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The Skorohod integral

How does one generalize the Itô integral?

By L. Llamazares-Elias
Posted on June 10, 2022

Three line summary

  • By fixing $t$, one can obtain a chaos expansion for (possibly non-adapted) square integrable stochastic processes $X(t)$.

  • The Itô integral of an Itô integrable process $X(t)$ has a chaos expansion.

  • This chaos expansion can converge even when $X(t)$ is not adapted to the filtration $\mathcal{F}\U t$ (and thus not Itô integrable). This allows us to extend the Itô integral to non-adapted processes.

Why should I care?

The Malliavin derivative is the adjoint operator of the Skorohod integral.

Notation

The same as in the previous post 1 on the chaos expansion. We will also write $\mathbb{L}^2(I\times \Omega)$ for the space of Itô integrable functions (this was defined in 2).

The chaos expansion of an Itô integral

Our goal in this post is to construct the Skorohod integral. This serves as a generalization of the Itô integral and the starting point for the definition of the Malliavin derivative. How is this done? Let us first consider a (not-necessarily adapted) stochastic process $X$ such that $X(t)\in L^2(\Omega,\mathcal{F}\U T)$ for each $t \in I$. Then we know that by the chaos expansion proved in the previous post, for each $t$ there exists $f\U {n,t}\in L^2(S\U n)$ such that

$$X(t)=\sum\U {n=0}^{\infty} I\U n(f\U {n,t}).$$

Let us write $f\U {n}(\cdot,t):=f\U {n,t}$. Note that we are now considering $f\U n$ as a function of $n+1$ variables instead of $n$. Furthermore, if we also impose that $X\in L^2(I\times \Omega,\mathcal{B}(I)\otimes \mathcal{F}\U T)$. Then by approximating $X$ by a sequence of functions of the form

$$X_k(t,\omega)=\sum_{i=1}^{N_k} a_i^{(k)}(t) b^{(k)}_i(\omega),$$

and using the linearity of $I\U n$, one can to show that $f\U n\in L^2(I^{n+1})$. In particular, we will be able to consider expressions like $I\U {n+1}(f\U n)$ later on. The first thing we do is study what the adaptedness of $X$ means in term of the functions $f\U n$ appearing in its chaos expansion.

Lemma 1. Let $X(t) \in L^2(\Omega,\mathcal{F}\U T)$ for each $t \in I$, then $X$ is adapted iff

$$f\U n(t\U 1,\ldots,t\U n,t)=0,\quad\forall t\leq\max\U {i=1,\ldots,n} t\U i .$$

Proof. Firstly, we note that a stochastic process $X$ is adapted iff

$$X(t)=\mathbb{E}\U {\mathcal{F}\U t}\zl X(t)\zr \quad\forall t\in I.$$

By commuting the sum (which we recall converges in $L^2(\Omega)$ and thus also in $L^1(\Omega)$) and using the uniqueness of the chaos expansion this is equivalent to requiring that, for all $t\in I$,

$$\begin{gathered} I\U n(f\U n(\cdot,t))= n!\mathbb{E}\U {\mathcal{F}\U t} \left[ \int\U {I} \left( \int\U {0}^{t\U n}\cdots \int\U {0}^{t\U 2}f\U n(t\U 1\ldots t\U n,t) dW(t\U 1) \cdots dW(t\U {n-1}) \right)dW(t\U n)\right]\\ =n!\int\U {0}^t \int\U {0}^{t\U n}\cdots \int\U {0}^{t\U 2}f\U n(t\U 1\ldots t\U n,t) dW(t\U 1) \cdots dW(t\U {n-1})dW(t\U n)= I\U n(f\U n(\cdot,t) 1\U {\max\U {t\U i\leq t}}) \end{gathered}$$

Where in the second equality we used that the Itô integral is a martingale. ◻

In particular, we obtain that, since $f\U n$ is already symmetric in its first n-coordinates, its symmetrization verifies that

$$f\U {n,S}(t\U 1,\ldots,t\U n,t\U {n+1})=\frac{1}{n+1}f\U n(t\U 1,\ldots\hat{t\U {j}},\ldots,t\U {n+1},t\U j),\quad \text{where } j=\text{arg}\max\U i t\U i.$$

Using this relationship we can directly calculate the Itô integral of a stochastic process to obtain that.

