souganidis (2019), and Davis and Burstein (1992), Buckdahn and Ma (2007), and Diehl et al. (2017) for applications in pathwise stochastic control problems), the situation is quite different. In fact, in such a case one can hardly expect (global) "classical" solutions, even in the Sobolev sense. Some other forms of solutions will have to come into play.
In a series of works, Lions-Souganidis (1998a;1998b;2000a;2000b) initiated the notion of "stochastic viscosity solutions" for fully nonlinear SPDEs, especially in the case when g = g(∂ x u), along the following two approaches. One is to use the method of stochastic characteristics (cf. Kunita (1997)) to remove the stochastic integrals of SPDE (1.1), and define the (stochastic) viscosity solution by considering test functions along the characteristics (whence randomized) for the transformed ω-wise (deterministic) PDEs. The other approach is to approximate the Brownian sample paths by smooth functions and define the (weak) solution as the limit, whenever it exists, of the solutions to the approximating equations, which are standard

The Main contributions of this work
The main purpose of this paper is to establish the viscosity theory for general fully nonlinear parabolic SPDEs and path-dependent PDEs through a unified framework based on the combined rough path and Dupire's pathwise analysis, as well as the idea of stochastic characteristics. We consider the most general case where the diffusion coefficient g is a nonlinear function of all variables (t, ω, x, u, ∂ x u). We shall first obtain the existence of local (in time) classical solutions when all the coefficients are sufficiently smooth. We remark that these results, although not surprising, seem to be new in the literature, to the best of our knowledge. More importantly, assuming that g is smooth enough, we shall establish most of the important issues in viscosity theory. These include: 1) consistency (i.e., smooth viscosity solutions must be classical solutions); 2) the equivalence of the notions of stochastic viscosity solutions using test functions and by semi-jets; 3) stability; and 4) a partial comparison principle (between a viscosity semi-solution and a classical semi-solution). Finally, in the case when g is linear in ∂ x u (but nonlinear in u, and f can be nonlinear in (u, ∂ x u, ∂ x x u)), we prove the full comparison principle for viscosity solutions and thus establish the complete theory.
To be more precise, let us briefly describe alternative forms of SPDEs that are equivalent to the underlying one (1.1) in some specific pathwise senses. First, note that Buckdahn et al. (2015) established the connection between (1.1) and the following path-dependent PDE (PPDE): (t, x, ω, u, ∂ x u). (1.2) Here, ∂ ω t and ∂ ω are temporal and spatial path derivatives in the sense of Dupire (2019). On the other hand, Keller and Zhang (2016) showed that the PPDE (1.2) can also be viewed as a rough PDE (RPDE): where ω is a geometric rough path corresponding to Stratonovic integration. We should note that the connection between SPDE (1.1) and RPDE (1.3) has been known in the rough path literature, see, e.g., Friz and Hairer (2014). Bearing these relations in mind, we shall still define the (stochastic) viscosity solutions via the method of characteristics. More precisely, we utilize PPDE (1.2) by requiring that smooth test functions ϕ satisfy ∂ ω ϕ(t, x) = g(t, x, ϕ, ∂ x ϕ). (1.4) It should be noted that the involvement of g in the definition of test functions is not new (see, e.g., the notion of "g-jets" and the g-dependence of "path derivatives" in Buckdahn and Ma (2001b;2002) and Buckdahn et al. (2015)). The rough-path language then enables us to define viscosity solutions directly for RPDE (1.3) as well as PPDE (1.2) in a completely local manner in all variables (t, x, ω). We should note that, barring some technical conditions as well as differences in language, our definition is very similar or essentially equivalent to the ones in, say, Lions and Souganidis (1998a;2000a); and when f does not depend on ∂ 2 x x u (i.e., in the case of first-order RPDEs), our definition is essentially the same as the one in Gubinelli et al. (2014). Furthermore, we show that our definition is equivalent to an alternative definition through semi-jets (such an equivalence was left open in Gubinelli et al. (2014)). Moreover, by using pathwise characteristics, we show that RPDE (1.3) can be transformed into a standard PDE (with parameter ω) without the dω t term. When g is semilinear (i.e., linear in ∂ x u), our definition is also equivalent to the viscosity solution of the transformed PDE in the standard sense of Crandall et al. (1992), as expected. In the general case when g is nonlinear on all (x, u, ∂ x u), the issue becomes quite subtle due to the highly convoluted system of characteristics and some intrinsic singularity of the transformed PDE, and thus we are not able to obtain the desired equivalence for viscosity solutions. In fact, at this point it is not even clear to us how to define a notion of viscosity solution for the transformed PDE.
Besides clarifying the aforementioned connections among different notions, the next main contribution of this paper is to establish some important properties of viscosity solutions, including consistency, stability, and a partial comparison principle. Our arguments follow some of our previous works on backward PPDEs (e.g., Ekren et al. (2014) and Ekren et al. (2016a;2016b)). However, unlike the backward case, the additional requirement (1.4) leads to some extra subtleties when small perturbations on the test function ϕ are needed, especially in the case of general g. Some arguments for higher-order pathwise Taylor expansions along the lines of Buckdahn et al. (2015) prove to be helpful.
As in all studies involving viscosity solutions, the most challenging part is the comparison principle. The main difficulty, especially along the lines of stochastic characteristics, is the lack of Lipschitz property on the coefficients of the transformed ω-wise PDE in the variable u, except for some trivial linear cases. Our plan of attack is the following. We first establish a comparison principle on small time intervals. Then we extend our comparison principle to arbitrary duration by using a combination of uniform a priori estimates for PDEs and BMO estimates inspired by the backward SDEs with quadratic growth. Such a "cocktail" approach enables us to prove the comparison principle in the general fully nonlinear case under an extra condition, see (6.13). In the case when g is semilinear however, even when f is fully nonlinear (e.g., of Hamilton-Jacobi-Bellman type), we verify the extra condition (6.13) and establish a complete theory including existence and a comparison principle. Thereby, we extend the result of Diehl and Friz (2012), which follows the second approach proposed by Lions and Souganidis (1998a;1998b) and studies the case when both g and f are semilinear. However, the verification of (6.13) in general cases is a challenging issue and requires further investigation.
Another contribution of this paper is the local (in time) well-posedness of classical solutions in the general fully nonlinear case. We first establish the equivalence between local classical solutions of RPDE (1.3) and those of the corresponding transformed PDE. Next, we provide sufficient conditions for the existence of local classical solutions to this PDE, similar to that of Da Prato and Tubaro (1996) when g is linear in u and ∂ x u. To the best of our knowledge, these results for the general fully nonlinear case are new. We emphasize again that our PDE involves some serious singularity issues so that the local existence interval depends on the regularity of the classical solution (which in turn depends on the regularity of u 0 ). Consequently, these results are only valid for classical solutions.

