Wiener Chaos and iterated Wiener–Itô integrals

Show that the definition of the double Itô integral for off - diagonal simple functions (Definition 20.4) is independent of the representation of the simple function.

Repeat the calculation from the end of Example 20.5 for a general measure $\mu$. What happens on the diagonal?

Show that the definition of the iterated Itô integral for $f \in L^{2}\left(\mathbb {R}_{+}^{2}\right)$ (Definition 20.9) is independent of the approximating sequence.

Show that the Hermite Polynomials are linearly independent.

Let $f, g \in L^{2}\left(\mathbb {R}_{+}\right)$and assume that $f \; \perp \! \! \! \perp \; g$, i.e. $\int_{\mathbb {R}_{+}} f(t) g(t) d t = 0$. Show that $I_{1}(f) \; \perp \! \! \! \perp \; I_{1}(g)$.

Let $(X, Y)$ be a mean zero Gaussian random variable. Show that $\mathbb {E}\left[X^{2} Y\right] = 0$. Use this observation to show that $I^{2}(f) - \left\| f\right\|_{L^{2}}^{2}$ is and $\mathcal{K}_{1}: = \left\{ I_{1}(g): g \in L^{2}\left(\mathbb {R}_{+}\right)\right\}$are orthogonal.

Pick any orthonormal basis $\left(e_{k}\right)_{k \geqslant 1}$ of $H$. Let $\alpha \in \mathbb {N}_{0}^{(\infty )}$, i.e. $\alpha = \left(\alpha_{1}, \ldots , \alpha_{N}, 0,0, \ldots \right)$ where $N \in \mathbb {N}$ and $\alpha_{1}, \ldots , \alpha_{N} \in \mathbb {N}_{0}$, and define $e_{\alpha }: = \bigotimes_{k} e_{k}^{\otimes \alpha_{k}}$, i.e.

We agree that $e^{\otimes 0}$ means "leave this factor out" and $\left|\alpha \right| = \sum_{i = 0}^{\infty } \alpha_{i}$. By $\widehat{e}_{\alpha }$ we denote the symmetrization of $e_{\alpha }$.

Let $F, G$ be smooth, bounded cylindrical functions. Show that

Let $f \in L^{2}\left(\mathbb {R}_{m}^{+}\right), g \in L^{2}\left(\mathbb {R}_{n}^{+}\right)$and assume that the functions are symmetric: $f = \widehat{f}$, $g = \widehat{g}$. We define the contraction as

Show the following formula:

Use Itô's formula to show that for any two symmetric $f, g \in L^{2}\left(\mathbb {R}_{+}^{2}\right)$

Prove (20.20), i.e. show that $\mathbb {E}\left[B_{T}^{2 n} e^{ - B_{T}^{2}}\right] = \frac{(2 n - 1) ! !}{\sqrt{2 T+1}}\left(\frac{T}{2 T+1}\right)^{n}$. Here, $( - 1) ! !: = 1$, $(2 n - 1) ! ! = 1 \cdot 3 \cdot \ldots \cdot (2 n - 1)$, is the double factorial.

Let $F = \exp \left[I_{1}(g) - \frac{1}{2} \int_{0}^{\infty } g^{2}(s) d s\right]$ for some $g \in L^{2}\left(\mathbb {R}_{+}\right)$. Find all derivatives $D^{k} F$ as well as the Wiener - Itô chaos representation.

Find the Wiener - Itô chaos decomposition for the following random variables ( $T > 0$ is fixed):

Find the Wiener - Itô chaos decomposition for the following random variables ( $T > 0$ is fixed):

Let $\mathcal{D} \subset L^{2}(\mu )$ and write $\Sigma : = \operatorname {span}(\mathcal{D})$. Show that $\Sigma$ is dense if, and only if $\mathcal{D}$ is total, i.e.