Math Problems Vault

Problem 13.8.1

(a)(b)(c)

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(a)

Verify directly that $\int_{0}^{t} s d W_{s} = t W_{t}-\int_{0}^{t} W_{s} d s$ .

(b)

Verify directly that $\int_{0}^{t} W_{s}^{2} d W_{s} = \frac{1}{3} W_{t}^{3}-\int_{0}^{t} W_{s} d s$ .

(c)

Show that $\mathbb {E}\left[\left[\int_{0}^{t} W_{s} d W_{s}\right]^{2}\right] = \int_{0}^{t} \mathbb {E}\left[W_{s}^{2}\right] d s$ .

Problem 13.8.2

Let $X_{t} = \int_{0}^{t} W_{s} d s$ . Show that $X$ is a Gaussian process, and find its autocovariance and autocorrelation function.

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Problem 13.8.3

Let $(\Omega , \mathcal{F}, \mathbb {P})$ be a probability space, and suppose that $X_{n} \xrightarrow {\text{ m.s. }} X$ as $n \rightarrow \infty$ . If $\mathcal{G} \subseteq \mathcal{F}$ , show that $\mathbb {E}\left[X_{n} \mid \mathcal{G}\right] \xrightarrow {\text{ m.s. }} \mathbb {E}\left[X \mid \mathcal{G}\right]$ .

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Problem 13.8.4

Let $\psi_{1}$ and $\psi_{2}$ be predictable step functions, and show that

\mathbb {E}\left[I\left(\psi _{1}\right) I\left(\psi _{2}\right)\right] = \mathbb {E}\left[\int _{0}^{\infty } \psi _{1}(t) \psi _{2}(t) d t\right]

whenever both sides exist.

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Problem 13.8.5

Assuming that Gaussian white noise $G_{t} = d W_{t} / d t$ exists in sufficiently many senses to appear as an integrand, show by integrating the stochastic differential equation $d X_{t} = -\beta X_{t} d t+d W_{t}$ that

X_{t} = W_{t}-\beta \int _{0}^{t} e^{-\beta (t-s)} W_{s} d s

if $X_{0} = 0$ .

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Problem 13.8.6

Let $\psi$ be an adapted process with $\left\| \psi \right\| < \infty$ . Show that $\left\| I(\psi )\right\|_{2} = \left\| \psi \right\|$ .

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