Semigroup Solution of Path-Dependent Second-Order Parabolic Partial Differential Equations

We apply a new series representation of martingales, developed by Malliavin calculus, to characterize the solution of the secondorder path-dependent partial differential equations (PDEs) of parabolic type. For instance, we show that the generator of the semigroup characterizing the solution of the path-dependent heat equation is equal to one-half times the second-order Malliavin derivative evaluated along the frozen path.


Introduction
In this paper we consider semilinear second-order pathdependent PDEs (PPDEs) of parabolic type.These equations were first introduced by Dupire [1] and Cont and Fournie [2] and will be defined properly in the next section.
To motivate our result, we first consider the heat equation expressed in terms of a backward time variable.For  ≤  we look for a function V(, ) that solves V (, ) = Ψ () . ( It is well known (see, e.g., [3], chapter 9.2 or [4]) that the solution is given by the flow of the semigroup (); that is, V(⋅, ) = ()Ψ, where The differential operator (1/2)( 2 / 2 ) is said to be the (infinitesimal) generator of the semigroup ().Consider now the path-dependent version of the heat equation: where    is a continuous path on the interval [0, ] and the derivatives are Dupire's path derivatives.Our goal is to find the generator of the semigroup (flowing the solution) of PPDEs, which we will refer to as the semigroup of the PPDE.It turns out that 1/2  , that is, one-half times the secondorder vertical derivative, is not the appropriate infinitesimal generator, because of path dependence.Indeed, the vertical derivative is the rate of change of the functional V(⋅, ) for a change at time .The correct infinitesimal generator is equal to (1/2)  ∘ D 2  , where D 2  is the second-order Malliavin derivative of () ≡ Ψ(   ()).An important difference is that  is now viewed as a random variable, and the (first-order) Malliavin derivative is a stochastic process in 2 International Journal of Stochastic Analysis the canonical probability space for Brownian motion.The stopping path operator   was introduced in [5].Informally, the action of the stopping path operator (which we define rigorously later) is to freeze the path after time :   ∘  () =  (  ) , (5) where   is the stopped path.The stopped Malliavin derivative   ∘ D  is thus an extension of both (i) the Dupire derivative; while the Dupire derivative corresponds to changes of the path at only one time, the iterated derivatives   ∘ D   1 ,...,  are taken with respect to changes of the canonical path at many different times  1 , . . .,   ; (ii) the Malliavin derivative; while the Dupire derivative can be taken pathwise, as far as we know, the construction of the Malliavin derivative necessitates the introduction of a probability space.
The proof of the representation result is straightforward.Let us consider the path-independent case (1).Let  be Brownian motion.By Itô's lemma, it is obvious that V((), ) is a martingale, say   , and that the value of this martingale is the conditional expectation at time  of Ψ(()).Consider now a general path-dependent terminal condition Ψ(), in [5], Jin et al. gave a new representation of Brownian martingales   (with  ≤ ) as an exponential of a timedependent generator, applied to the terminal value   ≡ Ψ(): By the functional Feynman-Kac formula introduced in . is the generator of the semigroup of the PPDE.
The main advantage of the semigroup method is that the solution of the PPDE can be constructed semianalytically: indeed, the method is similar to the Cauchy-Kowalewsky method, of calculating iteratively all the Malliavin derivatives of Ψ; (6) can be rewritten indeed as The main disadvantage can be seen immediately by considering (7): the terminal condition Ψ must be infinitely Malliavin differentiable.In contradistinction, the viscosity solution given in [7] necessitates Ψ to be only bounded and continuous.However, compared to the result shown in [6], Ψ needs only to be defined on continuous paths.
This paper is composed of two parts.In the first part, we give a rigorous proof of the result (7).Indeed, we complete the proof of Theorem 2.3 in our article [5]; although the statement was correct in that paper, one step of the proof was not obvious to finish.In the second part we characterize the generator of the semilinear PPDE.

