On rough differential equations.
On the constructions of the skew Brownian motion.
On the convergence of stochastic integrals driven by processes converging on account of a homogenization property.
Semi-martingales and rough paths theory.
Stochastic differential equations driven by processes generated by divergence form operators II: convergence results
We have seen in a previous article how the theory of “rough paths” allows us to construct solutions of differential equations driven by processes generated by divergence form operators. In this article, we study a convergence criterion which implies that one can interchange the integral with the limit of a family of stochastic processes generated by divergence form operators. As a corollary, we identify stochastic integrals constructed with the theory of rough paths with Stratonovich or Itô integrals...
Stochastic differential equations driven by processes generated by divergence form operators I: a Wong-Zakai theorem
We show in this article how the theory of “rough paths” allows us to construct solutions of differential equations (SDEs) driven by processes generated by divergence-form operators. For that, we use approximations of the trajectories of the stochastic process by piecewise smooth paths. A result of type Wong-Zakai follows immediately.
Matrices aléatoires : statistique asymptotique des valeurs propres
Perturbed linear rough differential equations
We study linear rough differential equations and we solve perturbed linear rough differential equations using the Duhamel principle. These results provide us with a key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebras more general than tensor algebras and to consider extensions of classical results such as the Magnus...
A Donsker theorem to simulate one-dimensional processes with measurable coefficients
In this paper, we prove a Donsker theorem for one-dimensional processes generated by an operator with measurable coefficients. We construct a random walk on any grid on the state space, using the transition probabilities of the approximated process, and the conditional average times it spends on each cell of the grid. Indeed we can compute these quantities by solving some suitable elliptic PDE problems.
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