Pathwise Taylor Expansions for It\^o Random Fields

In this paper we study the {\it pathwise stochastic Taylor expansion}, in the sense of our previous work \cite{Buckdahn_Ma_02}, for a class of It\^o-type random fields in which the diffusion part is allowed to contain both the random field itself and its spatial derivatives. Random fields of such an"self-exciting"type particularly contains the fully nonlinear stochastic PDEs of curvature driven diffusion, as well as certain stochastic Hamilton-Jacobi-Bellman equations. We introduce the new notion of"$n$-fold"derivatives of a random field, as a fundamental device to cope with the special self-exciting nature. Unlike our previous work \cite{Buckdahn_Ma_02}, our new expansion can be defined around any random time-space point $(\t,\xi)$, where the temporal component $\t$ does not even have to be a stopping time. Moreover, the exceptional null set is independent of the choice of the random point $(\t,\xi)$. As an application, we show how this new form of pathwise Taylor expansion could lead to a different treatment of the stochastic characteristics for a class of fully nonlinear SPDEs whose diffusion term involves both the solution and its gradient, and hence lead to a definition of the {\it stochastic viscosity solution} for such SPDEs, which is new in the literature.


Introduction
In our previous work [4] we studied the so-called pathwise stochastic Taylor expansion for a class of Itô-type random fields. The main result can be briefly described as follows. Suppose that {u(t, x), (t, x) ∈ [0, T ] × R n } is an Itô-type random field of the form where B is a 1-dimensional standard Brownian motion, defined on a complete probability space (Ω, F, P ). If we denote F = {F t } t≥0 to be the natural filtration generated by B and augmented by all P -null sets in F, then under reasonable regularity assumptions on the integrands u 1 and u 2 , the following stochastic "Taylor expansion" holds: For any stopping time τ and any F τ -measurable, square-integrable random variable ξ, and for any sequence of random variables {(τ k , ξ k )} where τ k 's are stopping times such that either τ k > τ , τ k ↓ τ ; or τ k < τ , τ k ↑ τ , and ξ k 's are all F τ k ∧τmeasurable, square integrable random variables, converging to ξ in L 2 , it holds almost surely that where (a, b, c, p, q, X) are all F τ -measurable random variables, and the remainder o(ζ k ) are such that o(ζ k )/ζ k → 0 as k → ∞, in probability. Furthermore, the six-tuple (a, b, c, p, q, X) can be determined explicitly in terms of u 1 , u 2 and their derivatives.
By choosing u 1 and u 2 in different forms, we then extended the Taylor expansion to solutions of stochastic differential equations with initial state as parameters, and to solutions of nonlinear stochastic PDEs. In the latter case we further introduced the notion of the stochastic super-(sub-)jets using the Taylor expansion, from which the definition of stochastic viscosity solution was produced. We should note that in [4] all the SDEs and SPDEs have the diffusion coefficient in the form g(t, x, u(t, x)), that is, they only involve the solutions themselves. Such a structure turns out to be essential for the so-called Doss-Sussmann transformation, and in that case the stochastic viscosity solution became natural.
In this paper we are interested in the stochastic Taylor expansion for random fields of the following form: u(t, x) = u(0, x) + where u 1 is a random field, and g is a deterministic function. We shall assume that they are "smooth" in the sense that all the desired derivatives exist, and the degree of smoothness will be specified later.
Given such a random field, we again consider the possibility to expand u in the sense of (1.2), but this time in a more natural way: We shall allow the pair (τ, ξ) : Ω → [0, T ] × R d to be arbitrary random points. Furthermore, we should note that although the Taylor expansion (1.2) holds almost surely, in general the null set may depend on the choice of random point (τ, ξ). In this paper we shall look for a universal expansion, in the sense that there is a subset Ω ⊂ Ω with full probability measure, on which the stochastic Taylor expansion holds for all choices of random points (τ, ξ). These technical improvements make the stochastic Taylor expansions much more "user-friendly", and more importantly, it will be more effectively used in our study of stochastic viscosity solution, especially in the proof of uniqueness, as we shall see in our forthcoming publications on that subject.
