Lecture 34. The Wiener chaos expansion

As in the previous Lectures, we consider a filtered probability space (\Omega, (\mathcal{F}_t)_{0 \le t \le 1}, \mathbb{P}) on which is defined a Brownian motion (B_t)_{0 \le t \le 1}, and we assume that (\mathcal{F}_t)_{0 \le t \le 1} is the usual completion of the natural filtration of (B_t)_{0 \le t \le 1}. Our goal is here to write an orthogonal decomposition of the space L^2(\mathcal{F}_1) that is particularly suited to the study of the space \mathbb{D}^{1,2}. For simplicity of the exposition, we restrict ourselves to the case where the Brownian motion (B_t)_{0 \le t \le 1} is one-dimensional.

In the sequel, for n \ge 1, we denote by \Delta_n the simplex \Delta_n =\{ 0\le t_1 \le \cdots \le t_n \le 1\} and if f_n \in L^2( \Delta_n),
I_n (f_n) =\int_0^1 \int_0^{t_n} \cdots \int_0^{t_2} f_n(t_1,\cdots,t_n)  dB_{t_1}...dB_{t_n}
=\int_{\Delta_n}  f_n(t_1,\cdots,t_n)  dB_{t_1}...dB_{t_n}.

The set
\mathbf{K}_n=\left\{\int_{\Delta_n}  f_n(t_1,\cdots,t_n)  dB_{t_1}...dB_{t_n}, f_n \in L^2( \Delta_n)  \right\}
is called the space of Wiener chaos of order n. By convention the set of constant random variables shall be denoted by \mathbf{K}_0.

By using the Itō’s isometry, we readily compute that
\mathbb{E} \left(I_n (f_n)I_p (f_p) \right)=  \begin{cases}  0 & \text{if }p \neq n \\  \| f_n \|^2_{L^2(\Delta_n)} & \text{if }p=n.  \end{cases}
As a consequence, the spaces \mathbf{K}_n are orthogonal in L^2. It is easily seen that \mathbf{K}_n is the closure of the linear span of the family
\left\{ I_n (f^{\otimes n}), f \in L^2([0,1]) \right\},
where for f \in L^2([0,1]), we denoted by f^{\otimes n} the map \Delta_n \to \mathbb{R} such that f^{\otimes n}(t_1,\cdots,t_n)=f(t_1)\cdots f(t_n). It turns out that I_n (f^{\otimes n}) can be computed by using Hermite polynomials. The Hermite polynomial of order n is defined as
H_n (x)=(-1)^n \frac{1}{n!} e^{\frac{x^2}{2}} \frac{d^n}{dx^n} e^{-\frac{x^2}{2}}.
By the very definition of H_n, we see that for every t, x \in \mathbb{R},
\exp \left( t x -\frac{t^2}{2}\right)=\sum_{k=0}^{+\infty} t^k H_k(x).

Lemma. If f \in L^2([0,1]) then I_n (f^{\otimes n})=\| f \|^n_{L^2([0,1])} H_n \left(\frac{ \int_0^1 f(s) dB_s}{\| f \|_{L^2([0,1])} } \right) .

Proof. On one hand, we have for \lambda \in \mathbb{R},
\exp \left( \lambda \int_0^1 f(s) dB_s-\frac{\lambda^2}{2} \int_0^1 f(s)^2 ds  \right)=\sum_{n=0}^{+\infty} \lambda^n \| f \|^n_{L^2([0,1])} H_n \left(\frac{ \int_0^1 f(s) dB_s}{\| f \|_{L^2([0,1])} } \right) .

On the other hand, for 0 \le t \le 1, let us consider
M_t(\lambda)=\exp \left( \lambda \int_0^t f(s) dB_s-\frac{\lambda^2}{2} \int_0^t f(s)^2 ds  \right).
From Itō’s formula, we have
M_t(\lambda)=1+\lambda \int_0^t M_s f(s) dB_s.
By iterating the previous linear relation, we easily obtain that for every n \ge 1,
M_1(\lambda)=1+\sum_{k=1}^n \lambda^k I_k( f^{\otimes k})+\lambda^{n+1} \int_0^1 M_tf(t)\left(\int_{\Delta_n([0,t])}  f(t_1)\cdots f(t_n)  dB_{t_1}...dB_{t_n}\right) dB_t.
We conclude,
I_n( f^{\otimes n})=\frac{1}{n!} \frac{ d^k M_1}{d \lambda^n}(0)=\| f \|^n_{L^2([0,1])} H_n \left(\frac{ \int_0^1 f(s) dB_s}{\| f \|_{L^2([0,1])} } \right) \square

As we pointed it out, for p \neq n, the spaces \mathbf{K}_n and \mathbf{K}_p are othogonal. We have the following orthogonal decomposition of L^2:

Theorem.[Wiener chaos expansion]
L^2 =\bigoplus_{n \ge 0} \mathbf{K}_n.

Proof. As a by-product of the previous proof, we easily obtain that for f \in L^2([0,1]),
\exp \left( \lambda \int_0^1 f(s) dB_s-\frac{\lambda^2}{2} \int_0^1 f(s)^2 ds  \right)=\sum_{n=1}^{+\infty}  I_n( f^{\otimes n}),
where the convergence of the series is almost sure but also in L^2. Therefore, if F \in L^2 is orthogonal to \bigoplus_{n \ge 1} \mathbf{K}_n, then F is orthogonal to every \exp \left( \lambda \int_0^1 f(s) dB_s-\frac{\lambda^2}{2} \int_0^1 f(s)^2 ds  \right), f \in L^2([0,1]). This implies that F=0 \square

As we are going to see, the space \mathbb{D}^{1,2} or more generally \mathbb{D}^{k,2} is easy to describe by using the Wiener chaos expansion. The keypoint is the following proposition:

Proposition. Let F=I_n(f_n) \in \mathbf{K}_n, then F \in \mathbb{D}^{1,2} and \mathbf{D}_t F=I_{n-1} ( \tilde{f}_n (\cdot, t)), where for 0\le t_1 \le \cdots \le t_{n-1} \le 1,
\tilde{f}_n (t_1,\cdots, t_{n-1},t)  =f_n (t_1,\cdots, t_k, t, t_{k+1}, \cdots, t_{n-1}) \quad  \text{if } t_{k} \le t \le t_{k+1}.

Proof. Let f \in L^2([0,1]). We have
I_n (f^{\otimes n})=\| f \|^n_{L^2([0,1])} H_n \left(\frac{ \int_0^1 f(s) dB_s}{\| f \|_{L^2([0,1])} } \right) .
Thus F=I_n (f^{\otimes n}) is a smooth cylindric functional and
\mathbf{D}_t F =\| f \|^{n-1}_{L^2([0,1])} f(t) H'_n \left(\frac{ \int_0^1 f(s) dB_s}{\| f \|_{L^2([0,1])} } \right).
It is easy to see that H_n'=H_{n-1}, therefore we have
\mathbf{D}_t F  =\| f \|^{n-1}_{L^2([0,1])} f(t) H_{n-1} \left(\frac{ \int_0^1 f(s) dB_s}{\| f \|_{L^2([0,1])} } \right)  =f(t) I_{n-1} (f^{\otimes {(n-1)}}).
As a consequence, we compute that \mathbb{E} \left(\int_0^1 (\mathbf{D}_t F)^2 dt  \right)=n \mathbb{E} (F^2). We now observe that \mathbf{K}_n is the closure in L^2 of the linear span of the family
\left\{ I_n (f^{\otimes n}), f \in L^2([0,1]) \right\}
to conclude the proof of the proposition \square

We can finally turn to the description of \mathbb{D}^{1,2} using the chaos decomposition:

Theorem. Let F \in L^2 and let
F=\mathbb{E}(F) +\sum_{m \ge 1} I_m (f_m),
be the chaotic decomposition of F. Then F \in \mathbb{D}^{1,2} if and only if
\sum_{m \ge 1} m \mathbb{E}\left(  I_m (f_m)^2\right) < +\infty,
and in that case,
\mathbf{D}_t F= \mathbb{E}(\mathbf{D}_tF) + \sum_{m \ge 2} I_{m-1} ( \tilde{f}_m (\cdot, t)).

Proof. It is a consequence of the fact that for F \in \mathbf{K}_n, \mathbb{E} \left(\int_0^1 (\mathbf{D}_t F)^2 dt  \right)=n \mathbb{E} (F^2) \square.

An immediate but useful corollary of the previous theorem is the following result:

Corollary. Let (F_n)_{n \ge 0} be a sequence in \mathbb{D}^{1,2} that converges to F in L^2 and such that
\sup_{ n \ge 0} \mathbb{E} \left(\int_0^1 (\mathbf{D}_t F_n)^2 dt  \right) < +\infty.
Then, F \in \mathbb{D}^{1,2}.

Exercise. Let F \in L^2 and let
F=\mathbb{E}(F) +\sum_{m \ge 1} I_m (f_m),
be the chaotic decomposition of F. Show that that F \in \mathbb{D}^{k,2}, k \ge 1 if and only if
\sum_{m \ge 1} m^k \mathbb{E}\left(  I_m (f_m)^2\right) < +\infty.

Exercise. Let L=\delta \mathbf{D} . Show that for F \in \mathbf{K}_n, LF=nF.

This entry was posted in Stochastic Calculus lectures. Bookmark the permalink.

2 Responses to Lecture 34. The Wiener chaos expansion

  1. alabair says:

    A typo on line constant instead of contant.

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