A 2-dimensional SDE whose solutions are not unique.
A characterization of Markov solutions for stochastic differential equations with jumps
A comparison theorem for semimartingales and its applications
A converse comparison theorem for BSDEs and related properties of -expectation.
A deterministic discretisation-step upper bound for state estimation via Clark transformations.
A general stochastic maximum principle for singular control problems.
A general stochastic target problem with jump diffusion and an application to a hedging problem for large investors.
A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations
A generalized mean-reverting equation and applications
Consider a mean-reverting equation, generalized in the sense it is driven by a 1-dimensional centered Gaussian process with Hölder continuous paths on [0,T] (T> 0). Taking that equation in rough paths sense only gives local existence of the solution because the non-explosion condition is not satisfied in general. Under natural assumptions, by using specific methods, we show the global existence and uniqueness of the solution, its integrability, the continuity and differentiability of the...
A horizontal Lévy process on the bundle of orthonormal frames over a complete riemannian manifold
A linear numerical scheme for nonlinear BSDEs with uniformly continuous coefficients.
A Malliavin calculus method to study densities of additive functionals of SDE’s with irregular drifts
We present a general method which allows to use Malliavin Calculus for additive functionals of stochastic equations with irregular drift. This method uses the Girsanov theorem combined with Itô–Taylor expansion in order to obtain regularity properties for this density. We apply the methodology to the case of the Lebesgue integral of a diffusion with bounded and measurable drift.
A martingale on the zero-set of a holomorphic function.
A Milstein-type scheme without Lévy area terms for SDEs driven by fractional brownian motion
In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second-order Taylor expansion, where the usual Lévy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds...
A new proof of Kellerer’s theorem
In this paper, we present a new proof of the celebrated theorem of Kellerer, stating that every integrable process, which increases in the convex order, has the same one-dimensional marginals as a martingale. Our proof proceeds by approximations, and calls upon martingales constructed as solutions of stochastic differential equations. It relies on a uniqueness result, due to Pierre, for a Fokker-Planck equation.
A new proof of Kellerer’s theorem
In this paper, we present a new proof of the celebrated theorem of Kellerer, stating that every integrable process, which increases in the convex order, has the same one-dimensional marginals as a martingale. Our proof proceeds by approximations, and calls upon martingales constructed as solutions of stochastic differential equations. It relies on a uniqueness result, due to Pierre, for a Fokker-Planck equation.
A Note on an Application of the Lasota-York Fixed Point Theorem in the Turbulent Transport Problem
We study a model of motion of a passive tracer particle in a turbulent flow that is strongly mixing in time variable. In [8] we have shown that there exists a probability measure equivalent to the underlying physical probability under which the quasi-Lagrangian velocity process, i.e. the velocity of the flow observed from the vintage point of the moving particle, is stationary and ergodic. As a consequence, we proved the existence of the mean of the quasi-Lagrangian velocity, the so-called Stokes...
A note on approximation for stochastic differential equations
A note on estimation in controlled diffusion processes
A note on one-dimensional stochastic equations
We consider the stochastic equation where is a one-dimensional Brownian motion, is the initial value, and is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients , beyond the homogeneous case there is no general result on the uniqueness in law of the solution. The purpose of the present note is to give conditions on ensuring the existence as well as the uniqueness in law of the solution.