Euler–Maruyama method

[1] Consider the stochastic differential equation (see Itô calculus) with initial condition X0 = x0, where Wt denotes the Wiener process, and suppose that we wish to solve this SDE on some interval of time [0, T].

Then the Euler–Maruyama approximation to the true solution X is the Markov chain Y defined as follows: The random variables ΔWn are independent and identically distributed normal random variables with expected value zero and variance Δt.

As many biological processes are both stochastic and continuous in nature, numerical methods of solving SDEs are highly valuable in the field.

The graphic depicts a stochastic differential equation solved using the Euler-Maruyama method.

The following Python code implements the Euler–Maruyama method and uses it to solve the Ornstein–Uhlenbeck process defined by The random numbers for