Adaptive numerical methods for stochastic differential equation models with non-Lipschitz coefficients

We present adaptive timestepping strategies for stochastic differential equation models with non-Lipschitz coefficients: consider for example the superlinear diffusion coefficient of the 3/2 stochastic volatility model. These... [ view full abstract ]

We present adaptive timestepping strategies for stochastic differential equation models with non-Lipschitz coefficients: consider for example the superlinear diffusion coefficient of the 3/2 stochastic volatility model. These strategies manage highly nonlinear coefficient response in order to control potential runaway growth in numerical solutions. We demonstrate strong convergence of the explicit and semi-implicit Euler-Maruyama method for equations with one-sided drift and globally Lipschitz diffusion and for equations with monotone coefficients respectively. Such an approach can improve multi-level Monte Carlo simulation. A strategy that preserves almost sure stability / instability and positivity for equations with positive, locally Lipschitz coefficients is also presented.