Numerical solution of stochastic DEs with the exponential method

Grant Lythe, Kalvis Jansons

Department of Applied Mathematics, University of Leeds, LS2 9JT, UK

grant@maths.leeds.ac.uk

http://www.maths.leeds.ac.uk/~grant

Contributed talk

We present new numerical methods for scalar stochastic differential equations. Successive time increments are independent random variables with an exponential distribution.

One of the most difficult tasks for existing numerical methods is the measurement of exit times, where the quantity of interest is the first time that a process attains a given value or exits a region. Even if updates of the process are generated with good accuracy, large errors can result from the possibility that the boundary is reached during a timestep although the process is within the boundary at both the beginning and the end of the timestep. At the end of each exponential timestep, we perform a simple a posteriori test for this possibility, based on the value of the process at the start and end of the timestep. The required conditional probabilities have a simple form under exponential timestepping.

• The probability that the process attains a value during a timestep given that its values at the beginning and end of the timestep are and , is a function only of and of the closer of and to .
• The distribution of the value taken by the process at the end of the timestep, given that the point is attained during the timestep, is the same as if the timestep had started at .