Stochastic analysis
 Calculate the solution of a (partial) stochastic differential equation numerically.
Unfortunately, Euler's method fails for many applied stochastic differential equations
and we need to find suitable algorithms instead
 Regularity of the semigroup of solutions of (partial) stochastic differential equations
 Perturbations of (partial) stochastic differential equations
Biomathematics

What is the influence of a randomly changing environment on the DNA sequence diversity?
Gillespie writes in the Preface of his 1991 book The Causes of Molecular
Evolution:
"If we are to propose that molecular evolution is due to
the action of natural selection, we need a mathematical
theory to demonstrate that the dynamics of selection
are compatible with the observations of molecular
variation. It is my conviction that the only viable
model of selection is one based on temporal and spatial
fluctuations in the environment. The mathematics of
selection in a random environment have never been
systematically developed or brought to a point where
they serve as a model of molecular evolution."
To answer this question, we study branching diffusions in random environment
and WrightFisher diffusions with random selection.

What is the footprint of selection and, in particular, of selective sweeps on sequence diversity?
There are different effects due to hard sweeps, soft sweeps and overlapping sweeps which are still
to be understood.

Can the positive effect of kin selection make up for a negatively selected sideeffect of a new mutation?