Table of Ito Integrals

This actually isn't a bad start anyway if anyone is looking for the same...

Stochastic Calculus Cheat Sheet

$\newcommand{\d}{\mathrm{d}}$Here is a short list of useful correspondences, especially for certain useful martingales such as $e^{B_t - \tfrac{1}{2}t}$:

\begin{array} {|r|r|} \hline X_t & \d X_t = u \d t + v \d B_t\\ \hline B_t & \d B_t\\ B_t^2 & 2 B_t \d B_t + \d t\\ B_t^2 - t & 2 B_t \d B_t\\ B_t^3 & 3 B_t^2 \d B_t + 3 B_t \d t\\ e^{B_t} & e^{B_t}\d B_t + \tfrac{1}{2}e^{B_t}\d t \\ e^{B_t - \tfrac{1}{2}t} & e^{B_t - \tfrac{1}{2}t} \d B_t\\ e^{\tfrac{1}{2}t}\sin B_t & e^{\tfrac{1}{2}t} \cos B_t \d B_t\\ e^{\tfrac{1}{2}t}\cos B_t & -e^{\tfrac{1}{2}t} \sin B_t \d B_t\\ (B_t + t) e^{-B_t - \tfrac{1}{2}t} & (1 - B_t - t) e^{-B_t - \tfrac{1}{2}t} \d B_t\\\hline \end{array}

All the above can be verified using Itô's formula and alongside the Itô isometry and integration by parts can be used to evaluate several stochastic integrals, their expectations and variances.

Here is a table of stochastic integrals...

\begin{array} {|r|r|r|} \hline \text{Stochastic Integral} & \text{Result} & \text{Variance}\\ \hline \int_0^t \d B_s & B_t & t \\ \int_0^t s \d B_s & tB_t - \int_0^t B_s \d s & \tfrac{1}{3}t^3 \\ \int_0^t B_s \d B_s & \tfrac{1}{2}B_t^2 - \tfrac{1}{2}t & \tfrac{1}{2}t^2\\ \int_0^t B_s^2 \d B_s & \tfrac{1}{3}B_t^3 - \int_0^t B_s \d s& 3t^2\\ \int_0^t e^{B_s - \tfrac{1}{2}s}\d B_s & e^{B_t - \tfrac{1}{2}t} - 1 & e^{t}-1\\ \hline \end{array}