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

Quantum version

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

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