Solution stability and phase transition for two SDEs by a fixed time step integration scheme

The fixed time step integration method proposed in Grigoriu (2009) [2] is used to construct recurrence formulas for generating samples of processes X(t)X(t) satisfying stochastic differential equations (SDEs) with Gaussian (GWN) and Poisson white noise (PWN). Theoretical arguments and numerical examples are employed to show that the sequence of processes Xn(t)Xn(t) defined by these recurrence formulas can be used to assess the stability of the trivial solution of SDEs with linear drift and diffusion coefficients driven by GWN and/or PWN and capture the phase transition phenomenon exhibited by the state of a randomized Verhulst model for population growth.

► Stochastic differential equations are solved with a fixed time step integration method.
► Gaussian and Poisson white noise input processes are considered.
► Weak convergence of Monte Carlo solutions.
► Stability of randomized Verhulst model for population growth.