Two models for making population forecasts are presented and discussed. One of them is based on an analytical Gaussian solution of the Kolmogorov-Fokker-Planck equation and regression model for risk factor dynamics. The second is based on more general diffusion type stochastic differential equations. Numerical schemes for parameter estimation from longitudinal data and for forecasting are discu...

The seminar gives an introduction to turbulent motions of incompressible fluids. Due to the nature of the subject, most of the explanations are qualitative, although references to the actual fundamental equations underlying the processes are also made. Special emphasis is made on the dynamics of vorticity in turbulent flows and the main results of the celebrated Kolmogorov 1941 theory.

Stochastic differential equations have a built-in direction of time flow since future increments in the driving process are assumed independent of present and past values of the process defined by the solution of the equation. The differential equations are thought of as evolving forward in time, normally from some fixed initial time, and the integral representation of a solution, involving as ...

In this paper we study ergodic backward stochastic differential equations (EBSDEs) dropping the strong dissipativity assumption needed in [12]. In other words we do not need to require the uniform exponential decay of the difference of two solutions of the underlying forward equation, which, on the contrary, is assumed to be non degenerate. We show existence of solutions by use of coupling esti...

Extended systems governed by partial differential equations can, under suitable conditions, be approximated by means of sets of ordinary differential equations for global quantities capturing the essential features of the systems dynamics. Here we obtain a small number of effective equations describing the dynamics of single-front and localized solutions of Fisher-Kolmogorov type equations. The...

We study regularizing properties for transition semigroups related to Ornstein Uhlenbeck processes with values in a Banach space E which is continuously and densely embedded in a real and separable Hilbert space H . Namely we study conditions under which the transition semigroup maps continuous and bounded functions into differentiable functions. Via a Girsanov type theorem such properties exte...

Recently, research that focuses on the rigorous understanding of the relation between simulation and/or exact models on graphs and approximate counterparts has gained lots of momentum. This includes revisiting the performance of classic pairwise models with closures at the level of pairs and/or triples as well as effective-degree-type models and those based on the probability generating functio...

In this paper we propose a new homogeneous stochastic Gompertz diffusion model with a threshold parameter. This can be considered an extension of the homogeneous three parameter Gompertz process with the addition of a fourth parameter. From the corresponding Kolmogorov equations and Ito’s stochastic differential equations, we obtain the transition probability density function and the moments of...

My research interests are in stochastic processes and mathematical finance. In particular, my dissertation provides a weak existence result which has application to the financial engineer’s calibration problem and essentially generalizes an earlier result of Gyöngy [9]. In this research statement I will briefly review the calibration problem, show how my work fits into the literature, and discu...

We extend many of the classical results for standard one-dimensional diffusions to a diffusion process with memory of the form dXt = σ(Xt, Xt)dWt, where Xt = m ∧ inf0≤s≤t Xs. In particular, we compute the expected time for X to leave an interval, classify the boundary behavior at zero and we derive a new occupation time formula for X. We also show that (Xt,Xt) admits a joint density, which can ...