MC 2: The Skorokhod Integral
1 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 \(\Ff_t\) (and thus not Itô integrable). This allows us to extend the Itô integral to non-adapted processes.
2 Why should I care?
If you want to define the Malliavin derivative you need the Skorokhod integral.
3 Notation
The same as in the previous post on the chaos expansion. We will also write \(\mathbb{L}^2(I\times \Omega)\) for the space of Itô integrable functions.
4 The chaos expansion of an Itô integral
Our goal in this post is to construct the Skorokhod 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 \(X_t\in L^2(\Omega,\Ff_\infty)\) for each \(t \in I\). Then we know that by the chaos expansion proved in the previous post, for each \(t\) there exists \(f_{n,t}\in L^2(S_n)\) such that
\[ X=\sum_{n=0}^{\infty} I_n(f_{n,t}). \]
Let us write \(f_{n}(\cdot,t):=f_{n,t}\). Note that we are now considering \(f_n\) as a function of \(n+1\) variables instead of \(n\). In particular, we will be able to consider expressions like \(I_{n+1}(f_n)\) later on. The first thing we do is study what the adaptedness of \(X\) means in term of the functions \(f_n\) appearing in its chaos expansion.
Lemma 1 Let \(X(t) \in L^2(\Omega,\Ff_\infty)\) for each \(t \in I\), then \(X\) is adapted iff
\[ f_n(t_1,\ldots,t_n,t)=0,\quad\forall t\leq\max_{i=1,\ldots,n} t_i . \]
Proof. Firstly, we note that a stochastic process \(X\) is adapted iff \[X_t=\E_{\Ff_t}[X(t)]\quad\forall t\in I.\] Since the Itô integral is a martingale, we obtain that, by commuting the sum and using the uniqueness of the chaos expansion this is equivalent to requiring that, for all \(t\) \[\begin{multline*} I_n(f_n(\cdot,t))\\= n!\E_{\Ff_t} \left[\int_{I} \left( \int_{0}^{t_n}\cdots \int_{0}^{t_2}f(t_1\ldots t_n,t) dW(t_1) \cdots dW(t_{n-1}) \right)dW(t_n)\right]\\ =n!\int_{0}^t \int_{0}^{t_n}\cdots \int_{0}^{t_2}f(t_1\ldots t_n,t) dW(t_1) \cdots dW(t_{n-1})dW(t_n)\\= I_n(f_n(\cdot,t) 1_{\max_{t_i\leq t}}), \end{multline*}\] where the commutation of the sum and the integral is justified by the \(L^2(\Omega)\) convergence of the chaos expansion (\(L^1(\Omega)\) convergence would have been enough).
In particular, we obtain that, since \(f_n\) is already symmetric in its first n-coordinates, its symmetrization verifies that
\[ f_{n,S}(t_1,\ldots,t_n,t_{n+1})=\frac{1}{n+1}f_n(t_1,\ldots\hat{t_{j}},\ldots,t_{n+1},t_j),\quad \text{where } j=\text{arg}\max_i t_i. \]
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
\[ \int_{I} X(t) dW(t)=\sum_{n=0}^{\infty} I_{n+1}(f_{n,S}). \]
Proof. This is a direct calculation using the previous result as
\[\begin{multline*} \int_{I} X(t) dW(t)=\sum_{n=0}^{\infty}\int_{I} I_n(f_{n,t})dW(t)\\=\sum_{n=0}^{\infty}n! \int_{I}\int_{S_n}f_{n,t}(t_1,\ldots,t_n) dW(t_1)\ldots dW(t_n) dW(t)\\=\sum_{n=0}^{\infty}(n+1)! \int_{I}\int_{S_n}f_{n,S}(t_1,\ldots,t_n,t) dW(t_1)\ldots dW(t_n) dW(t)\\=\sum_{n=0}^{\infty}(n+1)! J_{n+1}(f_{n,S}) =\sum_{n=0}^{\infty} I_{n+1}(f_{n,S}). \end{multline*}\]
5 The Skorokhod integral
The last term appearing in the equality is what we will call the Skorokhod integral.
