The topic of this thesis is a diffusion process on a potential landscape which is given by a smooth Hamiltonian function in the regime of small noise. The work provides a new proof of the Eyring-Kramers formula for the Poincaré inequality of the associated generator of the diffusion. The Poincaré inequality characterizes the spectral gap of the generator and establishes the exponential rate of ...

In this paper, we introduce a new class of processes which are diffusions with jumps driven by multivariate nonlinear Hawkes process. Our goal is to study their long-time behavior. the case exponential memory kernels for underlying process establish conditions positive Harris recurrence couple $(X,Y )$, where $X$ denotes diffusion and $Y$ piecewise deterministic Markov (PDMP) defining stochasti...

We consider a stochastic volatility model with jumps where the underlying asset price is driven by a process sum of a 2-dimensional Brownian motion and 2-dimensional compensated Poisson process. The market is incomplete, there is an infinity of Equivalent Martingale Measures (E.M.M) and an infinity of hedging strategies. We characterize the set of E.M.M, and we hedge by minimizing the variance ...

The intent of these essays is to study the minimal entropy martingale measure, to examine some new martingale representation theorems and to discuss its related Kunita-Watanabe decompositions. Such problems arise in mathematical finance for an investor who is confronted with the issues of pricing and hedging in incomplete markets. We adopt the standpoint of a ra­ tional investor who principally...

X iv :m at h/ 04 09 13 9v 1 [ m at h. FA ] 8 S ep 2 00 4 A WEAK TYPE INEQUALITY FOR NON-COMMUTATIVE MARTINGALES AND APPLICATIONS NARCISSE RANDRIANANTOANINA Abstract. We prove a weak-type (1,1) inequality for square functions of noncommutative martingales that are simultaneously bounded in L and L. More precisely, the following non-commutative analogue of a classical result of Burkholder holds: ...

Abstract -In this paper, the problem of H∞ filtering design is studied for nonlinear sampled-data systems using the Takagi-Sugeno (T-S) fuzzy model approach. Traditionally, the sufficient conditions for the existence of such H∞ filter are characterized in terms of the solution of a differential Hamilton-Jacobi inequality with jumps, which is equivalent to solving the partial differential inequa...