25.4 Concluding thoughts

If you start with a known differential equation and want to add stochasticity to a parameter, here is a process:

• Replace whatever parameter with a “parameter + Noise” term (i.e $a \rightarrow a + \mbox{ Noise }$).
• Collect terms multiplied by Noise - they will form the stochastic part of the differential equation.
• The deterministic part of the differential equation should be your original differential equation.

The most general form of the stochastic differential equation is: $\displaystyle d\vec{y} = f(\vec{y},\vec{\alpha},t) \; dt + g(\vec{y},\vec{\alpha},t) \; dW(t)$, where $\vec{y}$ is the vector of state variables you want to solve for, and $\vec{\alpha}$ is your vector of parameters, and $dW(t)$ is the stochastic noise from the random walk.

At a given initial condition, the Euler-Maruyama method applies locally linear approximations to forecast the solution forward $\Delta t$ time units: $\displaystyle \vec{y}_{n+1} = y_{n} + f(\vec{y}_{n},\vec{\alpha},t_{n}) \cdot \Delta t + g(\vec{y}_{n},\vec{\alpha},t_{n}) \cdot \sigma \cdot \mbox{rnorm(N)} \cdot \sqrt{\Delta t}$, where rnorm(N) is $N$ dimensional random variable from a normal distribution with mean 0.