Sylvain Rubenthaler

Stability and Uniform Particle Approximation of Nonlinear Filters in Case of Non Ergodic Signals

Abstract In this paper, we propose a new approach to study the stability of the optimal filter w.r.t. its initial condition in introducing a “robust filter, ” which approximates the optimal filter uniformly in time. This approach allows us to prove, in some cases, when the signal is nonergodic, the stability of the optimal filter in mean over the observations and the uniform convergence in mean over the observations of a special interacting particle filter to the optimal filter.

Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process

The Euler scheme is a well-known method of approximation of solutions of stochastic differential equations (SDEs). A lot of results are now available concerning the precision of this approximation in case of equations driven by a drift and a Brownian motion. More recently, people got interested in the approximation of solutions of SDEs driven by a general Levy process. One of the problem when we use Levy processes is that we cannot simulate them in general and so we cannot apply the Euler scheme. We propose here a new method of approximation based on the cutoff of the small jumps of the Levy process involved. In order to find the speed of convergence of our approximation, we will use results about stability of the solutions of SDEs.

Global solvability of a networked integrate-and-fire model of McKean-Vlasov type

We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality is of great importance as the resulting system is known to blow-up as this becomes large. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when the coefficient of proportionality is small enough.

Tree based functional expansions for Feynman-Kac particle models.

We design a theoretic tree-based functional representation of a class of Feynman-Kac particle distributions, including an extension of the Wick product formula to interacting particle systems. These weak expansions rely on an original combinatorial, and permutation group analysis of a special class of forests. They provide refined non asymptotic propagation of chaos type properties, as well as sharp Lp-mean error bounds, and laws of large numbers for U-statistics. Applications to particle interpretations of the top eigenvalues, and the ground states of Schrodinger semigroups are also discussed.

Path storage in the particle filter

This article considers the problem of storing the paths generated by a particle filter and more generally by a sequential Monte Carlo algorithm. It provides a theoretical result bounding the expected memory cost by T+CNlogN where T is the time horizon, N is the number of particles and C is a constant, as well as an efficient algorithm to realise this. The theoretical result and the algorithm are illustrated with numerical experiments.

Monte Carlo Approximations of American Options that Preserve Monotonicity and Convexity

It can be shown that when the payoff function is convex and decreasing (respectively increasing) with respect to the underlying (multidimensional) assets, then the same is true for the value of the associated American option, provided some conditions are satisfied. In such a case, all Monte Carlo methods proposed so far in the literature do not preserve the convexity or monotonicity properties. In this paper, we propose a method of approximation for American options which can preserve both convexity and monotonicity. The resulting values can then be used to define exercise times and can also be used in combination with primal-dual methods to get sharper bounds. Other application of the algorithm include finding optimal hedging strategies.

Optimal hedging in discrete time

Building on the work of Schweizer (1995) and Cern and Kallseny (2007), we present discrete time formulas minimizing the mean square hedging error for multidimensional assets. In particular, we give explicit formulas when a regime-switching random walk or a GARCH-type process is utilized to model the returns. Monte Carlo simulations are used to compare the optimal and delta hedging methods.

Improved convergence rate for the simulation of stochastic differential equations driven by subordinated Lévy processes

We consider the Euler approximation of stochastic differential equations (SDEs) driven by Levy processes in the case where we cannot simulate the increments of the driving process exactly. In some cases, where the driving process Y is a subordinated stable process, i.e., Y=Z(V) with V a subordinator and Z a stable process, we propose an approximation Y by Z(Vn) where Vn is an approximation of V. We then compute the rate of convergence for the approximation of the solution X of an SDE driven by Y using results about the stability of SDEs.

The convergence to equilibrium of neutral genetic models

This article is concerned with the long-time behavior of neutral genetic population models with fixed population size. We design an explicit, finite, exact, genealogical tree based representation of stationary populations that holds both for finite and infinite types (or alleles) models. We analyze the decays to the equilibrium of finite populations in terms of the convergence to stationarity of their first common ancestor. We estimate the Lyapunov exponent of the distribution flows with respect to the total variation norm. We give bounds on these exponents only depending on the stability with respect to mutation of a single individual; they are inversely proportional to the population size parameter.

Dispersion and Collapse in Stochastic Velocity Fields on a Cylinder

The dynamics of fluid particles on cylindrical manifolds is investigated. The velocity field is obtained by generalizing the isotropic Kraichnan ensemble, and is therefore Gaussian and decorrelated in time. The degree of compressibility is such that when the radius of the cylinder tends to infinity the fluid particles separate in an explosive way. Nevertheless, when the radius is finite the transition probability of the two-particle separation converges to an invariant measure. This behavior is due to the large-scale compressibility generated by the compactification of one dimension of the space.