Remarks
As the first step towards a unified treatment of stochastic viscosity solutions for fully nonlinear SPDEs, in this paper we still need some extra conditions on the coefficients f and g. For example, even in the case when g is semilinear, we need to assume that f is uniformly non-degenerate and convex in ∂ x x u. It would be interesting to remove either one, or both constraints on f. Also, as we point out in Remark 7.5, in the general fully nonlinear case the equivalence between our rough PDE and the associated deterministic PDE in the viscosity sense is by no means clear. Consequently, a direct approach for the comparison principle for RPDE (3.6), which is currently lacking, would help greatly. It would also be interesting to investigate the alternative approach by using rough path approximations as in Caruana et al. (2011) and many other aforementioned papers, in the case when g is fully nonlinear. We hope to investigate some of these issues in our future publications.
We would also like to mention that, although the SPDEs in Buckdahn and Ma 2007, Davis and Burstein 1992, Diehl et al. (2017 for pathwise stochastic control problems appear with terminal conditions, they fall into our realm of forward SPDEs with initial conditions by a simple time change (which is particularly convenient here since our rough path integrals correspond to Stratonovic integrals). However, many SPDEs arising in stochastic control theory with random coefficients and in mathematical finance, see, e.g., Peng (1992) and Musiela and Zariphopoulou (2010), have different nature and are not covered by this paper. The main difference lies in the time direction of the adaptedness of the solution with respect to the random noise(s), as illustrated by Pardoux and Peng (1994).
Finally, for notational simplicity throughout the paper, we consider the SPDEs on a finite time horizon [0, T ] and in a one-dimensional setting. Our results can be easily extended to the infinite horizon in most of the cases. But the extension to multidimensional rough paths, albeit technical, is more or less standard. We shall provide further remarks when the extension to the multidimensional case requires extra care. For example, Proposition 4.1 relies on results for multidimensional RDEs. Finally, some of the results in this paper involve higher-order derivatives and related norms. For simplicity, we shall use the norms involving all partial derivatives up to the same order; and our estimates, although sufficient for our purpose, will often contain a generic constant, and are not necessarily sharp. This paper is organized as follows. In Section 2, we review the basic theory of rough paths and rough differential equations (RDEs). Furthermore, we introduce our function spaces and the crucial rough Taylor expansions. In Section 3, we set up the framework for SPDEs, RPDEs, and PPDEs. In Section 4, we introduce the crucial characteristic equations and transform our main object of study, the RPDE (3.6), into a PDE. We establish the equivalence of their local classical solutions and provide sufficient conditions for their existence. Sections 5 and 6 are devoted to viscosity solutions in the general case. In Section 7, we establish the complete viscosity theory in the case that g is semilinear. Finally, in the Appendix (Section 8), we provide the proofs of the results from Section 2 that go beyond the standard literature.

Preliminary results from rough path theory
We begin by briefly reviewing the framework for rough path theory that is used in this paper, mainly following Keller and Zhang (2016) (see Friz and Hairer (2014) and the references therein for the general theory).
To this purpose, we introduce some general notation first. For normed spaces E and V, put When V = R, we omit V and just write L ∞ (E). For a constant α > 0, set Given functions u : [0, T ] → R and u : [0, T ] 2 → R, we write the time variable as subscript, i.e., u t = u(t) and u s,t = u(s, t), and we define Moreover, we shall use C to denote a generic constant in various estimates, which will typically depend on T and possibly on other parameters as well. Furthermore, we define the standard Hölder spaces and parabolic Hölder spaces (cf. Lunardi (1995, Chapter 5)): Given k ∈ N 0 and β ∈ (0, 1], set

Rough path differentiation and integration
Rough path theory makes it possible to integrate with respect to non-smooth functions ("rough paths") such as typical sample paths of Brownian motions and fractional Brownian motions. In this paper, we use Hölder continuous functions as integrators. To this end, we fix two parameters α ∈ (1/3, 1/2] and β ∈ (0, 1] satisfying The parameter α denotes the Hölder exponent of our integrators. The parameter β will take the role of the exponent in the usual Hölder spaces C k+β . Later, we introduce modified Hölder type spaces suitable for our theory.
To be more precise, a rough path, in general, consists of several components, the first stands for the integrator whereas the additional ones stand for iterated integrals. Those additional components have to be given exogenously and a different choice leads to different integrals, e.g., those corresponding to the Itô and to the Stratonovic integral.
In our setting, the situation is relatively simple. We consider a rough pathω := (ω, ω) with only two components ω and ω that are required to satisfy the following conditions: (2.3) Note that ω s,t should not be understood as ω t − ω s as in (2.1).
(iii) In standard rough path theory, it is typically not required thatω is truly rough as defined in (2.3). But it is convenient for us because, under (2.3), the rough path derivatives we define next will be unique.
Next, we introduce path derivatives with respect to our rough path. To this end, we introduce spaces of multi-indices

