Stochastic Differential Equation

${\dot{X}}_t=F(X_t)dt+G(X_t)d{W}_t$ is the most frequently seen definition of a Stochastic Differential Equation, where:

• $W_t$ is the Wiener Process
• $F,G:\mathbb{R}\to \mathbb{R}$ are real valued functions
• $X_t$ is a Semimartingale.

The above is syntactic sugar for the expression $X_t=X_0+{\int }_{t_0}^{t}F(X_s)ds+{\int }_{t_0}^{t}G(X_s)d{W}_s$

Info

ETH Zurich provides a useful overview of SDEs: https://ethz.ch/content/dam/ethz/special-interest/mavt/dynamic-systems-n-control/idsc-dam/Lectures/Stochastic-Systems/SDE.pdf

Table

Note	Equation
Heston Model	$\{\begin{array}{rl}d{S}_t& =S_t\sqrt{V_t}d{W}_t\\ d{V}_t& =\lambda \left(\theta -V_t\right)dt+\lambda \nu \sqrt{V_t}d{B}_t\\ ⟨d{W}_t,d{B}_t⟩& =\rho dt\end{array}$
Fokker-Planck Equation	$d{X}_t=\mu (X_t,t)dt+\sigma (X_t,t)d{W}_t$
Geometric Brownian Motion	$d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$
Ornstein-Uhlenbeck process	$d{X}_t=\theta (\mu -x_t)dt+\sigma d{W}_t$
[[Lindbladian\	Lindblad Master Equation]]

Quantum version

The above is often seen as $d|{\psi }_t⟩=[-\frac{i}{\hslash }Hdt+i\sqrt{\lambda }V(\hat{q})d{W}_t]|{\psi }_t⟩$

• Separate this section with a new note