Theorem 1. Let $X \in \mathbb{L}^2(I\times\Omega)$ then the Itô integral of $X$ is

$$\int\U {I} X(t) dW(t)=\sum\U {n=0}^{\infty} I\U {n+1}(f\U {n,S}).$$

Proof. This is a direct calculation using the previous result as

$$\begin{gathered} \int\U {I} X(t) dW(t)=\sum\U {n=0}^{\infty}\int\U {I} I\U n(f\U {n,t})dW(t)=\sum\U {n=0}^{\infty}n! \int\U {I}\int\U {S\U n}f\U {n,t}(t\U 1,\ldots,t\U n) dW(t\U 1)\ldots dW(t\U n) dW(t)\\=\sum\U {n=0}^{\infty}(n+1)! \int\U {I}\int\U {t\U 2\leq\cdots\leq t\U n\leq t}f\U {n,S}(t\U 1,\ldots,t\U n,t) dW(t\U 1)\ldots dW(t\U n) dW(t)=\sum\U {n=0}^{\infty}(n+1)! J\U {n+1}(f\U {n,S}) =\sum\U {n=0}^{\infty} I\U {n+1}(f\U {n,S}).\end{gathered} $$

◻

In the above theorem we commuted the sum with the Itô integral (note the difference with the commutation in Lemma 1). We didn’t completely justify this, we do so now.

Lemma 3 Consider $X\U n\in L^2(I \times \Omega)$ such that for each $t\in I$ we have the convergence

$$\sum\U {n=0}^\infty \norm{X\U n(t)}\U{L^2(\Omega)}\in L^2(I). $$

Denote by $X(t):=\sum\U {n=0}^\infty X\U n(t)$. Then if $X \in L^2(I\times \Omega)$ the convergence is also in $L^2(I\times\Omega)$. That is, it holds that

$$\lim\U{N\to\infty}\norm{X-\sum\U{n=0}^N X\U n}\U {L^2(I\times\Omega)}=0. $$

Proof. We have that

$$ \lim \U {N\to\infty}\norm{X-\sum\U{n=0}^N X\U n}\U{L^2(I\times\Omega)}=\lim \U {N\to\infty}\left\|\norm{\sum\U{n=N}^\infty X\U n}\U{L^2(\Omega)}\right\|\U{L^2(I)}=\left\|\lim \U {N\to\infty}\norm{\sum\U{n=N}^\infty X\U n}\U{L^2(\Omega)}\right\|\U{L^2(I)}=0.$$

Where in the commutation of the limit with the $L^2(I)$ norm we used the bounded convergence theorem, which we may apply as the sum of $X\U n$ is normally convergent by hypothesis.  ◻

Let us now set $X\U n(t):=I \U n(f\U {n,t})$. We already noted that $f\U n \in L^2(I^{n+1})$. Thus, as we showed in the first post of the series), $I\U n(f\U n)$ is in $\mathbb{L}^2(I\times \Omega)$. As a result, by Itô’s $n$-th isometry this sequence meets the conditions of the above lemma. in consequence, the sum converges in $L^2(I\times\Omega)$ and also in $\mathbb{L}^2(I\times\Omega)$ . The commutation with the sum in Theorem $1$ is now justified by using that Itô integration is a continuous (isometry) on $\left(\mathbb{L}^2(I\times\Omega),\norm{\cdot}\U{ L^2(I\times\Omega)}\right)$.

The Skorohod integral

The last term appearing in the equality is what we will call the Skorohod integral.

Definition 1. Let $X\in L^2(I\times \Omega)$ be a stochastic process such that

$$\delta(X):=\int\U {I} X(t)\delta W(t):=\sum\U {n=0}^{\infty} I\U {n+1}(f\U {n,S})\in L^2(\Omega).$$

Then we will say that $X$ has Skorohod integral $\delta(X)$ and write $X\in dom(\delta)$.

As we saw in the previous theorem the Skorohod integral is equal to the Itô integral for all stochastic processes in $\mathbb{L}^2(I\times\Omega)$. However, it may also be defined for non-adapted stochastic processes. In fact, by using the orthogonality of the iterated integrals (what we called Itô’s $n$-th isometry in the last post 1, we deduce the following).