Remark 2.3 (i)
In the rough path literature, a first-order spatial derivative ∂ ω u is typically called a Gubinelli derivative and the corresponding function u is called a controlled rough path. In our case, the path derivatives defined above are unique due toω being truly rough (Friz and Hairer 2014, Proposition 6.4).
(ii) The derivative ∂ ω u depends on ω, but not on ω. The derivative ∂ ω t u depends on ω as well and should be denoted by ∂ω t u. However, in our setting, ω is a function of ω and thus we write ∂ ω t u instead. (iii) When ∂ ω u = 0, it follows from (2.5) and (2.2) that u is differentiable in t and ∂ ω t u = ∂ t u, the standard derivative with respect to t. (iv) In the multidimensional case, ∂ ωω u ∈ R d×d could be symmetric if u is smooth enough (Buckdahn et al. 2015, Remark 3.3); i.e., ∂ ω i and ∂ ω j commute for 1 ≤ i, j ≤ d. However, typically ∂ ω t and ∂ ω do not commute, even when d = 1.

Remark 2.4
Note that in (2.5) the term t − s is the difference of the identity function t → t, which is Lipschitz continuous. For all estimates below, it suffices to assume ∂ ω t u ∈ C α(2+β)−1 ([0, T ]). However, to make the estimates more homogeneous, we only use the Hölder-2α regularity of t and thus require ∂ ω t u ∈ C αβ ([0, T ]). For this same reason, all of our estimates will actually hold true if we replace t with a Hölder-2α continuous path ζ ∈ C 2α ([0, T ]). To be more precise, we define a path derivative of u with respect to ζ as a function ∂ ω then Lebesgue integration dt should be replaced with Young integration dζ t . (2.7) We emphasize that, besides k, the norms depend on T, ω, α, and β as well. To simplify the notation, we do not indicate these dependencies explicitly. In some places we restrict u to some subinterval [t 1 , t 2 ] ⊂ [0, T ]. Corresponding spaces C k α,β ([t 1 , t 2 ]) are defined in an obvious way. To not further complicate the notation, the corresponding norm is still denoted by · k . Note that, for u ∈ C 1 α,β ([0, T ]) and for a constant C depending on ω, Finally, we define the rough integral of u ∈ C 1 α,β ([0, T ]). Let π : 0 = t 0 < · · · < t n = T be a time partition and |π | := max 0≤i≤n−1 |t i+1 − t i |. By Gubinelli (2004), exists and defines the rough integral. The integration path U t := t 0 u s dω s belongs to C 1 α,β ([0, T ]) with ∂ ω U t = u t and we define t s u r dω r := U s,t . In this context, we define iterated integrals as follows. For ν ∈ V n , set (μ 1 ,···,μ n ) s,r d μ n+1 r for μ = (μ 1 , · · ·, μ n+1 ) ∈ V n+1 . In the multidimensional case, defining iterated integrals is not trivial. Nevertheless, by Lyons (1998, Theorem 2.2.1), this can be accomplished via uniquely determined (higher-order) extensions of the geometric rough pathω = (ω, ω).
By (2.5) and (2.2), the following result is obvious and we omit the proof.

Rough differential equations
We start with controlled rough paths with parameter x ∈ R d . They serve as solutions to RPDEs and coefficients for RDEs and RPDEs. For this purpose, we have to allow d > 1 here. Consider a function u : [0, T ] × R d → R. If, for fixed x ∈ R d , the mapping t → u(t, x) is a controlled rough path, we use the notations ∂ ω u, ∂ ω t u, D ν u to denote the path derivatives as in the previous subsection. For fixed t, we use ∂ x u, ∂ 2 x x u, etc., to denote the derivatives of x → u(t, x) with respect to x. Now, we introduce the appropriate spaces, extending Definition 2.2.
(iii) We say u ∈ C 2,loc α,β ([t 1 , t 2 ] × O) if the following holds: We first show that the differentiation and integration operators are commutative. (2.14) The next result is the crucial chain rule (Keller and Zhang 2016, Theorem 3.4). (2.16) (2.17) Our study relies heavily on the following rough Taylor expansion. The result holds true for multidimensional cases as well and we emphasize that the numbers δ below can be negative.

Lemma 2.9
Let u ∈ C k,loc α,β ([0, T ] × R) and K ⊂ R be compact. Then, for every Proof See the Appendix.
To study RDEs, uniform properties for the functions in C k,loc α,β ([t 1 , t 2 ] × O) are needed. In the next definition, we abuse the notation · k from (2.7).

Definition 2.10 (i) We say that u
(ii) For solutions to standard PDEs (recall Remark 2.3 (iii)), we use (2.20) We remark that in (i) we do not require sup t∈[t 1 ,t 2 ] [∂ k x u(t, ·)] β < ∞, but restrict ourselves to local Hölder continuity with respect to x (uniformly in t), which suffices for our rough Taylor expansion above.
Although functions in C k,0 α,β ([0, T ] × R) are, in general, only at most once differentiable in time, they behave in our rough path framework as if they were k times differentiable in time (Friz and Hairer 2014, section 13.1).

