Oxford-Man Institute

The Cameron-Martin theorem is a fundamental result in stochastic analysis. We will show that the Wiener measure on a geometrically and stochastically complete Riemannian manifold is quasi-invariant. This is a complete a complete generalization of the classical Cameron-Martin theorem for Euclidean space to Riemannian manifolds. We do not impose any curvature growth conditions.

We construct a kinetic SDE in the state variables (position,velocity), where the spatial dependency in the drift term of the velocity equation is a conditional expectation with respect to the position. Those systems are introduced in fluid mechanic by S. B. Pope and are used in the simulation of complex turbulent flows. Such simulation approach is known as Probability Density Function (PDF) method .

We construct a PDF method applied to a dynamical downscaling problem to generate fine scale wind : we consider a bounded domain D. A weather prediction model solves the wind field at the boundary of D (coarse resolution). In D, we adapt a Lagrangian model to the atmospheric flow description and we construct a particles algorithm to solve it (fine resolution).

In the second part of the talk, we give a (partial) construction of a Lagrangian SDE confined in a given domain and such that the corresponding Eulerian velocity at the boundary is given. This problem is related to stochastic impact problem and existence of trace at the boundary for the McKean-Vlasov equations with specular boundary condition

The McKean stochastic game (MSG) is a two-player version of the perpetual American put option. The MSG consists of two agents and a certain payoff function of an underlying stochastic process. One agent (the seller) is looking for a strategy (stopping time) which minimises the expected pay-off, while the other agent (the buyer) tries to maximise this quantity.

For Brownian motion one can find the value of the MSG and the optimal stopping times by solving a free boundary value problem. For a Lévy process with jumps the corresponding free boundary problem is more difficult to solve directly and instead we use fluctuation theory to find the solution of the MSG driven by a Lévy process with no positive jumps. One interesting aspect is that the optimal stopping region for the minimiser "thickens" from a point to an interval in the presence of jumps. This talk is based on joint work with Andreas Kyprianou (University of Bath).

In the first part of the talk, we fit the Hua-Pickrell measure (which is a two parameters deformation of the Haar measure) on the unitary group and the Ewens measure on the symmetric group in a same framework. We shall see that in the unitary case, the eigenvalues follow a determinantal point process with explicit hypergeometric kernels. We also study asymptotics of these kernels. The techniques used rely upon splitting of the Haar measure and sampling techniques. In the second part of the talk, we provide a matrix model for the circular Jacobi ensemble, which is the sampling used for the Hua-Pickrell measure but this time on Dyson's circular ensembles. In this case, we use the theory of orthogonal polynomials on the unit circle. In particular we prove that when the parameter of the sampling grows with n, both the spectral measure and the empirical spectral measure converge weakly in probability to a non-trivial measure supported only by one piece of the unit circle.

We consider a critical, measure-valued branching diffusion ξ in Rd, where the branching is continuous and the spatial motion is given by the heat flow. For d ≥ 2 and fixed t > 0, ξt is known to be an a.s. singular random measure of Hasudorff dimension 2. We explain how it can be approximated by Lebesgue measure on ε-neighbourhoods of the support. Next we show how ξt can be approximated in total variation near n points, and how the associated Palm distributions arise in the limit from elementary conditioning. Finally we hope to explan the duality between moment and Palm measures, and to show how the latter can be described in terms of discrete “Palm trees.”

Euler equations for incompressible fludis describe the evolution of the divergence-free velocity of a non-viscous fluid (when viscosity is present, we have the well-known Navier-Stokes equations). V. Arnold discovered that they correspond to geodesic equations in the space of volume-preserving diffeomorphisms but several exemples show that it is not always possible to solve the corresponding variational problems inducing minimal energy displacements. A solvable relaxed version, in a non-deterministic setting (measures on the path space, with possible splitting of the particles), has been introduced by Y. Brenier who intensively studied the problem. Together with M. Bernot and A. Figalli we founded new solutions and characterization results. In the talk I'll present the most interesting features of the problem and of its solutions.

Mike Giles recently came up with a very general technique that improves the fundamental complexity of Monte Carlo simulation in the context where stochastic differential equations are simulated numerically. I will discuss some work with Mike Giles and Xuerong Mao that extends the theoretical support for this approach to the case of financial options without globally Lipschitz payoff functions. I will also suggest other application areas where this multi-level approach might prove valuable, including stochastic computation in cell biology.

Probabilistic methods for the solution of Backward Stochastic Differential Equations (BSDE) provide us with a new approach to the problem of approximating the solution of a semi-linear PDE. Utilizing on the Markovian nature of these BSDE’s we show how one may consider the problem of numerical solutions to BSDEs within the area of weak approximations of diffusions. To emphasize this point, we suggest an algorithm based on the Cubature method on Wiener space of Lyons - Victoir. Instead of using standard discretization techniques of BSDE’s, we choose to work with the actual flow. This allows to take advantage of estimates on the derivatives of the solution of the associated semi-linear PDE and hence, we recover satisfactory convergence estimates.

An important and challenging problem in mathematical finance is how to choose a pricing measure in an incomplete market, i.e. how to find a probability measure under which expected payoffs are calculated and fair option prices are derived under some notion of optimality.

The notion of q-optimality is linked to the unique equivalent martingale measure (EMM) with minimal q-moment (if q > 1) or minimal relative entropy (if q=1). Hobson's (2004) approach to identifying the q-optimal measure (through a so-called fundamental equation) suggests a relaxation of an essential condition appearing in Delbaen & Schachermayer (1996). This condition states that for the case q=2, the Radon-Nikodym process, whose last element is the density of the candidate measure, is a uniformly integrable martingale with respect to any EMM with a bounded second moment. Hobson (2004) alleges that it suffices to show that the above is true only with respect to the candidate measure itself and extrapolates for the case q>1. Cerny & Kallsen (2008) however presented a counterexample (for q=2) which demonstrates that the above relaxation does not hold in general.

The speaker will present the general form of the q-optimal measure following the approach of Delbaen & Schachermayer (1994) and prove its existence under mild conditions. Moreover, in the light of the counterexample in Cerny & Kallsen (2008) concerning Hobson's (2004) approach, necessary and sufficient conditions will be presented in order to determine when a candidate measure is the q-optimal measure.