Define a process by its stochastic differential equation:

Simulate the process:

Compute the mean function:

Compute the covariance function:

Define a vector process corresponding to iterated Stratonovich integrals , , , , , :

Compute its mean function:

And its covariance function:

The dynamics of a free particle under the effect of thermal fluctuation can be modeled by the Langevin equation of motion, , where is the standard WienerProcess and is the strength of the thermal noise. Here it is assumed that can only depend on and focus on the equation of velocity. There are two common ways to integrate the equation of motion: Ito formulation and Stratonovich formulation:

When is a constant, the two formulations are identical and lead to the same stationary distribution as :

If is velocity dependent, then due to the nature of the WienerProcess, has nonzero quadratic variation and the two formulations lead to different results. Convert Ito formulation to the equivalent Stratonovich formulation:

The drift under Stratonovich formulation is different from the drift under Ito formulation:

Create an OrnsteinUhlenbeckProcess and represent it with StratonovichProcess:

Obtain Kolmogorov forward equation:

Solve the equation numerically in with a localized initial condition at and Dirichlet boundary conditions:

Plot the solution of Kolmogorov forward equation at and compare it with the closed-form density function:

Visualize the dynamic of the solution with Animate:

Convert ItoProcess to StratonovichProcess:

Convert back:

StratonovichProcess does not support random initial conditions, so cannot be represented:

But it supports processes with fixed initial condition:

Initial time of the driven process needs to match with StratonovichProcess:

With matching initial time, this can be represented: