Backward stochastic differential equations associated to jump Markov processes and applications

In this paper we study backward stochastic differential equations (BSDEs) driven by the compensated random measure associated to a given pure jump Markov process X$X$ on a general state space K$K$. We apply these results to prove well-posedness of a class of nonlinear parabolic differential equations on K$K$, that generalize the Kolmogorov equation of X$X$. Finally we formulate and solve optimal control problems for Markov jump processes, relating the value function and the optimal control law to an appropriate BSDE that also allows to construct probabilistically the unique solution to the Hamilton–Jacobi–Bellman equation and to identify it with the value function.     

On the existence of smooth densities for jump processes

Summary We consider a Lévy processX _t and the solutionY _t of a stochastic differential equation driven byX _t; we suppose thatX _t has infinitely many small jumps, but its Lévy measure may be very singular (for instance it may have a countable support). We obtain sufficient conditions ensuring the existence of a smooth density forY _t: these conditions are similar to those of the classical Malliavin calculus for continuous diffusions. More generally, we study the smoothness of the law of variablesF defined on a Poisson probability space; the basic tool is a duality formula from which we estimate the characteristic function ofF.     

Coupling and strong Feller for jump processes on Banach spaces

By using lower bound conditions of the Lévy measure w.r.t. a nice reference measure, the coupling and strong Feller properties are investigated for the Markov semigroup associated with a class of linear SDEs driven by (non-cylindrical) Lévy processes on a Banach space. Unlike in the finite-dimensional case where these properties have also been confirmed for Lévy processes without drift, in the infinite-dimensional setting the appearance of a drift term is essential to ensure the quasi-invariance of the process by shifting the initial data. Gradient estimates and exponential convergence are also investigated. The main results are illustrated by specific models on the Wiener space and separable Hilbert spaces.     

Martingale solutions and Markov selection of stochastic 3D Navier-Stokes equations with jump

In this paper, we study the existence of martingale solutions of stochastic 3D Navier-Stokes equations with jump, and following Flandoli and Romito (2008) [7] and Goldys et al. (2009) [8], we prove the existence of Markov selections for the martingale solutions.     

An inverse first-passage problem for one-dimensional diffusions with random starting point

We consider an inverse first-passage time (FPT) problem for a homogeneous one-dimensional diffusion X(t), starting from a random position η. Let S(t) be an assigned boundary, such that P(η≥S(0))=1, and F an assigned distribution function. The problem consists of finding the distribution of η such that the FPT of X(t) below S(t) has distribution F. We obtain some generalizations of the results of Jackson et al., 2009, which refer to the case when X(t) is Brownian motion and S(t) is a straight line across the origin.     

Winding numbers for 2-dimensional,positive recurrent diffusions

A well-known theorem by Spitzer states that the winding number of a standard Brownian motion around the origin is asymptotically Cauchy-distributed. A similar result is derived for positive recurrent diffusions in the plane given by a non-degenerate stochastic equation.     

Gradient estimate for Ornstein-Uhlenbeck jump processes

By using absolutely continuous lower bounds of the Lévy measure, explicit gradient estimates are derived for the semigroup of the corresponding Lévy process with a linear drift. A derivative formula is presented for the conditional distribution of the process at time t under the condition that the process jumps before t. Finally, by using bounded perturbations of the Lévy measure, the resulting gradient estimates are extended to linear SDEs driven by Lévy-type processes.     

On solutions to backward stochastic partial differential equations for Lévy processes

In this paper, we prove the existence and uniqueness of the solution for a class of backward stochastic partial differential equations (BSPDEs, for short) driven by the Teugels martingales associated with a Lévy process satisfying some moment conditions and by an independent Brownian motion. An example is given to illustrate the theory.     

On the small-time behaviour of Lévy-type processes

We show some Chung-type lim inf$lim\; inf$ law of the iterated logarithm results at zero for a class of (pure-jump) Feller or Lévy-type processes. This class includes all Lévy processes. The norming function is given in terms of the symbol of the infinitesimal generator of the process. In the Lévy case, the symbol coincides with the characteristic exponent.     

Constructions of coupling processes for Lévy processes

We construct optimal Markov couplings of Lévy processes, whose Lévy (jump) measure has an absolutely continuous component. The construction is based on properties of subordinate Brownian motions and the coupling of Brownian motions by reflection.     

Invariant measures for stochastic evolution equations of pure jump type

In this paper, we obtain a characterization of invariant measures of stochastic evolution equations and stochastic partial differential equations of pure jump type. As an application, it is shown that the equation has a unique invariant probability measure under some reasonable conditions.     

On the Gerber-Shiu function and change of measure

We consider several models for the surplus of an insurance company mainly under some light-tail assumptions. We are interested in the expected discounted penalty at ruin. By a change of measure we remove the discounting, which simplifies the expression. This leads to (defective) renewal equations as they had been found by different methods in the literature. If we use the change of measure such that ruin becomes certain, the renewal equations simplify to ordinary renewal equations. This helps to discuss the asymptotics as the initial capital goes to infinity. For phase-type claim sizes, explicit formulae can be derived.     

Measure-valued branching diffusions with spatial interactions

Summary A strong equation driven by a historical Brownian motion is used to construct and characterize measure-valued branching diffusions in which the spatial motions obey an Itô equation with drift and diffusion depending on the position of an individual and the entire population.     

The cone of positive harmonic functions for scale-invariant diffusions

Consider a scale invariant diffusion whose state space is a closed cone in R ^d , minus the vertex. Then the process is either recurrent, transient to ∞ or transient to the vertex of the cone. In the latter case, the diffusion has finite lifetime (a.s.) and converges to the vertex at the lifetime. The Martin boundary consists of two points, and the corresponding minimal harmonic functions are of the form 1 and |x| ^α ψ(x/|x|).