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We apply a new series representation of martingales, developed by Malliavin calculus, to characterize the solution of the second-order 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.

In this paper we consider semilinear second-order path-dependent PDEs (PPDEs) of parabolic type. These equations were first introduced by Dupire [

To motivate our result, we first consider the heat equation expressed in terms of a backward time variable. For

It is well known (see, e.g., [

The differential operator

the Dupire derivative; while the Dupire derivative corresponds to changes of the path at only one time, the iterated derivatives

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 (

By the functional Feynman-Kac formula introduced in [

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

The main disadvantage can be seen immediately by considering (

This paper is composed of two parts. In the first part, we give a rigorous proof of the result (

We first introduce some basic notations of Malliavin calculus. For a detailed introduction, we refer to [

We denote the Malliavin derivative of order

For any deterministic function

From the definition, it is not hard to obtain that, for any

Let

Let

there exist

We introduce the derivative

For

Then our main theorem is the integral version of this operator differential equation. We first introduce the convergence condition.

For any

According to isometry (

We claim that other conditions exist which are easier to check than Condition

Moreover, with different structures of

If

Therefore, if there exists a constant

with the help of Stirling approximation

If

Then according to (

with some constant

Then we have the following main result.

Suppose that

The importance of the exponential formula (

We now introduce some key concepts of the functional Itô calculus introduced by Dupire [

Given a function

Then we say that

For a given

The function

For each

The function

As in [

We will make the following assumptions:

We now assume that

The drift

We can now define the following quasilinear parabolic path-dependent PDE:

Theorem 4.2 in [

Suppose that, for each

According to (2.20) in [

We note that, in the case of no drift (

This proof is made up by several inductions. Therefore we separate them into several steps.

We first apply Itô’s lemma and integration by parts formula of the Skorohod integral of Brownian motion to provide an explicit expansion for

By the integration by parts formula (see (1.49) in [

Based on this idea, for

To prove (

To prove (

Now we are going to consider the action of the freezing path operator. We first prove that for all

Now we can prove recurrence formula (

By (

Now apply (

We now use induction to prove (

The proof is constructive. For any fixed

For any fixed

Now we construct

For any

For

The authors declare that there is no conflict of interests regarding the publication of this paper.