Stochastic Processes and Financial Mathematics
(part two)

12.5 Exercises on Chapter 12

In all the following questions, $B_t$ denotes a Brownian motion and ${\mathcal{F}}_t$ denotes its generated filtration.

On Ito integration

• 12.1 Using (12.8), find ${\int }_v^{t}1d{B}_u$, where $0\le v\le t$.

• 12.2 Show that the process $e^{B_t}$ is in ${\mathcal{H}}^2$. (Hint: Use (11.2).)

• 12.3

  • (a) Let $Z\sim N(0,1)$. Show that the expectation of $e^{\frac{Z^2}{2}}$ is infinite.

  • (b) Give an example of a continuous, adapted, stochastic process that is not in ${\mathcal{H}}^2$.

• 12.4 Let $X_t$ be an Ito process satisfying

  $$X_t=2+{\int }_0^{t}t+B_u^{2}du+{\int }_0^t{B}_u^{2}d{B}_u.$$

  Find $\mathbb{E}[X_t]$.

• 12.5 Which of the following stochastic processes are Ito processes?

  • (a) $X_t=0$,

  • (b) $Y_t=t^2+B_t$,

  • (c) The symmetric random walk from Section 4.1.

• 12.6 Let $V_t$ be the stochastic process given by

  $$V_t=e^{-kt}v+\sigma e^{-kt}{\int }_0^t{e}^{ku}d{B}_u$$

  where $k,\sigma ,v>0$ are deterministic constants. Find the mean and variance of $V_t$.

• 12.7 Suppose that $\mu >0$ is a deterministic constant and that ${\sigma }_t\in {\mathcal{H}}^2$. Let $X_t$ be given by

  $$X_t={\int }_0^t\mu du+{\int }_0^t{\sigma }_{u}d{B}_u.$$

  Show that $X_t$ is a submartingale.

• 12.8

  • (a) Give an example of a stochastic process $F_t$ such that ${\int }_0^t\mathbb{E}[F_s]ds=\mathbb{E}[{\int }_0^t{F}_{s}d{B}_s]$.

  • (b) Give an example of a stochastic process $F_t$ such that ${\int }_0^t\mathbb{E}[F_s]ds\ne \mathbb{E}[{\int }_0^t{F}_{s}d{B}_s]$.

Challenge Questions

• 12.9

  • (a) Let $X$ and $Y$ be random variables in $L^2$. Show that

    $$2|\mathbb{E}[XY]|\le \mathbb{E}[X^2]+\mathbb{E}[Y^2]$$

  • (b) Show that ${\mathcal{H}}^2$ is a real vector space.

• 12.10 $\text{offsyl}$ Prove Lemma 12.3.3.