Stochastic exponentials and exponential martingales

Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be a Brownian motion with the natural filtration $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$.

Find all $\alpha , \beta \in \mathbb {R}$ for which the process

is a martingale.

Let $W = \left\{ W_{t}, t \geqslant 0\right\}$ be a Brownian motion with the natural filtration $\mathbb {F} = \left(\mathscr {F}_{t}\right)_{t \geqslant 0}$.

For $\alpha , x \in \mathbb {R}$ and $t \geqslant 0$ define the function

Set

Prove that $\left(h_{k}\left(W_{t}, t\right), \mathscr {F}_{t}\right)_{t \geqslant 0}$ is a martingale for each $k \geqslant 1$ (note also that, by Exercise 13, the process $\left\{ h\left(\alpha , W_{t}, t\right), t \geqslant 0\right\}$ is a martingale for each $\alpha \in \mathbb {R}$ ). Thus the processes $W_{t}, W_{t}^{2}-t, W_{t}^{3}-3 t W_{t}$, $W_{t}^{4}-6 t W_{t}^{2}+3 t^{2}, \ldots$ are martingales $(t \geqslant 0)$.

Definition 10. We say that a function $f$ belongs to the class $J_{1}$ if $f:[0, \infty ) \times \Omega \rightarrow \mathbb {R}$ is progressively measurable and

Consider, for $f \in J_{1}$, the process

Prove that $d Z_{t} = Z_{t} f(t, \omega ) d W_{t}$ (as before, all differential relations are shorthand for the corresponding integral ones).

Exponential martingale, geometric Brownian motion. Let $W$ be a standard Wiener process and define $S(t) = e^{a t+b W(t)}$. Show that:

Let $W$ be a standard Wiener process and let $X(t) = \exp \left\{ i \theta W(t)+\frac{1}{2} \theta^{2} t\right\}$ where $i = \sqrt{-1}$. Show that $X$ is a martingale with respect to the filtration given by $\mathcal{F}_{t} = \sigma (\left\{ W(u): u \leq t\right\} )$.

Let $\left(W_{t}, t \geq 0\right)$ be a Wiener process. Find all pairs $(\alpha , \beta ) \in \mathbb {R}^{2}$ such that the process

is a martingale (submartingale, supermartingale) with respect to the natural filtration of the process $W_{t}$.

Let $\left(X_{t}, t \geqslant 0\right)$ be a process from $L^{2}\left((0,+\infty ) \times \Omega , \mathcal{PRED}, \text{mes} \times \mathbb {P}\right)$. Prove that then the process

satisfies the relation $\mathbb {E}\left[Y_{t}\right] \leqslant 1$ for all $t>0$ (we assume that all necessary processes are integrable).

Let $\left(X_{t}, t \geqslant 0\right)$ be a bounded process from $L^{2}((0,+\infty ) \times \Omega , \mathcal{P R E D}$, mes $\times \mathbb {P})$, i.e., there exists $C>0$ such that $\left|X_{t}\right| \leqslant C$ for all $t \geqslant 0$. Prove that then the process

is a martingale.

Let $\left(B_{t}, \mathscr {F}_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$. Show that $X_{t} = \exp \left(a B_{t}+b t\right), t \geqslant 0$, is a martingale if, and only if, $a^{2} / 2+b = 0$.

Let $\left(B_{t}\right)_{t \geqslant 0}$ be a $\mathrm{BM}^{1}$. Apply Doob's maximal inequality (A.13) to the exponential martingale $M_{t}^{\xi }: = \exp \left(\xi B_{t} - \frac{1}{2} \xi^{2} t\right)$ to show that for all $x, \xi > 0$

(This inequality can be used for the upper bound in the proof of Theorem 12.1, avoiding the combination of (12.2) and the upper bound in (12.1).)

State and prove the $d$ - dimensional version of Lemma 19.1.