As one can easily observe from (1.3) that the random field u is actually already a solution to a first order stochastic PDE. An immediate difficulty in deriving the Taylor expansion is the characterization of the derivatives of the random field, as it will increase in a "bootstrap" way, very similar to those that one has often seen in the anticipating calculus. As a consequence, the original Doss-Sussmann type transformation used in our previous works [4,5,6] no longer works in this case. To overcome this difficulty we introduce the notion of "n-fold derivative" of a random field, which essentially takes (u, Du, · · · , D n−1 u) as a vector-valued random field, and define its derivative in a recursive way. Such a definition turns out to be very close to the idea of converting a higher order ordinary differential equation to a first order system, and is mainly motivated by the "stochastic characteristics" of a stochastic PDE (cf. e.g., Kunita [12]). In fact, by combining the definition of stochastic characteristics in [12] and the stochastic Taylor expansion developed in this paper, we are able to rigorously define a stochastic diffeomorphism that relates the Stochastic PDE of the form to a PDE without Brownian components in their Taylor expansions around any temporal-spatial point, generalizing the Doss-Sussmann transformation in our previous works to the present case.
We should note that one of the main difficulties in the study of the stochastic viscosity solution can be described as "local" vs. "global". That is, the local nature of the viscosity solution vs.
the global nature of the stochastic analysis (e.g., stochastic integrals). Our idea is to "localize" the stochastic integral, or solution to the SPDE via the stochastic Taylor expansion that we established in the previous section. It should be noted that the universal set "Ω ′ " that we found in these expansions is the essential point here.
Finally, we would like to point out that the topic of stochastic Taylor expansion has been explored by many authors in various forms, and used to provide numerical and other approximation schemes for SDEs and SPDEs or randomized ODEs and PDEs (see, for example, [1,3,10,11], to mention a few). These expansions often use either the Lie-algebraic structure of the path space or the chaotic type expansion of multiple stochastic integrals. As a consequence it is hard to deduce the simultaneous spatial-temporal expansions that we are pursuing in this work, especially when the remainders are estimated in a pathwise manner. We should also note that the pathwise version of stochastic viscosity solutions, suggested by Lions-Souganidis [13,14], has found an effective framework recently, using the theory of rough path (cf. Caruana-Friz-Oberhauser [7]).
However, due to the special nature of the rough path integrals, the arguments seem to depend heavily on the fact that there exist stochastic characteristics in the form of C 3 -diffeomorphisms which transform the SPDE to a pathwise PDE. Consequently, the SPDE studied in [7], while fully nonlinear in the drift, seems to be restricted to the cases when the diffusion coefficient depends only (linearly) on Du, the gradient of the solution, so that a chain-rule type of argument could be applied. The generality of the diffusion part in the fully non-linear PDE suggested in this paper, and the stochastic characteristics related to it, does not seem to be an easy consequence of such a method. Furthermore, our Taylor expansions are constructed within a more "elementary" stochastic analysis framework, by exploiting the properties of Brownian motion without using the advanced algebraic geometric structure of the path spaces, therefore we believe that it provides a more accessible alternative.
This paper is organized as follows. In Section 2 we clarify all the necessary notations, and state the main theorem. In section 3 we give a fundamental estimate of this paper, regarding multiple stochastic integrals. In sections 4 and 5 we study the forward and backward Taylor expansion, respectively. We note that in the stochastic case, the temporal direction of the expansion does affect the outcome. Finally in section 6 we try to apply the Taylor expansion to the solution of a class of nonlinear SPDEs, and propose a possible definition of the stochastic viscosity solution in this case.

Preliminaries and Statement of the Main Theorem
Throughout this paper we denote (Ω, F, P ) to be a complete probability space on which is defined be the natural filtration generated by B, augmented by the P -null sets of F; and let F B = F B ∞ . We denote M B 0,T to be the set of all F B -stopping times τ such that 0 ≤ τ ≤ T , P -a.s., where T > 0 is some fixed time horizon; and denote M B 0,∞ to be all F B -stopping times that are almost surely finite.
