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.

1. Introduction

In this paper we consider semilinear second-order path-dependent 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 t ≤ T we look for a function v(x, t) that solves

()

It is well known (see, e.g., [3], chapter 9.2 or [4]) that the solution is given by the flow of the semigroup S(t); that is, v(·, t) = S(t)Ψ, where

()

The differential operator (1/2)(∂²/∂x²) is said to be the (infinitesimal) generator of the semigroup S(t). Consider now the path-dependent version of the heat equation:

()

where

is a continuous path on the interval [0, t] 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 1/2D_xx, that is, one-half times the second-order 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(·, t) for a change at time t. The correct infinitesimal generator is equal to

, where

is the second-order Malliavin derivative of

. An important difference is that F is now viewed as a random variable, and the (first-order) Malliavin derivative is a stochastic process in the canonical probability space for Brownian motion. The stopping path operator ω^t was introduced in [5]. Informally, the action of the stopping path operator (which we define rigorously later) is to freeze the path after time t:

()

where ω_t is the stopped path. The stopped Malliavin derivative

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 are taken with respect to changes of the canonical path at many different times s₁, …, s_n;
(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 B be Brownian motion. By Itô’s lemma, it is obvious that v(B(t), t) is a martingale, say M_t, and that the value of this martingale is the conditional expectation at time t of Ψ(B(T)). Consider now a general path-dependent terminal condition Ψ(B), in [5], Jin et al. gave a new representation of Brownian martingales M_t (with t ≤ T) as an exponential of a time-dependent generator, applied to the terminal value M_T ≡ Ψ(B):

()

By the functional Feynman-Kac formula introduced in [1, 6], it is immediate that 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.

2. Martingale Representation

We first introduce some basic notations of Malliavin calculus. For a detailed introduction, we refer to [8] and our paper [5]. Let and be the complete filtered probability space, where the filtration is the usual augmentation of the filtration generated by Brownian motion B on . The canonical Brownian motion can be also denoted by B(t) = B(t, ω) = ω(t), t ∈ [0, T], ω ∈ Ω, by emphasizing its sample path. We denote by the space of square integrable random variables. For simplicity, we denote (du)^⊗k≔du₁ ⋯ du_k.

We denote the Malliavin derivative of order l at time t₁, …, t_n by

. We call

the set of random variables which are infinitely Malliavin differentiable and

-measurable, that is, for any integer n and

()

Definition 1. For any deterministic function f ∈ L²([0, T]), we define the “stopping path” operator ω^t for t ≤ T as

()

In particular, ω^t∘B(s) = B(s∧t) that is to “freeze” Brownian motion after time t.

From the definition, it is not hard to obtain that, for any n-variable smooth function g, ω^t∘g(B(s₁), …, B(s_n)) = g(B(s₁∧t), …, B(s_n∧t)). For a general random variable , ω^t∘F refers to the value of F along the stopping scenario ω_t ≡ ω^t(ω) of Brownian motion. According to the Wiener-Chaos decomposition, for any , there exists a sequence of deterministic function {f_n} _n≥1 such that with convergence in L²([0, T] ⁿ). Therefore, in order to obtain an explicit representation of ω^t acting on a general variable F, we first show the following proposition.

Proposition 2. Let f_n ∈ L²([0, T] ⁿ), an n-variable square integrable deterministic function; then

()

Therefore

()

as well as the isometry:

()

Theorem 3. Let . Then for any fixed time t and t ≤ s < T, there exists a sequence {F^N} _N≥0 that satisfies the following:

(i)
F^N → F in ;
(ii)
D_uF^N = D_s+1/NF^N for any u ∈ s, s + 1/N];
(iii)
there exist ε ∈ (0,1) and a constant C which does not depend on N such that
()

We introduce the derivative d in

as, for any process F_s,

()

Then we can set up an operator differential equation for E_s. The following theorem is a generalization of Theorem 2.2. in [5] to functionals that are not discrete.

Theorem 4. For 0 ≤ t ≤ s ≤ T, assuming that , one has

()

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

Condition 1. For any n ≥ 0, F satisfies

()

According to isometry (12), this condition implies .

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,

for any s ∈ [t, T] and n ≥ 1, with a constant c.

Moreover, with different structures of F, we have different alternative conditions which are easier to check for practical calculations. Here we list two examples.

(1)
If with smooth deterministic function f and square integrable deterministic function g, it is not hard to obtain
()
Therefore, if there exists a constant C such that, for all n ≥ 1,
()
with the help of Stirling approximation , Condition 1 is satisfied.
(2)
If F has its chaos decomposition , we have
()
Then according to (12), Condition 1 can be replaced by
()
with some constant C or some much stronger but easier conditions like the following: for m ≥ 1
()

Then we have the following main result.

