Oxford-Man Institute

We describe individual based continuous models of random evolutions and discuss some effects of competitions in these models. The range of applications includes models of spatial ecology, genetic mutation-selection models and particular socio-economic systems. The main aim of our presentation is to establish links between local characteristics of considered models and their macroscopic behaviour

In this talk we revisit the setting of Bouchard, Touzi, and Zeghal (2004).

For an incomplete market and a non-smooth utility function U defined on the whole real line we study the problem:

sup E [U(XTx,θ – B)]

θΘ(S)

Here B is a bounded contingent claim and Xx,θ represents the wealth process with initial capital x generated by portfolio θ. We study the case when the portfolios are constrained in a closed convex cone.

For the case without constraints and with a smooth utility function the solution method is to approximate the utility function and look at the same problem on a bounded negative domain. However, when one attempts to solve this bounded domain problem for a non-smooth utility function, the standard methods of proof cannot be applied. To circumvent this difficulty the idea of quadratic inf-convolution was introduced in Bouchard, Touzi, and Zeghal (2004). This method is mathematically appealing but leads to lengthy and technical proofs.

We will show that despite the presence of constraints, the dependence on quadratic inf-convolution can be removed. We will also show the existence of a constrained replicating portfolio for the optimal terminal wealth when the filtration is generated by a Brownian motion. This provides a natural generalisation of the results of Karatzas and Shreve (1998) to the whole real line.

The convergence of Markov processes to stationary distributions is a basic topic of introductory courses in stochastic processes, and the theory has been thoroughly developed. What happens when we add killing to the process? The process as such will not converge in distribution, but the survivors may; that is, the distribution of the process, conditioned on survival up to time t, converges to a "quasistationary distribution" as t goes to infinity.

This talk presents recent work with Steve Evans, proving an analogue of the transience-recurrence dichotomy for killed one-dimensional diffusions. Under fairly general conditions, a killed one-dimensional diffusion conditioned to have survived up to time t either escapes to infinity almost surely (meaning that the probability of finding it in any bounded set goes to 0) or it converges to the quasistationary distribution, whose density is given by the top eigenfunction of the adjoint generator.

These theorems arose in solving part of a longstanding problem in biological theories of ageing, and then turned out to play a key role in a very different problem in population biology, the effect of unequal damage inheritance on population growth rates.

The classical maximum principle for optimal control of solutions of stochastic differential equations (developed by Pontryagin (deterministic case), Bismut, Bensoussan, Haussmann and others), assumes that the system is Markovian and that the controller has access to full, updated information about the system at all times. The classical solution method involves an adjoint process defined as the solution of a backward stochastic differential equation, which is often difficult to solve.

We apply Malliavin calculus for Lévy processes to obtain a generalized maximum principle valid for non-Markovian systems and with (possibly) only partial information available for the controller. The backward stochastic differential equation is replaced by expressions involving the Malliavin derivatives of the quantities of the system.

The results are illustrated by some applications to finance

This talk gives a survey on a series of work which I and co-authors have been doing for 10 years. I will start from the Feynman-Kac type formula for Dirichlet forms. Then a necessary and sufficient condition is given to characterize the killing transform of Markov processes. Lastly we shall discuss the regular subspaces of linear transform and answer some problems related to the Feynman-Kac formula

Counting paths, or walks, is an important ingredient in the classical representation theory of compact groups. Using Brownian paths gives a new flexible and intuitive approach, which allows to extend some of this theory to the non- cristallographic case. This is joint work with P. Biane and N. O'Connell

The Bakry Emery criterion asserts that a probability measure with a strictly positive generalized curvature satisfies a logarithmic Sobolev inequality, and by results of Bakry and Ledoux an isoperimetric inequality of Gaussian type. These results were complemented by a theorem of Wang: if the curvature is bounded from below by a negative number, then under an additional Gaussian integrability assumption, the log-Sobolev inequality is still valid.

The goal of this joint work with A. Kolesnikov is to provide an extension of Wang's theorem to other integrability assumptions. Our results also encompass a theorem of Bobkov on log-concave measures on normed spaces and allows us to deal with non-convex potentials when the convexity defect is balanced by integrability conditions. The arguments rely on optimal transportation and its connection to the entropy functional

This is a joint work with Bruno Schapira, and it is a work in progress.

We study recurrence and transience properties of some edge reinforced random walks on the integers: the probability to go from $x$ to $x+1$ at time $n$ is equal to $f(\alpha_n^x)$ where $\alpha_n^x=\frac{1+\sum_{k=1}^n 1_{(X_{k-1},X_k)=(x,x+1)}}{2+\sum_{k=1}^n 1_{X_{k-1}=x}}$. Depending on the shape of $f$, we give some sufficient criteria for recurrence or transience of these walks