In what follows we write E (also E 1 , · · · ) for a generic Euclidean space, whose inner products and norms will be denoted as the same ones ·, · and | · |, respectively; and write B for a generic Banach space with norm · . Moreover, we shall denote G ⊆ F B to be a sub-σ-field of F B , and for any x ∈ R d and constant r > 0 we denote B r (x) to be the closed ball with center x and radius r. Furthermore, the following spaces of functions will be frequently used in the sequel. We denote In particular, q = 0 stands for all B-valued, F B -progressively measurable processes; and q = ∞ denotes all processes in L 0 (F B , [0, T ]; B) that are uniformly bounded.
, where the subscript "b" means all functions and their partial derivatives are uniformly bounded; "l" means all functions are of at most linear growth; and "p" means all functions and their partial derivatives are of at most polynomial growth. The subspaces with the combined subscripts of b, l, and p are defined in an obvious way.
Finally, since the random fields that we are interested in are always of the form of (1.3), which is an SPDE already, the following definition in [4] is useful. Consider the fully nonlinear second-order SPDE: where (t, x) ∈ [0, T ] × R d , and f and g are functions with appropriate dimensions.
(ii) u is an Itô-type random field with the form In this paper we will consider a special type of "smoothness" for a random field, defined through what we shall call the "n-fold differentiability" below. Such a characterization of differentiability is mainly motivated by the "stochastic characteristics" for stochastic PDEs (cf. e.g., [12]), which often take the form of a system of first order Stochastic PDEs. The idea is actually quite similar to the well-known transformation from a higher order ordinary differential equation (ODE) to a first order system of ODEs.
To begin with, we recall that for any multi-index j = (j 1 , · · · , j d ), its "length" is defined by We have the following definition.

2)
for all (t, y) (T3) For any 1 ≤ i ≤ n + 1, m ≥ 1, and multi-index j = (j 1 , ..., j d ) with |j| = j 1 + ... + j d ≤ n, it holds that We shall call ζ i , i = 2, · · · , n + 1 the "generalized derivatives" of ζ = ζ 1 , with "coefficients" For notational convenience we will often write F = F 1 and G = G 1 when there is no danger of confusion. We denote the set of all n-fold differentiable random fields by C Before we state the main result of this paper, let us note again that the main feature of our stochastic Taylor expansion is that it can be expanded around any random point (τ, ξ), and that the expansion holds almost surely with an exceptional set that is independent of the choice of (τ, ξ). But this amounts to saying that the expansion can be performed around any deterministic point (τ, ξ) with any (deterministic) increments, outside a uniform exceptional set. In other words, the complicated approximating sequences (τ k , ξ k ) in (1.2) can be replaced by simple deterministic increments (t + h, x + k), for all (h, k) near zero. We should also note that for the purpose of our application, in this paper we consider only the Taylor expansion up to the second order, and for that purpose the 3-fold differentiability of the random field would suffice. The precise statement of our main result is the following theorem. Then, for every α ∈ 1 3 , 1 2 and every m ∈ N there exist a subsetΩ α,m ⊂ Ω with P {Ω α,m } = 1, such that, for all (t, y, ω) ∈ [0, T ] × B m (0) ×Ω α,m , we have the following Taylor expansion Furthermore, the remainder of Taylor expansion R α,m : Proof. Since the proof of this theorem is quite lengthy and technical, we shall split it into several cases and carry it out in the following sections.
Remark 2.4 (i) It is worth noting that the "universal" estimate for the remainder R α,m is the main reason why the Taylor expansion can now hold around any random space-time point. In other words, the points (t, y) and (h, k) in Theorem 2.3 can be replaced by any (τ, ξ), (σ, η) ∈ In what follows we denote R α,m to be a generic term satisfying (2.5), which is allowed to vary from line to line.
(ii) From the expressions (2.4) it is clear that the drift term F (= F 1 ) appears only in the coefficient a, and it is in its original form. In fact, as we shall see in the proof, the exact form of F (t, y)(= F (y, ζ 2 (t, y))) is not important at all. We could simply change it to F (t, y) and the results remains the same.