Remark 2.11
, is continuous under · k (as defined in (2.7)) and, for ν = k, D ν u(t, ·) is Hölder-β continuous, uniformly in t. Hence, the continuity required in the definition of C k,loc (as defined in (2.19)).
Now, we study rough differential equations of the form (

2.22)
Proof See the Appendix.
In the following linear case, we have a representation formula for u: This is a direct consequence of Lemma 2.8, and thus the proof is omitted.
Remark 2.14 This representation holds true only in the one-dimensional case. For multidimensional linear RDEs, Keller and Zhang (2016) derived a semi-explicit representation formula. Moreover, note that (2.23) actually does not satisfy the technical conditions in Lemma 2.12 (f and g are not bounded). But nevertheless, due to its special structure, RDE (2.23) is well-posed as shown in this lemma.
Finally, we extend Lemma 2.12 to RDEs with parameters of the form Proof See the Appendix.
Then, for u ∈ C( ), we have Here, the left-hand side is a Stratonovic integral while the right-hand side is a rough path integral. In this sense, we may write SPDE (3.1) as the RPDE (ii) In an earlier version of this paper (see arXiv:1501.06978v1), we studied pathwise viscosity solutions of SPDE (3.1) in the a.s. sense. In this version, we study instead the wellposedness of RPDE (3.5) for fixed ω. This is easier and more convenient. Moreover, the rough path framework allows us to prove crucial perturbation results such as Lemma 5.8.
(iii) If we have obtained a solution (in the classical or the viscosity sense) u(·, ω) of RPDE (3.5) for each ω, to go back to SPDE (3.1), one needs to verify the measurability and integrability of the mapping ω → u(·, ω). To do so, one can, in principle, apply the strategy by Da Prato and Tubaro (1996, section 3), which relies on construction of solutions to SDEs via iteration so that adaptedness is preserved. This strategy can be applied in our setting and does not require f and g to be continuous in ω. Another possible approach is to follow the argument by Friz and Hairer (2014, section 9.1), which is in the direction of stability and norm estimates but requires at least g to be continuous in ω. Since the paper is already very lengthy, we do not pursue these approaches here in detail.
From now on, we shall fix (α, β) and ω as in Section 2.1 and omit ω in f, g, and u. To be precise, the goal of this paper is to study the RPDE In particular, ∂ ω t u is different from ∂ t u in the standard PDE literature. Moreover, by Lemma 2.5, we may write (3.6) as the path-dependent PDE The arguments of f and g are implicitly denoted as f (t, x, y, z, γ ) and g (t, x, y, z). Throughout this paper, the following assumptions are employed.
Note that, for any bounded set Assumption 3.4 Let u 0 be continuous and u 0 ∞ ≤ K 0 .
We remark that for RPDE (3.6) there is no comparison principle in terms of g. Hence, a smooth approximation of g does not help for our purpose and thus we require g to be smooth. By more careful arguments, we may figure out the precise value of k 0 , but that would make the paper less readable. In the rest of the paper, we use k to denote a generic index for regularity, which may vary from line to line. We always assume that k is large enough so that we can freely apply all the results in Section 2, and we assume that the regularity index k 0 in Assumption 3.2 is large enough so that we have the desired k-regularity in the related results.
We say that u is a classical solution (resp., subsolution, supersolution) of RPDE (3.6) if Again, note that there is no comparison principle in terms of g. So the first line in (3.8) is an equality even for sub/super-solutions.

Classical solutions of rough PDEs
We establish wellposedness of classical solutions for RPDE (3.6). To this end, we must require that the coefficients f, g and the initial value u 0 are sufficiently smooth. For general RPDEs, most results are valid only locally in time. However, this is sufficient for our study of viscosity solutions in the next sections.

The characteristic equations
Our main tool is the method of characteristics (see Kunita (1997) for the stochastic setting). It will be used to get rid of the diffusion term g and to transform the RPDE into a standard PDE. Given θ := (x, y, z) ∈ R 3 , consider the coupled system of RDEs Proposition 4.1 Let Assumption 3.2 hold and let K 0 ≥ 0 be a constant. Then there exist constants δ 0 > 0 and C 0 , depending only on K 0 and the k 0 -th norm of g (in the sense of Definition 2.10 (i)) on [0, T ] × Q, such that for all θ ∈ Q, the system (4.1) has a unique solution (θ) such that Proof Uniqueness follows directly from an appropriate multidimensional extension of Lemma 2.12 for each θ ∈ Q. To prove existence, we note that the main difficulty here is that some coefficients in (4.1) are not bounded. To deal with this difficulty, we introduce, for each N > 0, a smooth truncation function ι N : Next, for each θ ∈ R 3 , consider the system Applying Lemma 2.15, but extended to the multidimensional case (using the extended Lemma 2.13 as shown in Remark 2.14), the RDE above has a unique solution Next, we linearize system (4.1). To this end, put The next result is due to Peter Baxendale. It is a slight generalization of Kunita (1997, (14), p. 291) (which corresponds to (4.15) below).

