Past Stochastic Analysis Seminar

TBA

/samath/seminars/njacob_abstract.pdf

Using the dyadic parametrization of curves, and elementary theorems and

probability theory, examples are constructed of domains having bad properties on

boundary sets of large Hausdorff dimension (joint work with F.D. Lesley).

Lambda-coalescents were introduced by Pitman in (1999) and Sagitov (1999). These processes describe the evolution of particles that

undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure Lambda has the Beta$(2-\alpha,\alpha)$ they are also known to describe the genealogies of large populations where a single individual can produce a large number of offsprings. Here we use a recent result of Birkner et al. (2005) to prove that Beta-coalescents can be embedded in continuous stable random trees, for which much is known due to recent progress of Duquesne and Le Gall. This produces a number of results concerning the small-time behaviour of Beta-coalescents. Most notably, we recover an almost sure limit theorem for the number of blocks at small times, and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the infinite site frequency spectrum associated with mutations in the context of population genetics.

Stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients

possessing unique solutions make up a very important class in applications. For

instance, Langevin-type equations and gradient systems with noise belong to this

class. At the same time, most numerical methods for SDEs are derived under the

global Lipschitz condition. If this condition is violated, the behaviour of many

standard numerical methods in the whole space can lead to incorrect conclusions.

This situation is very alarming since we are forced to refuse many effective

methods and/or to resort to some comparatively complicated numerical procedures.

We propose a new concept which allows us to apply any numerical method of weak

approximation to a very broad class of SDEs with nonglobally Lipschitz

coefficients. Following this concept, we discard the approximate trajectories

which leave a sufficiently large sphere. We prove that accuracy of any method of

weak order p is estimated by $\varepsilon+O(h^{p})$, where $\varepsilon$ can be

made arbitrarily small with increasing the radius of the sphere. The results

obtained are supported by numerical experiments. The concept of rejecting

exploding trajectories is applied to computing averages with respect to the

invariant law for Langevin-type equations. This approach to computing ergodic

limits does not require from numerical methods to be ergodic and even convergent

in the nonglobal Lipschitz case. The talk is based on joint papers with G.N.

Milstein.

A classical result due to Kunita says that if the coefficients are global

Lipschitzian, then the s.d.e defines a global flow of homeomorphisms. In this

talk, we shall prove that under suitable growth on Lipschitz constants, the sde

define still a global flow.