(iii) Theorem 2.3 remains true if the assumption (T1) is replaced by the weaker assumption: However, we shall not pursue this generality in this paper due to the length of the paper.
To end this section we give an example, which more or less motivated our study.
x)} be a regular solution of the SPDE (2.1), and for simplicity we assume that both f and g are "time-homogeneous" (i.e., they are independent of the variable t), and g is independent of x as well. We define ζ = ζ 1 = u, and Then, assuming that the coefficients f and g are sufficiently smooth and their derivatives of all order are bounded, one can show that (T1)-(T3) are satisfied. Furthermore, applying Theorem 2.3 we see that on some subset Ω of Ω with P ( Ω) = 1, independent of the expansion point for all h ∈ R and k ∈ R d such that t + h ∈ [0, T ] and |k| ≤ m, respectively. Here, the coefficients The explicit expressions of the pathwise Taylor expansion in (2.6) will be important for our study of the stochastic characteristics for SPDE (2.1), which will be discussed in the last section of this paper.

Some fundamental estimates
Before we prove the theorem, let us first introduce the so-called Wick-square of the Brownian motion, which is originated in the Wiener homogeneous chaos expansion (cf., e.g., [9], [19]). For where in this case D x u denotes the weak partial derivative of u with respect to x.
We begin by proving an important estimate for multiple Stochastic integrals. We consider the multiple stochastic integral defined recursively as follows.
t,s (x) ≡ 1, and for j = 1, · · · , N , It is also useful to define the similar multiple integrals: for k > 0, let X k,k−1 t,s (x) ≡ 1, and for Then it is clear that X 1,j t,s = X j t,s , for j ≥ 1. We have the following regularity result.
Proof. The proof is based on the Kolmogorov continuity criterion for random fields (cf. e.g., [23, Theorem I.2.1]), combined with an induction argument. We note that throughout the proof we shall use the notations C m , C m,p , C M,k , etc., to represent the generic constants depending only on f i 's, T , and the parameters in their subscripts, which are allowed to vary from line to line.
We begin with the case N = 1. In this case we write f 1 = f and X 1 = X for simplicity. Let m ∈ N, γ > 2, and 0 ≤ t < s ≤ T . Applying Sobolev's embedding theorem, Hölder Inequality, and noting that X 0 t,s = 1, we have Endowing the space C(B m ) of continuous functions defined on the closed m-ball B m := {x ∈ R d : Applying the Kolmogorov continuity criterion, we conclude that ζ β,m ∈ L γ (Ω, F, P ) for all β ∈ 0, 1 2 − d+2 γ(d+1) . Since we can choose γ > 2 arbitrarily large, (3.4) follows. We now prove the inductional step. That is, we assume that (3.4) is true for N − 1, and show that it is also true for N . To do this, we shall adapt the proof of the Kolmogorov continuity criterion given in [23] to our framework. Let 0 ≤ t < s ≤ T , γ > 2, and m ∈ N. First note that by a simple application of Burkholder-Davis-Gundy and Hölder inequalities we have By simply iterating the above argument we obtain that Furthermore, with a similar argument as above we can also show that for all M, k ∈ N, p > 1, and multi-index i satisfying |i| = 1, Here, we applied the Hölder inequality in the last step above. Consequently, if p ≤ M k 2 , then the assumption (3.3) implies that, for all m > 0, We can then conclude that, for all 0 ≤ t < s ≤ T , γ > 2,

Consequently, by Hölder inequality and (3.7) with