RPDEs and PDEs
Our goal is to associate RPDE (3.6) with a function v satisfying which would imply that v solves a standard PDE. To illustrate this idea, let us first derive the PDE for v heuristically. Assume that u is a classical solution of RPDE (3.6) with sufficient regularity. Recall (4.1). We want to find v satisfying (4.7) and In fact, recall (4.4) and writê (4.9) Applying the operator ∂ ω t on both sides of the first equality of (4.8) together with Lemma 2.8 yields We emphasize that the variable θ t (x) above is fixed when Lemma 4.2 is applied, while the variable t in V t is viewed as the running time. In particular, in the last term above s (θ t (x)) involves both times s and t. Then, by (4.10), By (4.8), u(t,X t ) and ∂ x u(t,X t ) are functions of (t, θ t (x)). Moreover, by applying the operator ∂ x on both sides of the second equality of (4.8), Therefore, formally v should satisfy the PDE Now, we carry out the analysis above rigorously. We start from PDE (4.10) and derive the solution for RPDE (3.6). Recall (2.20) and that k is a generic, sufficiently large regularity index that may vary from line to line.
Let δ 0 be determined by Proposition 4.1. Then there exists a constant δ ∈ (0, δ 0 ] such that the following holds: Recall that, by Definition 2.10 (i), the regularity here is uniform in x. Thus, together with the regularity of v, we have (4.14) has a unique solutionS ∈ C k,loc α,β ([0, δ] × R). Now, by (i), we see thatS actually satisfies RDE (4.12).
Theorem 4.4 Let Assumption 3.2 hold and v and δ be as in Lemma 4.3. Assume further that v is a classical solution (resp., subsolution, supersolution) of PDE (4.10). Since . We prove only the subsolution case. The other statements can be proved similarly.
and thus Since v is a classical subsolution of (4.10)-(4.11), the definition of F yields .
Now, we proceed in the opposite direction, namely deriving v from u. Assume that u ∈ C k α,β ([0, T ] × R) for some large k and define K 0 := u ∞ ∨ ∂ x u ∞ . Let Q 2 and Q be as in (4.2) and δ 0 as in Proposition 4.1. For any fixed (t, x) ∈ [0, δ 0 ] × R, consider the mapping from Q 2 to R 2 . The Jacobi matrix of this mapping is given by Note that det(J (0, x, y, z)) = 1. Thus, noting also that ∂ x u and ∂ 2 x x u are bounded, one can see, similarly to (4.13), that there exists a δ ≤ δ 0 such that det(J (t, x, y, z)) ≥ 1/2 for all (t, x, y, z) ∈ [0, δ] × Q. This implies that the mapping (4.21) is one to one and the inverse mapping has sufficient regularity. Denote by R(t, x) the range of the mapping (4.21). Then Thus, by (4.13) and the boundedness of ∂ x u, ∂ 2 x x u again, and by choosing a smaller δ if necessary, we may assume that (0, 0) ∈ R(t, x) for all (t, x) ∈ [0, δ] × R. Therefore, for any (t, Differentiating the first equation in (4.22) with respect to x and applying the second, we obtain where the last equality holds true thanks to Lemma 4.2. Then w(t, x) = ∂ x v(t, x) and thus (4.8) holds. In particular, we may use the notation θ t (x) in (4.8) again to replaceθ t (x).
We verify now that v indeed satisfies PDE (4.10).
Theorem 4.5 Let Assumption 3.2 hold, let u ∈ C k α,β ([0, T ] × R) for some large k, and let δ and v be determined as above. Assume further that u is a classical solution (resp., subsolution, supersolution) of RPDE (3.6). Then, for a possibly smaller δ > 0, Proof The regularity of v is straightforward. We prove only the case that u is a classical subsolution. The other cases can be proved similarly.
Recall the notations in (4.9). Differentiating the first equality of (4.8) with respect to ω and applying the second equality, we obtain By (3.8) and (4.8), ∂ ω u(t,X t ) = g(t,X t , u(t,X t ), ∂ x u(t,X t )) = g(t,ˆ t ). Then, by (4.1) and Lemma 4.2, Thus, ∂ ω v(t, x) = 0 and Lemma 4.3 can be applied. In particular, for a possibly Finally, following exactly the same arguments as for deriving (4.10), one can complete the proof that v is a classical subsolution of PDE (4.10).

Remark 4.6
We shall investigate the case with semilinear g in detail in section 7 below. Here, we consider the special case which has received strong attention in the literature. Let σ and σ denote the first-and second-order derivatives of σ , respectively. In this case, the system of characteristic equations (4.1) becomes which has the explicit global solution Moreover, in this case, (4.11) becomes

Local wellposedness of PDE (4.10)
To study the wellposedness of PDE (4.10) and hence that of RPDE (3.6), we first establish a PDE result. Let K 0 > 0 and, similar to (4.2), consider . The further regularity of v when k ≥ 2 follows from standard bootstrap arguments (Gilbarg and Trudinger 1983, Lemma 17.16) together with Remark 2.11. Since the proof is very similar to that of Lunardi (1995, Theorem 8.5.4), which considers a similar boundary-value problem, we shall present only the main ideas for the more involved existence part of the lemma. The first step is to linearize our equation and set up an appropriate fixedpoint problem. To this end, let δ > 0 and define an operator (4.28) Now given v ∈ B 1 , consider the solution w of the linear PDE with w(0, ·) = u 0 . Following the arguments by Lunardi (1995, Theorem 8.5.4), when δ > 0 is small enough, PDE (4.29) has a unique solution w ∈ B 1 . This defines a mapping (v) := w for v ∈ B 1 . Moreover, when δ > 0 is small enough, is a contraction mapping, and hence there exists a unique fixed point v ∈ B 1 . Then v = w and, by (4.29), v solves (4.10) on [0, δ] × R.
Proof Recall (4.11). By the uniform regularity of in Proposition 4.1, one can verify straightforwardly that, for δ > 0 small enough, F satisfies the conditions in Lemma 4.7 (ii). Then, by Lemma 4.7, PDE (4.10)-(4.11) has a classical solution v ∈ B 1 for a possibly smaller δ. Finally, it follows from Theorem 4.4 that RPDE (3.6) has a local classical solution.

The first-order case
We consider the case f being of first-order, i.e., (4.30) This case is completely degenerate in terms of γ . It is not covered by Theorem 4.8. However, in this case, PDE (4.10)-(4.11) is also of first-order, i.e., When f is smooth, so is F. Thus, we can modify the characteristic Eqs. 4.1 to solve PDE (4.10)-(4.31) explicitly. Put˜ = (X ,Ỹ ,Z ) and consider x)). Then one can see that (4.7) should be replaced with ∂ tṽ = 0, and thusṽ(t, x) = u 0 (x). By similar (actually easier) arguments as in previous subsections, one can prove the following statement. (ii) For each t ∈ [0, δ], the mapping x ∈ R →X t (x, u 0 (x), ∂ x u 0 (x)) ∈ R is invertible and thus possesses an inverse function, to be denoted byS t .

Viscosity solutions of rough PDEs: definitions and basic properties
We introduce a notion of viscosity solution for RPDE (3.6) and study its basic properties. For any (t 0 ,

ii) We say that u is a viscosity solution of RPDE (3.6) if it is both a viscosity supersolution and a viscosity subsolution of (3.6).
We remark that it is possible to consider semi-continuous viscosity solutions as in the standard literature. However, for simplicity, in this paper we restrict ourselves to continuous solutions only.
First, assume that u is a viscosity subsolution. By choosing u itself as a test function, we can immediately infer that u is a classical subsolution.