Theorem 6. Suppose that F satisfies Condition 1 and is -measurable. For t ≤ T, then, in ,

()

The importance of the exponential formula (23) stems from the Dyson series representation, which we rewrite hereafter in a more convenient way:

()

3. 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 T > 0 be fixed. For each t ∈ [0, T] we denote by Λ_t the set of càdlàg (right continuous with left limits)

-valued functions on [0, t]. For each γ_t ∈ Λ_t, the value of γ_t at s ∈ [0, t] is denoted by γ(s). Denote

. For each γ_t ∈ Λ, T ≥ s ≥ t, and

, we define

()

Definition 7. Given a function , there exists such that

()

Then we say that is vertically differentiable at γ_t ∈ Λ and define . The function is said to be vertically differentiable if exists for each γ_t ∈ Λ. The second-order derivative D_xx is defined similarly.

Definition 8. For a given γ_t ∈ Λ, if

()

then we say that

is horizontally differentiable at γ_t and define

. The function

is said to be horizontally differentiable if

exists for each γ_t ∈ Λ.

Definition 9. The function is said to be in if , , and exist and we have

()

where

, C and k are some constants depending only on φ, and

()

is the distance on Λ. The classes

and

are defined analogously.

For each t ∈ [0, T], we denote by Ω_t the set of continuous

-valued functions on [0, t]. We denote

. Clearly Ω⊆Λ. Given

and

, we say that u is consistent with

on Ω if (since we already use the symbol ω^t to denote our freezing path operator (see Definition 1), we here use ω_t to denote a sample path) for each ω_t ∈ Ω,

()

Definition 10. The function is said to be in if there exists a function such that (30) holds and for ω_t ∈ Ω we denote

()

Note. In the introduction, we use the notation {v(·, t)} for a family of nonanticipative functionals where . In order to highlight the symmetry between PDEs and PPDEs, the notation in PPDEs shows that is the counterpart of the argument x in PDEs and is used instead of ω_t. This is in spirit closer to the original notation of [1, 2]. The reader will have no problem identifying .

3.2. Non-Markovian BSDEs

As in [6], we use to denote the completion of the σ-algebra generated by B(s) − B(t) with s ∈ [t, r]. Then we introduce , the space of all -adapted -valued processes (X(s))_{s∈[t, T]} with , and S²(t, T), the space of all -adapted -valued continuous processes (X(s))_{s∈[t, T]} with E[sup_{s∈[t, T]}|X(s)|²] < ∞. Denote now .

We will make the following assumptions:

(H1) Φ is a -valued function defined on Λ_T. Moreover,

(H2) The drift a(γ_t) is a given

-valued continuous function defined on Λ (see [6] for a definition of continuity). For any γ_t ∈ Λ and s ∈ [0, t], the function

is differentiable and its derivative

satisfies

()

where C and k are constants depending only on φ.

We now assume that (H1) and (H2) hold. We consider a non-Markovian BSDE, which is a particular case of (3.2) in [6]. From Theorem 2.8 in [6], for any γ_t ∈ Λ, there exists a unique solution

of the following BSDE:

()

where

()

In particular,

defines a deterministic mapping from Λ to

3.3. Path-Dependent PDEs

The drift a 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):

(B1) The function Ψ is a -valued function defined on Ω_T. Moreover, there is a function such that Ψ = Φ on Ω_T.

(B2) The drift a(ω_t) is a given -valued continuous function defined on (see [6] for a definition of continuity). Moreover, there exists a function b satisfying (H2) such that a = b on Ω.

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

()

Theorem 4.2 in [6] states the following: let be a solution of the above equation. Then we have for each ω_t ∈ Ω, where is the unique solution of BSDE (33).

Theorem 11. Suppose that, for each t ∈ [0, T], the random variable

()

satisfies Condition 1. Then the solution of (35) is

()

Proof. According to (2.20) in [9] page 351, the solution of (33) is, for t ≤ s ≤ T,

()

The result now follows by Theorem 6 and the fact that

We note that, in the case of no drift (a = 0), we recover the result (6).

