Mini-workshop on Numerics for SDEs

Schedule:

14:35–15:20 Conall Kelly, University College Cork: Adaptive numerical solution of stochastic differential equations with Markovian switching (abstract below)

15:25–15:45 Fika

15:45–16:30 Gabriel Lord, Radboud University: A numerical method for an SDE with a WIS integral (abstract below)

16:35–17:20 Andreas Neuenkirch, University of Mannheim: Strong approximation of the CIR process (abstract below)

Contact David Cohen if you want password to join via zoom.

Adaptive numerical solution of stochastic differential equations with Markovian switching

Abstract: We demonstrate the use of adaptive discretisation strategies for the numerical solution of stochastic differential equations (SDEs) with coefficients that change at random times according to the evolution of a Markov chain.

We will show how an approach to adaptive mesh construction originally designed for SDEs with jump perturbations may be repurposed for SDEs with Markovian Switching. Implementation will be illustrated via a model of telomere shortening in jackdaws.

This is joint work with Kate O'Donovan (UCC), building upon a prior collaboration with Gabriel Lord (Radboud University) and Fandi Sun (Heriot-Watt University).

A numerical method for an SDE with a WIS integral

Abstract: This talk examines a quasilinear SDE forced by multiplicative fractional Brownian motion. I will introduce the notion of a Wick product and the WIS integral. This integral can be interpreted for all values of Hurst parameter H. We examine how a numerical metthod can be obtained and examine strong convergence. Our numerical results suggest that the theoretical result could be improved in particular for H<1/2.

Strong approximation of the CIR process

Abstract: The CIR process is the prototype stochastic differential equation (SDE) for the class of square root diffusions. These equations have widespread applications, in particular in finance, biology and chemistry. Moreover, since the diffusion coefficient contains a square root and is not Lipschitz continuous, the CIR process is also the prototype example for an SDE whose coefficients do not satisfy the so-called standard assumptions for numerical analysis.

Due to these reasons, the approximation of the CIR process has attracted a lot of attention in the last 20 years. In this talk, I will give a state-of-the-art summary and will present some of the latest developments for the strong approximation of the CIR process.