Equivalent definition through semi-jets
As in the standard PDE case (Crandall et al. 1992), viscosity solutions can also be defined via semi-jets. To see this, we first note that, for ϕ ∈ A 0 g u(t 0 , x 0 ; δ), our second-order Taylor expansion (Lemma 2.9) yields Motivated by this, we define semi-jets as follows. Given u ∈ C([0, T ] × R), We then define the g-superjet J g u(t 0 , x 0 ) and the g-subjet J g u(t 0 , x 0 ) by Nevertheless, we still have the following equivalence.

Proposition 5.3 Let Assumptions 3.2 and 3.3 be in force and let u ∈ C([0, T ] × R). Then u is a viscosity supersolution (resp., subsolution) of (3.6) at
Proof We prove only the supersolution case. The subsolution case can be proved similarly.

Remark 5.4 By Proposition 5.3 and its proof, we can see that, depending on the regularity order k 0 of g as specified in Assumption 3.2, it is equivalent to use test functions of class C k
α,β (D δ (t 0 , x 0 )) for any k between 2 and k 0 . This is crucial for Theorem 5.9 below.

Change of variables formula
Let λ ∈ C([0, T ]) and n ≥ 2 be an even integer. For any u : Clearly,f andg inherit the regularity of f and g. Whenever they are smooth, Then it is straightforward to verify thatf andg inherit most desired properties of f and g that we utilize later.
Lemma 5.5 (i) If g is of the form of (7.1) or (7.26), then so isg; and if f is of the form of (7.29), then so isf .
(ii) If f is convex in γ , then so isf .
(iii) If f is uniformly parabolic, then so isf .
(iv) If f is uniformly Lipschitz continuous in y, z, γ , then so isf .
In particular, if f and g satisfy Assumptions 3.2 and 3.3, then so dof andg. However, we remark thatg does not inherit the same form when g is in the form of (4.23). Now consider the RPDE forũ:

Proposition 5.6 Let Assumptions 3.2 and 3.3 be in force, λ ∈ C([0, T ]), n ≥ 2 even, and u ∈ C([0, T ] × R). Then u is a viscosity subsolution (resp., classical subsolution) of RPDE (3.6) if and only ifũ is a viscosity subsolution (resp., classical subsolution) of RPDE (5.11).
Proof The equivalence of the classical solution properties is straightforward. Regarding the viscosity solution properties, we prove the if part; the only if part can be proved similarly.
Assume thatũ is a viscosity subsolution of RPDE (5.11). For any (t 0 , 1+x n ϕ(t, x). It is straightforward to check thatφ ∈ Agũ(t 0 , x 0 ). Then, by the viscosity subsolution property ofũ at This implies that u is a viscosity subsolution of RPDE (3.6). (ii) If f is uniformly Lipschitz continuous in y, by choosing λ sufficiently large (resp., small), we havẽ f is strictly increasing (resp., decreasing) in y.

Remark 5.7 Let ( f, g) satisfy Assumptions 3.2 and 3.3 and let u be a viscosity
(5.13) In particular,f will be proper in the sense of Crandall et al. (1992).
Theorem 5.9 (Stability) Let Assumption 3.2 hold and ( f n ) n≥1 be a sequence of functions satisfying Assumption 3.3. For each n ≥ 1, let u n be a viscosity subsolution of RPDE (3.6) with generator ( f n , g). Assume further that, for some functions f and u, locally uniformly in (t, x, y, z, γ ) ∈ [0, T ] × R 4 . Then u is a viscosity subsolution of (3.6).
Proof By the locally uniform convergence, f and u are continuous. Let (t 0 , x 0 ) ∈ (0, T ] × R and ϕ ∈ A g u(t 0 , x 0 ). We apply Lemma 5.8 at (t 0 , x 0 ), but in the left neighborhood We emphasize that, while for notational simplicity we established Lemma 5.8 in the right neighborhood D + ε (t 0 , x 0 ), we may easily reformulate it to the left neighborhood by using the backward rough paths introduced in (2.12). By Remark 5.4, we may assume without loss of generality that ϕ ∈ C k α,β ([0, T ] × R) for some large k. Then, for any ε > 0 small, by Lemma 5.8, there exists ψ ε ∈ C 4 α,β (D − ε (t 0 , x 0 )) such that the following holds: This together with setting ϕ ε := ϕ + ψ ε yields Since u n converges to u locally uniformly, we have, for n = n(ε) large enough, Note that Then ϕ ε ∈ A g u n (t ε , x ε ). By the viscosity subsolution property of u n , x ε ) ≤ 0. Fix n and send ε → 0. Then, by the convergence of ψ ε and its derivatives, , u is a viscosity subsolution of (3.6).

Partial comparison principle
Here, we assume that at least one of the functions u 1 and u 2 is smooth. We need the following result (cf. Lemma 5.8). and ε > 0, recall (5.14), and consider the RPDE where C depends only on g and ϕ, but not on t 0 , ε, and δ. Moreover, ψ ε satisfies Proof The uniform regularity of ψ ε and the first line of (6.3) are clear. Note The second line of (6.3) follows from the Hölder continuity of the functions in terms of t. Moreover, since g ϕ (t, x, 0, 0) = 0, we may write it as g ϕ (t, x, ψ ε , ∂ x ψ ε ) = σ (t, x)ψ ε +b(t, x)∂ x ψ ε , where σ and b depend on ψ ε . Then we may view (6.2) as a linear RPDE with coefficients σ and b. Thus, by (7.31)-(7.32), we have a representation formula for ψ ε . The uniform regularity of ψ ε implies the uniform regularity of σ and b, which leads to the third line of (6.3).
Theorem 6.2 Let Assumptions 3.2 and 3.3 and (6.1) be in force. If one of u 1 and u 2 is in C k α,β ([0, T ] × R) for some large k, then u 1 ≤ u 2 .

