Discrete Euler Schemes For Killed Diffusions

Stochastic Differential Equations (SDEs) and their interface with Partial
Differential Equations (PDEs) is an interesting and broad research area with
applications in many scientific topics, varying from circuit analysis and finance
to fundamentals of quantum mechanics and modern cosmology. Specifically, in
the area of mathematical finance, an abundance of the fundamental results rely
on the assumption that the underlying dynamics of asset prices, currencies
exchange rates, interest rates, are continuous time stochastic processes
governed by SDEs. The present proposal is concerned with the area of
stochastic differential equations and its interface with partial differential
equations. In particular, we consider a partial differential equation associated
with the generator of a "killed" diffusion living in a bounded domain, i.e., a
diffusion which is stopped once it reaches the boundary of the domain. The
above correspondence is possible due to the well known Feynman-Kac
representation. The calculation of the solution and other relevant quantities,
poses a mathematical and computational problem especially in higher
dimensions. In fact, the usual methods such as the Walk on Spheres and the
standard Euler discretization scheme appear to be inefficient. To this end,
motivated by the recent works Cetin and Kardaras and Robertson, we propose
to study the development of an improved Euler scheme for killed diffusions
that will improve the computational efficiency and could potentially reduce the
problem's complexity. This new method would allow us to numerically
evaluate complex financial objects that would be otherwise intractable.
The present proposal was discussed with Professors U. Cetin and K. Kardaras.