Continuous Time Martingales

Continuous Martingale

Let $(X_t{)}_{t\ge 0}$ be a process with $X_t\in L^2(\Omega ,{\mathcal{F}}_t,\mathbb{P})$ for each $t\ge 0$. $(X_t{)}_{t\ge 0}$ is called a martingale with respect to the filtration $({\mathcal{F}}_t{)}_{t\ge 0}$ if for any $s,t\ge 0$ with $s\le t$ $\mathbb{E}(X_t\mid {\mathcal{F}}_s)=X_s.$ The martingale is called continuous if it almost surely has continuous paths. That is, $\mathbb{P}(t\mapsto X_t\text{ continuous})=1.$

Stopping Time

A random variable $\tau :\Omega \to [0,\infty ]$ is called a stopping time with respect to the filtration $({\mathcal{F}}_t{)}_{t\ge 0}$ if $\{\tau \le t\}\in {\mathcal{F}}_t$ for any $t\ge 0$.

Uniformly Integrable

A collection $\mathcal{C}$ of random variables in $L^1(\Omega ,\Sigma ,\mathbb{P})$ is called uniformly integrable if ${\lim }_{K\to \infty }{\sup }_{Z\in \mathcal{C}}\mathbb{E}\left(|Z|1_{|Z|\ge K}\right)=0.$

Proposition

Let $(Z_n{)}_{n\in \mathbb{N}}$ be uniformly integrable and $Z_n\to Z$ almost surely as $n\to \infty$. Then $Z\in L^1(\Omega ,\Sigma ,\mathbb{P})$ and $\Vert Z_n-Z{\Vert }_{L^1}\to 0$ as $n\to \infty$.

Theorem

Let $({\mathcal{F}}_t{)}_{t\ge 0}$ be a filtration and $(M_t{)}_{t\ge 0}$ be a continuous martingale with respect to the filtration $({\mathcal{F}}_t{)}_{t\ge 0}$. Let $\tau$ be a stopping time with respect to $({\mathcal{F}}_t{)}_{t\ge 0}$. Then the stopped process $(M_{t\wedge \tau }{)}_{t\ge 0}$ is a continuous martingale with respect to $({\mathcal{F}}_t{)}_{t\ge 0}$.

Doob's Maximal and $L^p$ Inequalities

Let $T>0$ and $(M_t{)}_{t\ge 0}$ be a continuous submartingale such that $M_t\ge 0$ almost surely for all $t\in [0,T]$. Then for any $\alpha >0$, $\alpha \cdot \mathbb{P}\left({\sup }_{0\le t\le T}{M}_t>\alpha \right)\le \mathbb{E}(|M_T|)$ and for any $p\in (1,\infty )$, ${\Vert {\sup }_{0\le t\le T}{M}_t\Vert }_p\le \frac{p}{p-1}\Vert M_T{\Vert }_p.$

Martingale Convergence Theorem

Let $(M_t{)}_{t\ge 0}$ be a continuous martingale with respect to the filtration $({\mathcal{F}}_t{)}_{t\ge 0}$. If ${\sup }_{t\ge 0}\mathbb{E}\left(|M_t{|}^2\right)<\infty$ then there exists $M_{\infty }\in L^2(\Omega ,\Sigma ,\mathbb{P})$ such that $M_t$ converges to $M_{\infty }$ almost surely and in the $L^2$-norm.

Proposition

Let $A,B>0$. Define $\tau :\Omega \to [0,\infty ]$ by $\tau (\omega ):=\inf \{t\ge 0:B_t=-B\text{ or }{B}_t=A\}$ where we take $\tau (\omega )=\infty$ if these values are never reached. Then: (i) $\mathbb{P}(\tau <\infty )=1$. (ii) $\mathbb{P}(B_{\tau }=A)=\frac{B}{A+B}$. (iii) $\mathbb{E}(\tau )=A\cdot B<\infty$.