Proposition 1. A stochastic process $X\in L^2(I\times \Omega)$ has a Skorohod integral iff

$$\sum\U {n=0}^{\infty} (n+1)!\|f\U {n,S}\|^2\U {L^2(I^{n+1})}<\infty.$$

Proof. By Itô’s $n$-th isometry we have that

$$\norm{\delta(X)}^2\U {L^2(\Omega)}=\sum\U {n=0}^{\infty} \norm{I\U {n+1}(f\U {n,S})}^2\U {L^2(\Omega)}=\sum\U {n=0}^{\infty} (n+1)!\norm{f\U {n,S}}^2\U {L^2(I^{n+1})}.$$

◻

An identical application of Itô’s $n$-th isometry proves that

Proposition 2 (Skohorod’s isometry). Given two stochastic processes $X,Y\in L^2(I\times \Omega)$ with chaos expansion $X(t)=\sum\U n I\U n(f\U n(\cdot,t))$ and $Y(t)=\sum\U n I\U n(g\U n(\cdot,t))$, it holds that

$$\langle \delta(X),\delta(Y)\rangle\U {L^2(\Omega)}=\sum\U {n=0}^{\infty} (n+1)!\langle f\U {n,S}, g\U {n,S}\rangle \U {L^2(I^{n+1})}<\infty.$$

 ◻

Of course, a priori the condition of Proposition $1$ is not that easy to check, as it involves calculating the chaos expansion for the given process $X$. In some cases, however, it is possible. Consider for example the stochastic process defined by $X(t)=W(T)$ on the interval $I=\zl 0,T\zr $. Then we have that

$$X(t)=\int\U {0}^T dW(t)=I\U 1(1).$$

Thus, for all $t\in I$ we have that

$$f\U 1=1;\quad f\U n=0\quad\forall n \in \mathbb{N}\setminus \{1\} .$$

So $X\in dom(\delta)$ with

$$\delta(X)= I\U 2(1)=2\int\U {0}^T\int\U {0}^t dW(t\U 1)dW(t)=2\int\U {0}^T W(t) dW(t)= W^2(T)-T.$$

Note however that the Itô integral of $W(T)$ is undefined as it is not $\mathcal{F}\U t$ adapted. Since the Skorohod integral of $1$ is equal to $W(T)$, the above example shows how one cannot simply “pull out constants in $t$” in the sense that, if $G$ is a random variable independent of $t$ and $X(t)=G\cdot u(t)$, then

$$\int\U {I} X(t) \delta W(t)=\int\U {I}G\cdot u(t)\delta W(t) \neq G\int\U {I}u(t) dW(t).$$

Though this may seem unintuitive, it is a consequence of the fact that, even though $f\U i$ may not depend on $t$, the terms

$$g(t):=\int\U {0}^t\int\U {0}^{t\U {n}}\cdots \int\U {0}^{t\U 2} f\U i dW(t\U 1)\cdots dW(t\U {n-1})dW(t\U n).$$

Can depend on $t$. Despite this, the Skorohod integral still maintains some of the natural properties we associate with integration.

Proposition 3. Let $X(t), Y(t)\in dom(\delta)$, $\lambda \in {\mathbb{R}}$. Then it holds that

  • $X(t)+ \lambda Y(t) \in dom(\delta)$ with $\delta(X+\lambda Y)=\delta(X)+\lambda \delta(Y)$.

  • $\mathbb{E}\zl \delta(X)\zr =0$.

  • $X\cdot 1\U A \in dom(\delta)$ for any measurable subset $A \subset I$. Furthermore, if $A \cup B =I$ then

$$\int\U {A}X(t) \delta(t)+ \int\U {B}X(t)\delta W(t):=\delta(X\cdot 1\U A)+\delta(X\cdot 1\U B)=\delta(X).$$

Proof. The first property is a consequence of the chaos expansion’s linearity (which is itself a consequence of the linearity of iterated Itô integration). The second is due to the expectation of the Itô integral being $0$. The final property is a consequence of the fact that the chaos expansion of $X\cdot 1\U A$ is

$$X\cdot 1\U A=\sum\U {n=0}^{\infty} I\U {n}(f\U n 1\U A).$$

Which shows by the equivalent characterization of Skorohod functions that $X\cdot 1\U A\in dom(\delta)$. The final property is a consequence of the previously proved linearity. ◻

We now know what the Skorohod expansion is, how to characterize it, and its main properties. In the next post, we will construct the Malliavin derivative as its adjoint.

Tags: Malliavin calculus
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