# Adaptive numerical methods for stochastic systems with non-globally Lipschitz coefficients

The department of Mathematics and Statistics at the University of Limerick invites you to a seminar by Dr Conall Kelly University College Cork.

Title: Adaptive numerical methods for stochastic systems with non-globally Lipschitz coefficients

Abstract: We present numerical schemes with adaptive timestepping for stochastic differential equation (SDE) models with non-Lipschitz coefficients. Such SDEs may be stiff, possibly via a linear operator in the drift and/or from the perturbation of nonlinear structures under discretisation. They arise, for example, in a financial setting from certain volatility models, and in a biological setting from models of neuronal activity.

We prove that a semi-implicit Euler-Maruyama scheme with an adaptive timestepping strategy of this kind is strongly convergent with order of convergence $1/2$ for equations with one-sided Lipschitz drift and globally Lipschitz diffusion, and with order of convergence $(1-\varepsilon)/2$ for equations with non-globally Lipschitz coefficients which together satisfy a Khasminskii-type monotone condition, where $\varepsilon\to 0$ as the number of available finite SDE moments increases without bound.

Numerically, we compare adaptive methods to several strongly convergent fixed-step methods, including a fully drift-implicit method, explicit taming-type methods and a truncated method. Our results indicate that the adaptive semi-implicit method is well suited as a general purpose solver.

This is joint work with Prof. Gabriel Lord, Heriot-Watt Universtiy, Edinburgh, UK.

This seminar will take place on Tuesday 1st May, at 2p.m. in Room A2-002

If you have any questions regarding this seminar, please direct them to Iain Moyles (061 233726, iain.moyles@ul.ie).

A full list of upcoming seminars can be found at http://www.ulsites.ul.ie/macsi/node/48011

Supported by Science Foundation Ireland funding, MACSI - the Mathematics Applications Consortium for Science and Industry (www.macsi.ul.ie), centred at the University of Limerick, is dedicated to the mathematical modelling and solution of problems which arise in science, engineering and industry in Ireland.