On pathwise uniqueness and expansion of filtrations
On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes
We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order with drift and diffusion coefficients , . When , we investigate pathwise uniqueness for this equation. When , we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether or and on whether the driving stable process is symmetric or not. Our assumptions...
On random boundary value problems for ordinary differential equations
On risk reserve under distribution constraints
The purpose of this work is a study of the following insurance reserve model: , t ∈ [0,T], P(η ≥ c) ≥ 1-ϵ, ϵ ≥ 0. Under viability-type assumptions on a pair (p,σ) the estimation γ with the property: is considered.
On rough differential equations.
On smoothing properties of transition semigroups associated to a class of SDEs with jumps
We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in driven by additive pure-jump Lévy noise. In particular, we assume that the Lévy process driving the SDE is the sum of a subordinated Wiener process (i.e. , where is an increasing pure-jump Lévy process starting at zero and independent of the Wiener process ) and of an arbitrary Lévy process independent of , that the drift coefficient is continuous (but not...
On solutions set of a multivalued stochastic differential equation
We analyse multivalued stochastic differential equations driven by semimartingales. Such equations are understood as the corresponding multivalued stochastic integral equations. Under suitable conditions, it is shown that the considered multivalued stochastic differential equation admits at least one solution. Then we prove that the set of all solutions is closed and bounded.
On Solutions to Stochastic Differential Equations with Discontinuous Drift in Hilbert Space.
On some fractional stochastic integrodifferential equations in Hilbert space.
On some limit theorems for solutions of stochastic differential equations
On some stability properties of stochastic differential equations of Itô's type
On stability for numerical approximations of stochastic ordinary differential equations.
On stochastic differential equations in a configuration space.
On stochastic differential equations with locally unbounded drift
We study the regularizing effect of the noise on differential equations with irregular coefficients. We present existence and uniqueness theorems for stochastic differential equations with locally unbounded drift.
On the approximate solutions of the Stratonovitch equation.
On the consistency of a least squares identification procedure
On the constructions of the skew Brownian motion.
On the control of the difference between two Brownian motions: a dynamic copula approach
We propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. Our approach is based on the copula between the Brownian motion and its reflection. We show that the class of admissible copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right...
On the control of the difference between two Brownian motions: an application to energy markets modeling
We derive a model based on the structure of dependence between a Brownian motion and its reflection according to a barrier. The structure of dependence presents two states of correlation: one of comonotonicity with a positive correlation and one of countermonotonicity with a negative correlation. This model of dependence between two Brownian motions B1 and B2 allows for the value of [...] to be higher than 1/2 when x is close to 0, which is not the case when the dependence is modeled by a constant...
On the convergence of generalized polynomial chaos expansions
A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement...