Stochastic calculus

Suppose we have a sample space $\Omega$ with prob $P$ and a random variable $Z ~ N(0,1)$ .

Example 1.

Let $\gamma \in \mathbb{R}$ be a constant. We can find equivalent probability measure $Q$ under which $Z ~ N(\gamma, 1)$ with

\[\frac{dQ}{dP} = \frac{f_\gamma (Z)}{f_0(Z)} = \frac{\frac{1}{\sqrt{2\pi}} exp(-\frac{1}{2}(Z - \gamma)^2)}{\frac{1}{\sqrt{2\pi}} exp(-\frac{1}{2}Z^2)} = exp(Z \gamma - \gamma^2 /2) \]

Example 2

Suppose $Z_1, ..., Z_n$ are iid N(0,1) under P. Let $\gamma_1, ..., \gamma_n$ be constants. We can definie a new probability measure Q by:

\[ \frac{dP}{dQ} = \frac{ f_\gamma(Z_1, ..., Z_n)}{f_0(Z_1, ..., Z_n)} = exp(\sum \gamma_i Z_i - \frac{1}{2} \sum \gamma_i^2) \]

Under $Q$ , $Z_1, ..., Z_n$ , are indepdendent with distributions $Z_i ~ N(\gamma_i , 1)$

Example 3

Let $\Omega$ be a sample space with prob $P$ and a Brownian motion $W$ . Let $(\gamma_s)$ be a process with $\gamma_t \in \mathcal{F}_t$ and $E[ exp{(\frac{1}{2} \int^T_0 \gamma_t^2 dt )}] \lt \infinity$ . Define a new prob measure $Q$ by:

\[ \frac{dQ}{dP} = exp {( \int^T_0 \gamma_t dW_t - \frac{1}{2} \int^T_0 \gamma_t^2 dt )} \]

Under $Q$ the process $\tilde{W}_t = W_t - \int^T_0 \gamma^2_t dt$ is a Brownian motion. ie. $\tilde{W}_t$ is a Ito diffusion with $dW_t = \gamma_t dt + d \tilde{W}_t$ (drift $\gamma_t$ , speed1).

If $(X_t)$ is an Ito diffusion under $P$ then under $Q$ :

\[ dX_t = U_t d_t + V_t (\gamma + d\tilde{W}_t) dX_t = (U_t + \gamma_t V_t) dt + V_t d\tilde{W}_t) \]

In particular if $\gamma_t = -\frac{U_t}{V_t}$ then $(X_t)$ is a martingale