### A Brownian population model.

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The biological theory of adaptive dynamics proposes a description of the long-time evolution of an asexual population, based on the assumptions of large population, rare mutations and small mutation steps. Under these assumptions, the evolution of a quantitative dominant trait in an isolated population is described by a deterministic differential equation called 'canonical equation of adaptive dynamics'. In this work, in order to include the effect of genetic drift in this model, we consider instead...

We present a simplified approach to the analytical approximation of the transition density related to a general local volatility model. The methodology is sufficiently flexible to be extended to time-dependent coefficients, multi-dimensional stochastic volatility models, degenerate parabolic PDEs related to Asian options and also to include jumps.

We prove existence and uniqueness for two classes of martingale problems involving a nonlinear but bounded drift coefficient. In the first class, this coefficient depends on the time t, the position x and the marginal of the solution at time t. In the second, it depends on t, x and p(t,x), the density of the time marginal w.r.t. Lebesgue measure. As far as the dependence on t and x is concerned, no continuity assumption is made. The results, first proved for the identity diffusion matrix,...

Understanding the evolution of individuals which live in a structured and fluctuating environment is of central importance in mathematical population genetics. Here we outline some of the mathematical challenges arising from modelling structured populations, primarily focussing on the interplay between forwards in time models for the evolution of the population and backwards in time models for the genealogical trees relating individuals in a sample from that population. In addition to classical...

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process $\left(K\right(t),i(t),Y(t\left)\right)$ on $({\mathbb{T}}^{2}\times \{1,2\}\times {\mathbb{R}}^{2})$, where ${\mathbb{T}}^{2}$ is the two-dimensional torus. Here $\left(K\right(t),i(t\left)\right)$ is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. $Y\left(t\right)$ is an additive functional of $K$, defined as ${\int}_{0}^{t}v\left(K\left(s\right)\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}s$, where $\left|v\right|\sim 1$ for small $k$. We prove that the rescaled process ${(NlnN)}^{-1/2}Y\left(Nt\right)$ converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately...