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
$\sum_{m \ge 1} m^k \mathbb{E}\left( I_m (f_m)^2\right) < +\infty.$