In other words we obtained that Next, recall the multiple integrals X k,l t,s (x) defined by (3.2). For 0 ≤ r 1 ≤ r 2 ≤ r 3 ≤ T , and An easy calculation shows that To simplify the further discussion, we now assume without loss of generality that T = 1. Let D n be the set of all t n i := 2 −n i, for some 0 ≤ i ≤ 2 n . The set D = n≥1 D n is then the set of all dyadic numbers in [0, 1]. Now, let n ∈ N and 0 ≤ t < s ≤ 1 be arbitrary dyadic numbers such that s − t ≤ 2 −n . Our aim is to estimate X t,s (x) uniformly in t, s ∈ D. To this end, we notice that there exists some k ≥ n such that t, s belong to D k . Moreover, denoting s j = sup{r ∈ D j , r ≤ s}, t j = sup{r ∈ D j , r ≤ t}, n ≤ j ≤ k, we have s n ≤ s n+1 ≤ . . . ≤ s k = s, t n ≤ t n+1 ≤ . . . ≤ t k = t. Therefore (see also Figure 1) Y N tn,s j ,s j+1 ,n 0 (x) , (3.11) In order to estimate Y N tn,s j ,s j+1 ,n 0 (x) and Y N t j ,t j+1 ,s,n 0 (x), we notice that for n ≤ j ≤ k. Moreover, recall that for every α ∈ (0, 1 2 ), there exists a θ α ∈ L ∞ − (Ω, F, P ) such that for all l, l ′ ∈ N with 0 ≤ l ′ − l ≤ N − 2, Similarly, we obtain Then, combining (3.11), (3.12), (3.15), and (3.14), we deduce Noting that by (3.8) we thus obtain for γ > 4 N −2β that Consequently, M β,m ∈ L ∞ − (Ω, F, P ), for all β ∈ (0, N α). But since α ∈ (0, 1 2 ) is arbitrary, we can extend the result to β ∈ (0, N/2). Finally, note that the definitions of M β,m and {ζ n β,m }, as well as the continuity of the mapping (t, s) → X t,s (x), imply that for all β ∈ (0, N/2) and m ∈ N. The proposition then follows from the recursive relation (3.10).
The following corollary can be easily obtained by adapting the proof of Proposition 3.1 in an obvious manner.

Forward Taylor expansion
In this and next section we shall provide a complete proof of Theorem 2.3. The results in each section, however, can also be applied independently, and therefore are of interest in their own right. We should note that, unlike the usual Taylor expansion, in the stochastic case the direction of the time increment makes significant difference in the argument, due to the "progressive measurability" of the random fields. We will thus separate the two cases: in this section we study the forward expansion, and leave the backward case to the next section.

Forward temporal expansion
We begin with the forward "temporal" expansion, that is, only the time variable has the increment.
Let us first introduce the following extra notations. For each T > 0 we define the 2-dimensional simplex For any real-valued measurable functional θ defined on Ω × △ [0,T ] × IR d+l and m ∈ N, we denote Furthermore, by a slight abuse of notations, in what follows we shall denote R α to be any measurable functional θ α , indexed by α ∈ 1 3 , 1 2 , such that ( θ α ) m ∈ L ∞ − (Ω, F, P ) for all m ∈ N, and again, it may vary from line to line.
Our main result of this section is the following Stochastic forward temporal Taylor expansion.
Then, for all m ∈ lN and α ∈ 1 3 , 1 2 there exists a subset Ω ⊂ Ω such that P ( Ω) = 1, and that on Ω, for all 0 ≤ t < t + h ≤ T , x ∈ R d , the following expansion holds Proof. Let h > 0 be such that 0 ≤ t ≤ t + h ≤ T . For any x ∈ R d , we write where I i = I i (t, h, x), i = 1, 2, are the two integrals. We shall study their expansions separately.
We begin by I 1 . The argument is very similar to that of [4], we provide a sketch for completeness. Let Then, applying Itô's formula, we have F (x, ζ 2 (s, x)) − F (x, ζ 2 (t, x)) = s t H 1 (x, ζ 2 (r, x), ζ 3 (r, x))dr for 0 ≤ t ≤ s ≤ T and x ∈ R d , P -a.s. We now show that there is a universal subset Ω ′ ⊆ Ω with P (Ω ′ ) = 1, on which (4.6) holds for all 0 ≤ t ≤ s ≤ T . But for this it suffices to prove the random ζ 2 (r, x)), G 2 (x, ζ 3 (r, x)) dB r have continuous versions. To see this, we try to make use of the Kolmogorov continuity criterion.
