Fabien Panloup

Approximation of the distribution of a stationary Markov process with application to option pricing

We build a sequence of empirical measures on the space D(R_+,R^d) of R^d-valued cadlag functions on R_+ in order to approximate the law of a stationary R^d-valued Markov and Feller process (X_t). We obtain some general results of convergence of this sequence. Then, we apply them to Brownian diffusions and solutions to Levy driven SDEs under some Lyapunov-type stability assumptions. As a numerical application of this work, we show that this procedure gives an efficient way of option pricing in stochastic volatility models.

Approximation of stationary solutions of Gaussian driven Stochastic Differential Equations

We study sequences of empirical measures of Euler schemes associated to some non-Markovian SDEs: SDEs driven by Gaussian processes with stationary increments. We obtain the functional convergence of this sequence to a stationary solution to the SDE. Then, we end the paper by some specific properties of this stationary solution. We show that, in contrast to Markovian SDEs, its initial random value and the driving Gaussian process are always dependent. However, under an integral representation assumption, we also obtain that the past of the solution is independent of the future of the underlying innovation process of the Gaussian driving process.

Stochastic Heavy Ball

This paper deals with a natural stochastic optimization procedure derived from the so-called Heavy-ball method differential equation, which was introduced by Polyak in the 1960s with his seminal contribution [Pol64]. The Heavy-ball method is a second-order dynamics that was investigated to minimize convex functions f. The family of second-order methods recently received a large amount of attention, until the famous contribution of Nesterov [Nes83], leading to the explosion of large-scale optimization problems. This work provides an in-depth description of the stochastic heavy-ball method, which is an adaptation of the deterministic one when only unbiased evalutions of the gradient are available and used throughout the iterations of the algorithm. We first describe some almost sure convergence results in the case of general non-convex coercive functions f. We then examine the situation of convex and strongly convex potentials and derive some non-asymptotic results about the stochastic heavy-ball method. We end our study with limit theorems on several rescaled algorithms.

Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise

We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter

Invariant measure of duplicated diffusions and application to Richardson–Romberg extrapolation

H\in (1/3,1)

Ergodic approximation of the distribution of a stationary diffusion : rate of convergence

and multiplicative noise component

Regret bounds for Narendra-Shapiro bandit algorithms

\sigma

Estimation of the instantaneous volatility

. When

Recursive computation of the invariant measure of a stochastic differential equation driven by a Lévy process.

\sigma

LONG TIME BEHAVIOR OF MARKOV PROCESSES AND BEYOND

is constant and for every