Martingale Representation
We first introduce some basic notations of Malliavin calculus.For a detailed introduction, we refer to [8] and our paper [5].Let Ω = ([0, ], R) and (Ω, F, {F  } ≥0 , P) be the complete filtered probability space, where the filtration {F  } ≥0 is the usual augmentation of the filtration generated by Brownian motion  on R. The canonical Brownian motion can be also denoted by () = (, ) = (),  ∈ [0, ],  ∈ Ω, by emphasizing its sample path.We denote by  2 (P) the space of square integrable random variables.For simplicity, we denote (d) ⊗ fl d 1 ⋅ ⋅ ⋅ d  .
We denote the Malliavin derivative of order  at time  1 , . . .,   by D   1 ,.
From the definition, it is not hard to obtain that, for any variable smooth function ,   ∘(( 1 ), . . ., (  )) = (( 1 ∧ ), . . ., (  ∧ )).For a general random variable  ∈  2 (P),   ∘  refers to the value of  along the stopping scenario   ≡   () of Brownian motion.According to the Wiener-Chaos decomposition, for any  ∈  2 (P), there exists a sequence of deterministic function {  } ≥1 such that  = ∑ ∞ =0   (  ) with convergence in  2 ([0, ]  ).Therefore, in order to obtain an explicit representation of   acting on a general variable , we first show the following proposition.Proposition 2. Let   ∈  2 ([0, ]  ), an -variable square integrable deterministic function; then International Journal of Stochastic Analysis 3 Therefore as well as the isometry: ( Then we can set up an operator differential equation for   .
The following theorem is a generalization of Theorem 2.2. in [5] to functionals that are not discrete.
Theorem 4. For 0 ≤  ≤  ≤ , assuming that  ∈ D 6 ([0, ]), one has Then our main theorem is the integral version of this operator differential equation.We first introduce the convergence condition.
Remark 5. We claim that other conditions exist which are easier to check than Condition 1.One of them is the convergence of the terms of series (23): To this " local" condition, that is, a condition based on the calculation along the frozen path only, one needs to add a "global" condition involving all the paths to make it sufficient; that is, [(D   ) 2 ] <  2 for any  ∈ [, ] and  ≥ 1, with a constant .
Moreover, with different structures of , we have different alternative conditions which are easier to check for practical calculations.Here we list two examples.
(1) If  = (∫  0 ()d()) with smooth deterministic function  and square integrable deterministic function , it is not hard to obtain Therefore, if there exists a constant  such that, for all  ≥ 1, with the help of Stirling approximation ! ∼ √ 2(/)  , Condition 1 is satisfied.

International Journal of Stochastic Analysis
(2) If  has its chaos decomposition  = ∑ ∞ =0   (  ), we have Then according to (12), Condition 1 can be replaced by with some constant  or some much stronger but easier conditions like the following: for  ≥ 1           sup Then we have the following main result.

Theorem 6. Suppose that 𝐹 satisfies Condition 1 and is
The importance of the exponential formula (23) stems from the Dyson series representation, which we rewrite hereafter in a more convenient way:

Representation of Solutions of Path-Dependent Partial Differential Equations
3.1.Functional Itô Calculus.We now introduce some key concepts of the functional Itô calculus introduced by Dupire [1].For more information, the reader is referred to [6], which we copy hereafter almost verbatim.Let  > 0 be fixed.For each  ∈ [0, ] we denote by Λ  the set of càdlàg (right continuous with left limits) R-valued functions on [0, ].For each where  = û,   û,   û,   û,  and  are some constants depending only on , and is the distance on Λ.The classes C 0,1 ,Lip and C 0,2 ,Lip are defined analogously.
Clearly Ω ⊆ Λ.Given û : Λ → R and  : Ω → R, we say that  is consistent with û on Ω if (since we already use the symbol   to denote our freezing path operator (see Definition 1), we here use   to denote a sample path) for each   ∈ Ω,  (  ) = û (  ) . (30) ,Lip (Λ) such that (30) holds and for   ∈ Ω we denote Note.In the introduction, we use the notation {V(⋅, )} for a family of nonanticipative functionals where V(⋅, ) : Λ  → R. In order to highlight the symmetry between PDEs and PPDEs, the notation V(   , ) in PPDEs shows that    is the counterpart of the argument  in PDEs and is used instead of   .This is in spirit closer to the original notation of [1,2].The reader will have no problem identifying (

Path-Dependent PDEs.
The drift  and terminal condition Ψ are required to be extended to the space of càdlàg paths because of the definition of the Dupire derivatives.We require the following (see [6] again): (B2) The drift (  ) is a given R-valued continuous function defined on Ω × R × R (see [6] for a definition of continuity).Moreover, there exists a function  satisfying (H2) such that  =  on Ω.
We note that, in the case of no drift ( = 0), we recover the result (6).