For notational simplicity let us denote Then, we can deduce from (T3) and the Burkholder-Davis-Gundy inequality that, for each k > 2, Hence, the Kolmogorov continuity criterion renders that the random field s t Γ(r, x)dB r , 0 ≤ t ≤ s ≤ T, |x| ≤ m possesses a version that is continuous in (t, s, x). A similar estimate allows to prove that also the first integral in (4.6) admits a version continuous in (t, s, x) as well. Hence, we conclude that on some Ω ′ ⊂ Ω of full probability measure the relation (4.6) holds for all 0 ≤ t ≤ s ≤ T and |x|, |x ′ | ≤ m.
Consequently, writing the integral I 1 as ζ 3 (r, x))dB r ds, and using the conclusion above we see that as a function of (t, h, x), I 1 is jointly continuous for all 0 ≤ t ≤ t + h ≤ T and x ∈ R d , over the universal set Ω ′ . It then follows from Corollary 3.2 that there exists an Ω ′′ ⊆ Ω ′ with P (Ω ′′ ) = 1 such that for all 0 ≤ t ≤ t + h ≤ T and x ∈ R d , it holds that In what follows we will not distinguish Ω ′′ from Ω ′ .
We now turn our attention to I 2 . Again, we begin by denoting, for (x, Then, for every x ∈ R d , we again apply Itô's formula to get, for 0 ≤ t ≤ s ≤ T , and P -a.s., Using the similar arguments as before we can find another universal subset, still denoted by Ω ′ ⊆ Ω with P (Ω ′ ) = 1, on which (4.9) holds for all 0 ≤ t ≤ s ≤ T .
Next, using (4.9) it is easy to see that I 2 can be written as and we can claim as before that (4.10) holds for all 0 ≤ t < t + h ≤ T , |x| ≤ m, over a universal subset of full probability measure, again denoted by Ω ′ .
We now analyze I 2,1 and I 2,2 separately. Using integration by parts we see that Then, following the argument developed in the previous part, we can show that the equality holds for all 0 ≤ t < t + h ≤ T , x ∈ R d , over an Ω ′ ⊆ Ω with P (Ω ′ ) = 1. Note that (4.11) it follows that over Ω ′ , The estimate for I 2,2 is slightly more involved. For notational simplicity let us define where z 2 = (z 2 , z 3 ) and z 3 = (z 3 , z 4 ). Moreover, we denote ζ i (s, x) = (ζ i (s, x), ζ i+1 (s, x)), i = 2, 3.
Then, applying Itô's formula we have, Again, we assume that the equality holds for all 0 ≤ s ≤ t ≤ T and x ∈ R d , on Ω ′ . Therefore

Forward temporal-spatial Taylor expansion
Based on the forward temporal Taylor expansion Proposition 4.1, we now add the spatial increment. Our main result of this section is the following proposition.
Then, for all m ∈ lN and α ∈ 1 3 , 1 2 there exists some subset Ω ⊂ Ω such that P ( Ω) = 1, and that on Ω, for all 0 ≤ t < t + h ≤ T and x, k ∈ R d , Proof. First let us write where the second [· · · ] above is the forward temporal expansion studied in the previous subsection.
In light of Proposition 4.1, we need only prove the following stochastic spatial Taylor expansion: To this end, we first apply the standard Taylor expansion and use the assumption (T3) to get for all 0 ≤ t < t + h ≤ T , x, k ∈ R d , P -a.s. Next, differentiating the equation for ζ = ζ 1 in (2.2) we have for all 0 ≤ t < t + h ≤ T , x ∈ R d , P -a.s. Consequently, it follows from (4.11) that s. Similarly, one shows that Combining (4.18), (4.19), and (4.20), we obtain that s. Finally, noting that for all h ∈ [0, T ], |k| ≤ m, we see that (4.17) holds, hence the proposition follows.

Backward Taylor expansion
In this section we treat the backward Taylor expansion, that is, when the temporal increments are negative. As a general belief such an expansion would be more difficult than the forward one, due to the obvious "adaptedness" issue. But we shall see, with our "pathwise" approach, such difficulty is eliminated. We nevertheless would like to separate its proof from the forward case because of the slight difference in the arguments. We again take two steps: first the backward temporal expansion, and then the mixed time-space expansion.