3.4. 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 I_n(f_n). The goal of the following step is to transform Skorohod integrals into time-integrals. For example, f(s₁, s₂) is symmetric:

()

By the integration by parts formula (see (1.49) in [8]),

()

Based on this idea, for n ≥ 1 and 1 ≤ r ≤ n, we define

()

and

. For n = 0,

. Then we are going to prove

()

based on the following recurrence formula of A_r: for any r = 0, …, n − 1

()

To prove (43), we apply the integration by parts formula. For simplicity, we only keep the variables s₁, …, s_k and s_r+1. The notation

means that the variable x is not an argument of a function. We also emphasize again the symmetricity of function f_n:

()

Observing the properties of the binomial coefficients,

()

We can see that, under the summation over k, (47) and (49) cancel each other, (45) and (46) combine into

, and (48) remains as the integral of

. Rigorously, we proved (43).

To prove (42), we use induction. Supposing that case n is correct, we observe case n + 1: by (43),

()

Step 2. Now we are going to consider the action of the freezing path operator. We first prove that for all r ≤ n

()

We only present the proof of r = n and the general case is the same. By definition, we know that ω^t∘B_s = B_sχ_{[0, t]}(s) + B_tχ_{[t, T]}(s). Therefore

()

Now we recall a basic integration rule for a smooth function g_n as

()

We apply (54) on (53) and obtain

()

Since the number of variable T is n − k + k − k₁ − j = n − k₁ − j, which does not depend on k, it enlightens us to change the order of summations. We want to sum over k first. Observe that

; we obtain

()

According to the property of binomial coefficient again

()

We claim that (56) is not 0 only when n = j + k₁. 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 m = k + k₁ and we continue the above formula:

()

Now we apply another basic rule of integration, for a m-variable symmetric function g_m

()

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 k ≤ ⌊n/2⌋. Then (10) implies

()

We calculate the right hand side of (11) with (65): let m = k + k₁

()

The proposition is proved.

3.5. Proof of Theorem 3

The proof is constructive. For any fixed t ∈ [0, T], if F has its chaos decomposition

, then for fixed N (depending on M), we will study

, where

()

In other words, the kernel

is constant when its arguments lie between s and s + 1/N. Then we have the following lemma.

Lemma 12. and in particular

()

where C is a constant which does not depend on N and n.

Proof. For any fixed n, we define a sequence of sets as

()

Observe that on

the kernels f_n and

coincide. According to (67), we obtain

()

To bound (70), we apply Proposition 2 to obtain

()

Now we apply (71) on (70) and by Cauchy-Schwartz inequality, we have

()

Since f_n is differentiable with respect to s₁, …, s_n, there exists a constant C_n such that

()

Therefore following (72), we obtain

()

where C is a constant which does not depend on n and N.

Now we construct F^N by . To prove the theorem, we introduce two subseries F^M,N and F^M by

()

For enough large N, we choose M such that

. Then by Lemma 12 and Cauchy-Schwarz inequality, there exists a constant ε ∈ (0,1) such that

()

Then using triangle inequality, we prove the theorem.

3.6. Proof of Theorem 4

For any

, s ∈ [t, T], we choose the sequence

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

()

Then we can establish the equation as

()

where the last equality follows from (79), Proposition 2. Thus combining (77), (79), (80), and (81) as well as the assumption

and Proposition 2, we have

()

Here, for simplicity, we define L² norm

. Then, from Theorem 3 and the closability of the Malliavin derivative operator, for some constant ε < 1,

()

With triangle inequality and Cauchy-Schwartz inequality, we finally have, using (81) and (83),

()

or in other words

()

3.7. Proof of Theorem 6

For i = 1, …, N(T − s), define

()

We rewrite (84) as

()

Jensen’s inequality states that

()

Since

is bounded in

, then

()

Using (88), we thus proved that, in

()

Then for positive integer n we define the operator

()

where

()

Then by iterating (90) we obtain the following: for n > 0

()

Thus according to Condition 1,

()

We now take s = t and obtain

()

Competing Interests

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

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

Abstract

1. Introduction

2. Martingale Representation

3. Representation of Solutions of Path-Dependent Partial Differential Equations

3.1. Functional Itô Calculus

3.2. Non-Markovian BSDEs

3.3. Path-Dependent PDEs

3.4. Proof of Proposition 2

3.5. Proof of Theorem 3

3.6. Proof of Theorem 4

3.7. Proof of Theorem 6

Competing Interests

References

References

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

Abstract

1. Introduction

2. Martingale Representation

3. Representation of Solutions of Path-Dependent Partial Differential Equations

3.1. Functional Itô Calculus

3.2. Non-Markovian BSDEs

3.3. Path-Dependent PDEs

3.4. Proof of Proposition 2

3.5. Proof of Theorem 3

3.6. Proof of Theorem 4

3.7. Proof of Theorem 6

Competing Interests

References

References

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