Remark 6.3
When g is independent of y, we can prove Proposition 6.2 much easier without invoking Lemma 6.1. In fact, in this case, assuming to the contrary that By (5.12) and [u 1 − u 2 ](0, ·) ≤ 0, there exists (t * , x * ) ∈ (0, t 0 ] × R such that Define ϕ = u 2 + c. Since g is independent of y, we have Then one can easily verify that ϕ ∈ A g u 1 (t * , x * ). Moreover, by Remark 5.7 (ii), we can assume without loss of generality that f is strictly decreasing in y. Now it follows from the classical supersolution property of u 2 and the viscosity subsolution property of u 1 that, taking values at (t * , x * ), , which is the desired contradiction since f is strictly decreasing in y.

Full comparison
We shall follow the approach of Ekren et al. (2014). For this purpose, we strengthen Assumption 3.2 slightly by imposing some uniform property of g in terms of y.
Assumption 6.5 The diffusion coefficient g belongs to C k 0 ,loc We remark that, under Assumption 3.2, all the results in this subsection hold true if we assume instead that T is small enough.
Proof We prove U = ∅ in several steps. The proof for U is similar.
We remark that it is possible to extend our definition of viscosity supersolutions to lower semi-continuous functions. However, here (i) shows that u is upper semicontinuous. So it seems that the continuity of u in (ii) is intrinsically required in this approach.
Proof By the proof of Lemma 6.6, u is bounded from above. Similarly, u is bounded from below. Then it follows from (6.11) that u and u are bounded.
We establish next the upper semicontinuity for u. The regularity for u can be proved similarly. Fix (t, x) ∈ [0, T ] × R. For any ε > 0, there exists ϕ ε ∈ U such that ϕ ε (t, x) < u(t, x)+ε. By the structure of U , it is clear that ϕ ε ≥ u on [0, T ] × R. Assume that ϕ ε ∈ U corresponds to the partition 0 = t 0 < · · · < t n = T as in (6.5). We distinguish between two cases.
Case 1. Assume t ∈ (t i−1 , t i ) for some i = 1, . . ., n. Since ϕ ε is continuous This implies that u is upper semi-continuous at (t, x).
We finally show that u is a viscosity subsolution provided it is continuous. The viscosity supersolution property of u follows similar arguments.
Proof By Lemma 6.7 and (6.13), it is clear that u = u is continuous and is a viscosity solution of RPDE (3.6). By Theorem 6.2 (partial comparison), u 1 ≤ u and u ≤ u 2 . Thus (6.13) leads to the comparison principle immediately.

Remark 6.9
The introduction of u and u is motivated from Perron's approach in PDE viscosity theory. However, there are several differences.
(i) In Perron's approach, the functions in U are viscosity supersolutions, rather than classical supersolutions. So our u is in principle larger than the counterpart in PDE theory. Similarly, our u is smaller than the counterpart in PDE theory. Consequently, it is more challenging to verify the condition (6.13).
(ii) The standard Perron's approach is mainly used for the existence of viscosity solution in the case the PDE satisfies the comparison principle. Here we use u and u to prove both the comparison principle and the existence.
(iii) In the standard Perron's approach, one shows directly that u is a viscosity solution, while in Lemma 6.7 we are only able to show u is a viscosity supersolution.
The condition (6.13) is in general quite challenging. In the next section, we establish the complete result when the diffusion coefficient g is semilinear.
Clearly, Assumption 7.1 implies Assumption 6.5. Note that in this section, we obtain a global result. Thus, we require that g 0 and its derivatives are uniformly bounded in y as well.
(7.5) (iii) For each t, the mapping x → X t (x) has an inverse function X −1 t (·); and for each (t, x), the mapping y → Y t (x, y) has an inverse function Y −1 t (x, ·).
We remark that the proof below uses (7.5). One can also use the backward rough path in (2.12) to construct the inverse functions directly. This argument works in multidimensional settings as well (Keller and Zhang 2016).
Proof (i) follows directly from Lemma 2.15, which also implies Then the representations in (7.5) follow from Lemma 2.13. Moreover, setX := x +X t (x)). Then, by the uniform regularity of σ , sup x∈R σ (·, x) k ≤ C. This implies that uniformly bounded, uniformly in (t, x). Therefore, we obtain the first estimate for ∂ x X in (7.5). The second estimate for ∂ y Y in (7.5) follows from the similar arguments.
Finally, for each t, the fact ∂ x X t (x) ≥ c implies that x → X t (x) is one to one and the range is the whole real line R. Thus X −1 t : R → R exists. Similarly, one can show that Y −1 t (x, ·) exists.
One can easily check, omitting (x, y, z) in X t (x), Y t (x, y), Z t (x, y, z), and then (4.11) becomes F(t, x, y, z, γ ) Under our conditions, F has typically quadratic growth in z and is not uniformly Lipschitz in y. Moreover, the first equality of (4.8) becomes By using similar arguments as in Section 4.2, we obtain the following result which is global in this semilinear case. The next result establishes equivalence in the viscosity sense.
Remark 7.5 In the general case, there are two major differences: (i) The transformation determined by (4.8) involves both u and ∂ x u, i.e., to extend Theorem 7.4, one has to assume that the candidate viscosity solution u is differentiable in x.
(ii) The transformation is local, in particular, the δ in Theorem 4.5 depends on ∂ 2 x x u ∞ , i.e., unless ∂ 2 x x u is bounded and the solution is essentially classical, we have difficulty to extend Theorem 7.4 to the general case, even in just a local sense.