Backward temporal expansion
We have the following analogy of Proposition 4.1.

Backward temporal-spatial expansion
We now give the complete statement of the backward temporal-spatial expansion.
Then, for all m ∈ N and α ∈ 1 3 , 1 2 there exists some subset Ω ⊂ Ω such that P ( Ω) = 1, and that on Ω, for all Proof. As in the forward expansion case, we need only show that for all m ∈ N and α ∈ 1 3 , 1 2 there exists some subset Ω ⊂ Ω of full probability such that on Ω, for all 0 ≤ t − h < t ≤ T , From the usual Taylor expansion with the remainder in the Lagrange form, we have for 0 ≤ t − h < t ≤ T , x, k ∈ R d , P -a.s. Moreover, from Proposition 5.1 it follows that and Consequently, Thus, by virtue of

Application to Stochastic PDEs
Having tried so hard to develop the various forms of stochastic Taylor expansion, as an application in this section we shall try to use it to study the stochastic viscosity solutions for fully nonlinear SPDE, following the idea that we developed in our earlier work [4]. In order not to over-complicate the computation we shall consider the following simpler version of the fully nonlinear SPDE (2.1): Compared to SPDE (2.1), as well as Example 2.5, we see that here the diffusion coefficient g in (6.1) is independent of u. The general case could be treated in a similar way but with more complicated expressions. Since our purpose here is to outline our idea of a new definition of stochastic viscosity solution, without adding too much technical complexity into this already lengthy paper, we shall leave the study of the general case to a forthcoming paper. We should also note that in (6.1) we are using the Stratonovich integral instead of the Itô integral for the simplicity of the presentation, the following relation is worth noting: It is worth pointing out that even in this simplified form, the nonlinearity of the function g on Du already makes it difficult to apply the rough path approach of [7] directly here.
To explain our idea of the definition of stochastic viscosity solution, let us first apply Theorem 2.3 to the regular solution u. Bearing the relation (6.2) in mind we have: It is interesting to note that in this simple case the terms involving D 2 u can be written collectively as (suppressing variables): Compared to the classical deterministic Taylor expansion, and the SPDEs studied in [4] in which g is independent of Du, we can see that in a general case the terms involving D 2 u becomes much more complicated, and our previous method (via Doss-Sussmann) will face a fundamental challenge, especially in the uniqueness proof. We therefore will try to find a different approach to define the stochastic viscosity solutions, using the stochastic characteristics introduced by Kunita [12], combined with our results on stochastic Taylor expansions. This new method also reflects the basic ideas of the works of Lions and Souganidis [13], [14], [16], and [17], and in a sense includes our previous work [4] as special case. For simplicity we shall now assume all processes involved are real valued.
To motivate our definition of the stochastic viscosity solution let us suppose in a first step that the coefficients of SPDE (1.4) are sufficiently smooth and that this equation admits a regular . Under this assumption, we compare the solution u with a smooth test field ϕ ∈ C 0,∞ (F B , [0, T ] × R) defined as the unique solution of the equation , and ϕ 0 ∈ C ∞ p (R). Let us now fix an F B -measurable [0, T ]-valued random variable τ and an F B -measurable R dvalued random variable ξ. We say that u − ϕ achieves a local left-maximum in (τ, ξ) if for almost all ω ∈ {τ < T } there is some ρ > 0 (which may depend on ω) such that Following the approach by Kunita (see Theorem 6.1.2 in [12]), we formally introduce the following stochastic characteristics. For the moment let us assume that the random time τ is actually deterministic to avoid further complications (keep in mind, however, that the Taylor expansion will hold even for the arbitrary random time τ !).
(ii) g(x, Du) = V (x)Du (see [7]). In this case one has D x g(x, z) thus the Taylor expansion (6.4) will also become much simpler. Furthermore, the characteristics (6.6) now become "disentangled" SDEs: and (6.7) takes the special form: ψ(t, x) = ϕ(t, φ t (x)). In other words, the transformation ψ is globally well-defined by an easy and explicit expression.