Definition 1 Let \(X(t)\in L^2(\Omega,\Ff_\infty)\) be a stochastic process such that
\[ \delta(X):=\int_{I} X(t)\delta W(t):=\sum_{n=0}^{\infty} I_{n+1}(f_{n,S})\in L^2(\Omega). \]
Then we will say that \(X\) has Skorokhod integral \(\delta(X)\) and write \(X\in \rm{dom} \delta)\).
As we saw in the previous theorem the Skorokhod 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 [@Chaosblog], we deduce the following).
Proposition 1 A stochastic process \(X(t)\in L^2(\Omega,\Ff_\infty)\) has a Skorokhod integral iff
\[ \sum_{n=0}^{\infty} (n+1)!\|f_{n,S}\|^2_{L^2([0,T]^{n+1})}<\infty. \]
Proof. By Itô’s \(n\)-th isometry we have that
\[ \norm{\delta(X)}^2_{L^2(\Omega)}=\sum_{n=0}^{\infty} \norm{I_{n+1}(f_{n,S})}_{L^2(\Omega)}^2=\sum_{n=0}^{\infty} (n+1)!\norm{f_{n,S}}_{L^2(I^{n+1})}^2. \]
Of course, a priori the above condition is not that easy to check for a given function as it involves calculating the chaos expansion for the given process \(X\). In some cases however it is possible, to consider for example the stochastic process defined by \(X(t)=W(T)\) on the interval \(I=[0,T]\). Then we have that
\[ X(t)=\int_{0}^T dW(t)=I_1(1). \]
Thus, for all \(t\in I\) we have that
\[ f_1=1;\quad f_n=0\quad\forall n \in \N \setminus \{1\} . \]
So \(X\in \rm{dom}(\delta)\) with
\[ \delta(X)= I_2(1)=\int_{0}^T\int_{0}^t dW(t_1)dW(t)=\int_{0}^T W(t) dW(t)= \frac{1}{2}(W^2(T)-T). \]
Note however that the Itô integral of \(W(T)\) is undefined as it is not \(\Ff_t\) adapted. Since the Skorokhod 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
\[ \int_{I} X(t) \delta W(t)=\int_{I}G\cdot u(t)\delta W(t) \neq G\int_{I}u(t) dW(t). \]
Though this may seem unintuitive, it is a consequence of the fact that, even though \(f_i\) may not depend on \(t\), the terms
\[ g(t):=\int_{0}^t\int_{0}^{t_{n}}\cdots \int_{0}^{t_2} f_i dW(t_1)\cdots dW(t_{n-1})dW(t_n) \]
can depend on \(t\). Despite this, the Skorokhod integral still maintains some of the natural properties we associate with integration.
Proposition 2 Let \(X(t), Y(t)\in \rm{dom} \delta)\),\(\lambda \in \R\). Then it holds that
\(X(t)+ \lambda Y(t) \in \rm{dom} \delta)\)with\(\delta(X+\lambda Y)=\delta(X)+\lambda \delta(Y)\).
\(\E[\delta(X)]=0\).
\(X\cdot 1_A \in \rm{dom} \delta)\)for any measurable subset\(A \subset I\). Furthermore, if\(A \cup B =I\) then
\[ \int_{A}X(t) \delta W(t)+ \int_{B}X(t)\delta W(t):=\delta(X\cdot 1_A)+\delta(X\cdot 1_B)=\delta(X). \]
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 \(X\cdot 1_A\) is
\[ X\cdot 1_A=\sum_{n=0}^{\infty} I_{n}([f_n 1_A]_S). \]
Which shown by the equivalent characterization of Skorokhod functions that \(X\cdot 1_A\in \rm{dom} \delta)\). The final property is a consequence of the previously proved linearity.
We now know what the Skorokhod expansion is, how to characterize it, and its main properties, in the next post we will construct the Malliavin derivative as its adjoint.
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