A modified Euler-Maruyama method to simulate a one-dimensional sticky diffusion

A sticky diffusion is a process that can stick to and detach from a lower-dimensional boundary.

A challenge in simulating such a process is in capturing the change in dimension in a dynamically consistent way.

We introduce a numerical algorithm to simulate a one-dimensional sticky diffusion, which sticks to and detaches from a point.

Our method is a simple modification of the standard Euler-Maruyama scheme, which chooses with some probability between a reflected Euler-Maruyama update and a jump to the sticky point.

We show how to choose this probability to be consistent with the generator of the desired dynamics, and we prove that our scheme converges weakly to a sticky diffusion with order 1.