Some a priori estimates
Here, we establish uniform a priori estimates for v that will be crucial for the comparison principle of viscosity solutions in the next subsection. First, we estimate the L ∞ -norm of v. Proof First, we write (4.10)-(7.6) as Since v is a classical solution, a and b are smooth functions. Reversing the time by ThenŶ t :=v(t,X t ) solves the BSDÊ , we have |F(t, x, y, 0, 0)| ≤ C[1 + |y|] (7.13) following from Lemma 7.2. Then, by standard BSDE estimates, which yields (7.11) for t = T . Along the same lines, one can prove (7.11) for all t > 0.
Remark 7.7 (i) We are not able to establish similar a priori estimates for ∂ x v. Besides the possible insufficient regularity of u 0 , we emphasize that the main difficulty here is not that F has quadratic growth in z, but that F is not uniformly Lipschitz continuous in y. Nevertheless, we obtain some local estimate for ∂ x v in Proposition 7.9, which will be crucial for the comparison principle of viscosity solutions later.
(ii) To overcome the difficulty above and apply standard techniques, Lions and Souganidis (2000a, (1.12)) imposed technical conditions on f in the case f = f (z, γ ): γ ∂ γ f + z∂ z f − f is either bounded from above or from below. (7.14) This is essentially satisfied when f is convex or concave in (z, γ ). Our f in (7.15) below does not satisfy (7.14), in particular, we do not require f to be convex or concave in z. See also Remark 7.13.
The next result relies on a representation of v and BMO estimates for BSDEs with quadratic growth. For this purpose, we restrict f to Bellman-Isaacs type with the Hamiltonian where E := E 1 × E 2 ⊂ R 2 is the control set and e = (e 1 , e 2 ). Lipschitz continuous in (x, y, z) with Lipschitz constant L 0 , and f 0 (t, x, 0, 0, e) is bounded by K 0 .
Remark 7.10 (i) We reverse the time in (7.19). Hence, in spirit of the backward rough path in (7.19), B and the rough path ω (or the original B in (3.1)) have opposite directions of time evolvement. Thus (7.19) is in the line of the backward doubly SDEs of Pardoux and Peng (1994). When E 2 is a singleton, Matoussi et al. (2018) provide a representation for the corresponding SPDE (3.1) in the context of secondorder backward doubly SDEs. We shall remark though, while the wellposedness of backward doubly SDEs holds true for random coefficients, its representation for solutions of SPDEs requires Markovian structure, i.e., the f and g in (3.1) depend only on B t (instead of the path B · ). The stochastic characteristic approach used in this paper does not have this constraint. Note again that our f and g in RPDE (3.6) and PPDE (3.7) are allowed to depend on the (fixed) rough path ω.
(ii) For (7.22), from a game theoretical point of view, it is more natural to use the so-called weak formulation (Pham and Zhang 2014). However, as we are here mainly concerned about the regularity, the strong formulation used by Buckdahn and Li (2008) is more convenient.

The global comparison principle and existence of viscosity solution
We need the following PDE result from Safonov (1988) (Mikulevicius and Pragarauskas (1994) have a corresponding statement for bounded domains and Safonov (1989) has one for the elliptic case).

Remark 7.13
The requirement that f is convex or concave is mainly to ensure the existence of classical solutions for PDE (7.23). Theorem 7.11 holds true for the multidimensional case as well. When the dimension of x is 1 or 2, Bellman-Isaacs equations may have classical solutions as well, see Lieberman (1996, Theorem 14.24) for d = 1 and Pham and Zhang (2014, Lemma 6.5) for d = 2 for bounded domains, and also Gilbarg and Trudinger (1983, Theorem 17.12) for elliptic equations in bounded domains when d = 2. We believe such results can be extended to the whole space and thus the theorem above as well as Theorem 7.14 will hold true when f is indeed of Bellman-Isaacs type. However, when the dimension is high, the Bellman-Isaacs equation, in general, does not have a classical solution (Nadirashvili and Vladut (2007) provide a counterexample).
Proof By Lemma 6.7, u and u are bounded by some C 0 .
When f is semilinear, i.e., linear in γ , clearly under natural conditions f satisfies the requirements in Theorem 7.14. We provide next a simple fully nonlinear example. satisfies the requirements in Theorem 7.14.
Remark 7.16 (i) As pointed out in Remark 7.5, for general g = g(t, x, y, z), the transformation is local and the δ in Theorem 4.5 depends on ∂ 2 x x u ∞ . Then the connection between RPDE (3.6) and PDE (4.10) exists only for local classical solutions, but is not clear for viscosity solutions. Since our current approach relies heavily on the PDE, we have difficulty in extending Theorem 7.4 to the general case, even in just the local sense. We will investigate this challenging problem by exploring other approaches in our future research.
(ii) When f is of first-order, i.e., σ f = 0 in (7.15), then (7.17) becomes F(t, x, y, z, γ ) = sup x, e)z + F 0 t, x, y, e , (7.25) Under Assumption 7.8, F 0 is uniformly Lipschitz continuous in y, and thus the main difficulty mentioned in Remark 7.7 (i) does not exist here. Then, following similar arguments as in this subsection, we can show that the results of Theorems 7.12 and 7.14 still hold true if we replace the uniform nondegeneracy condition σ f ≥ c 0 > 0 with σ f = 0.

The case that g is linear
In this subsection, we study the special case when g is linear in (y, z) (by abusing the notation g 0 ) 6.1) to obtain the Eq. 2.26. Hence ∂ x R 1,u s,t (x) = ∂ x u s,t (x) − g(s, x, u s (x))ω s,t = ∂ x u s,t (x) − ∂ x g(s, x, u s (x)) + ∂ y g (s, x, u s (x))∂ x u s (x) ω s,t = t s [∂ x f + ∂ y f ∂ x u r (x)](r, x, u r (x))dr + t s [∂ x g + ∂ y g∂ x u r (x)](r, x, u r (x))dω r − ∂ x g(s, x, u s (x)) + ∂ y g(s, x, u s (x))∂ x u s (x) ω s,t .