ItoProcess

ItoProcess[{a, b}, x, t] represents an Ito process x(t), where dx(t) = a(t,x(t))dt + b(t,x(t))·dw(t).

ItoProcess[{a, b, c}, x, t] represents an Ito process y(t) = c(t,x(t)), where dx(t) = a(t,x(t))dt + b(t,x(t))·dw(t).

ItoProcess[..., {x, x0}, {t, t0}] uses initial condition x(t0) = x0.

ItoProcess[..., ..., ..., Σ] uses a Wiener process w(t), with covariance Σ.

ItoProcess[proc] converts proc to a standard Ito process whenever possible.

ItoProcess[sdeqns, expr, x, t, w, dproc] represents an Ito process specified by a stochastic differential equation sdeqns, output expression expr, with state x and time t, driven by w following the process dproc.

Examples

Define a geometric Brownian motion:

proc = ItoProcess[{\[Mu] x[t], \[Sigma] x[t]}, x[t], {x, 1}, t]

Simulate the process:

RandomFunction[proc /. {\[Mu] -> 0.1, \[Sigma] -> 0.2}, {0, 1, 0.01}]

Please visit the official Wolfram Language Reference for more details.