In what follows we shall denote Θ △ = (l, h, k). Also, for any function γ = γ(x, y, z) we denote ∇γ = (D x γ, D y γ, D z γ) T . Then we can write, for example, the first equation in (6.6) in the Itô integral from: Now, treating the random fields η and χ in (6.6) the same way, and applying the stochastic (backward temporal) Taylor expansion to (φ, η, χ) we have, for all α ∈ ( 1 3 , 1 2 ), On the other hand, writing (6.5) in an Itô integral form we have Dϕ(s, x))]}ds + t τ g(x, Dϕ(s, x))dB s (6.11) where F and G are defined in an obvious way, and with Note that differentiating the both sides of (6.11) we have Applying the backward temporal Taylor expansion (Theorem 2.3) again to the random field ϕ around any point (τ, ς), with we obtain, after some simple cancelations, that We note that the above holds for all (τ, ς) and all ω ∈ Ω α,m .
Let us now define the desired random field ψ. We should note that our main purpose here is to find such a transformation so as to eliminate the stochastic integral. In other words, we shall look for such ψ that has the following first order Taylor expansion: (6.14) Here ∂ − t denotes the left partial derivative with respect to t. To this end, we note that (6.6) and (6.7) imply that for t = τ , one has ϕ(τ, x) = ϕ(τ, φ τ (x, Dψ(τ, x))) = η τ (x, ψ(τ, x), Dψ(τ, x)) = ψ(τ, x), x ∈ R, (6.15) and hence Dϕ(τ, x) = Dψ(τ, x) holds for all x ∈ R as well. Next, we look at the Taylor expansion for both φ and η. Recalling (6.8) and the Taylor expansions (6.10). We shall first take x = (x, ψ(t, x), Dψ(t, x))), and then replacing x by φ t (x, Dψ(t, x)) and ς by x in (6.13). It should be noted that after these substitutions the remainder will look like R α,m (t, τ, x, φ t (x, Dψ(t, x))), a slightly more complicated form than the original ones. To make sure the accuracy of the expansion in what follows we shall denote R α,m,loc = sup t,s∈[0,T ]; x∈Bm(0) |R α,m (t, s, x, φ t (x, Dψ(t, x)))|.
A random field u ∈ C([0, T ] × R d ) is called a stochastic viscosity solution of (1.4), if it is both a stochastic viscosity subsolution and a supersolution.

Remark 6.3
We would like to note that in Definition 6.2 the (possibly anticipating) test functions ϕ can be directly defined by the expansion (6.13) with ϕ(τ, ξ) being replaced by u(τ, ξ) and (θ(τ, ξ), D x ϕ(τ, ξ), D 2 x ϕ(τ, ξ)) being replaced by a triplet of random variables (β, p, A). The main advantage here is that the random field ϕ is now defined globally, overcoming the essential difficulties in the theory of stochastic viscosity solution thus far, and will significantly facilitate the uniqueness proof. Moreover, on the subset of Ω where u − ϕ achieves the local left-maximum (resp., left-minimum) at the point (τ, ξ), the triplet (β, p, A) can be considered as a stochastic sub-(resp.super-)jet, as it was traditionally done. These issues will be further explored in our forthcoming publications.
In the rest of this section we shall verify that a regular solution must be a stochastic viscosity solution in the sense of Definition 6.2, which will provide a justification for our new definition. To this end, let us assume that the coefficient f is proper in the following sense (H1) The function F (t, u, p, X) △ = −f (x, u, p, X) is "degenerate elliptic". That is, f is continuous in all variables, and is non-decreasing in the variable X.
For any given pair of random variables (τ, ξ) and an arbitrary test field ϕ such that u − ϕ attains a local left-maximum at (τ, ξ) on a subset of Ω with positive probability, let ψ be the process associated to ϕ by (6.18) (where the remainder can be chosen to be zero!), and φ be defined by (6.10).
This proves that the classical solution u is a stochastic viscosity subsolution. That u is also a supersolution can be proved using a similar argument. Therefore, u is a stochastic viscosity solution.