Proof of Proposition 2.
This proof is made up by several inductions.Therefore we separate them into several steps.
Step 1.We first apply Itô's lemma and integration by parts formula of the Skorohod integral of Brownian motion to provide an explicit expansion for   (  ).The goal of the following step is to transform Skorohod integrals into timeintegrals.For example, ( 1 ,  2 ) is symmetric: International Journal of Stochastic Analysis By the integration by parts formula (see (1.49) in [8]), and   0 ( 1 , . . .,   ) = 1.For  = 0,   0 = 1.Then we are going to prove based on the following recurrence formula of   : for any  = 0, . . .,  − 1 To prove (43), we apply the integration by parts formula.For simplicity, we only keep the variables  1 , . . .,   and  +1 .The notation x means that the variable  is not an argument of a function.We also emphasize again the symmetricity of function   : Observing the properties of the binomial coefficients, We can see that, under the summation over , (47) and ( 49) cancel each other, ( 45) and ( 46) combine into   +1 , and (48) remains as the integral of   −1 .Rigorously, we proved (43).To prove (42), we use induction.Supposing that case  is correct, we observe case  + 1: by (43), Step 2. Now we are going to consider the action of the freezing path operator.We first prove that for all  ≤    ∘    ( +1 , . . .,   ) =    ( +1 , . . .,   ) . (52) We only present the proof of  =  and the general case is the same.By definition, we know that   ∘   =    [0,] () +    [,] ().Therefore Now we recall a basic integration rule for a smooth function   as ) . (54) We apply (54) on ( 53) and obtain Since the number of variable  is  −  +  −  1 −  =  −  1 − , which does not depend on , it enlightens us to change the International Journal of Stochastic Analysis order of summations.We want to sum over  first.Observe that According to the property of binomial coefficient again We claim that (56) is not 0 only when  =  +  1 .Thus we have Step 3. Now we can prove recurrence formula (10).By ( 52) and (42), we have Now we calculate the right hand side of (10): Let  =  +  1 and we continue the above formula: Now we apply another basic rule of integration, for a - Now apply (62) in (61) and we finally obtain Step 4. We now use induction to prove (11), based on (10).For simplicity, we introduce for  ≤ ⌊/2⌋.Then (10) implies We calculate the right hand side of (11) with (65 The proposition is proved.
In other words, the kernel    is constant when its arguments lie between  and +1/.Then we have the following lemma.
Lemma 12.   ∘   (   ) 2 (P) where  is a constant which does not depend on  and .
Proof.For any fixed , we define a sequence of sets Observe that on   1 ,0 the kernels   and    coincide.According to (67), we obtain To bound (70), we apply Proposition 2 to obtain Therefore following (72), we obtain where  is a constant which does not depend on  and .Now we construct   by ∑ ∞ =0   (   ).To prove the theorem, we introduce two subseries  , and   by For enough large , we choose  such that ( 7 (!) 2 ) 1/3  ≤ .Then by Lemma 12 and Cauchy-Schwarz inequality, there exists a constant  ∈ (0, 1) such that Then using triangle inequality, we prove the theorem.

Proof of Theorem 4.
For any  ∈  2 (P),  ∈ [, ], we choose the sequence {  } ≥0 constructed in Theorem 3. Then by the Clark-Ocone formula, we obtain where On one hand, by Lemma 5.2 in [5], we obtain On the other hand, we can compute International Journal of Stochastic Analysis 11 Then we can establish the equation as where the last equality follows from (79), Proposition 2. Thus combining (77), ( 79