Mathematics
Probability
Featured Researches
"Pass the Buck" on a Rooted Tree
The Stochastic Abacus is can employed to compute winning probabilities for each vertex of a rooted tree in the game "Pass the Buck", with the starting vertex being the root of the tree. For all but the simplest trees, the abacus can't really be implemented due to the large number of steps needed for completion. In this paper, a technique for anticipating the outcome is introduced.
Read more2D Euler equations with Stratonovich transport noise as a large scale stochastic model reduction
The limit from an Euler type system to the 2D Euler equations with Stratonovich transport noise is investigated. A weak convergence result for the vorticity field and a strong convergence result for the velocity field are proved. Our results aim to provide a stochastic reduction of fluid-dynamics models with three different time scales.
Read moreA C 0,1 -functional Itô's formula and its applications in mathematical finance
Using Dupire's notion of vertical derivative, we provide a functional (path-dependent) extension of the Itô's formula of Gozzi and Russo (2006) that applies to C^{0,1}-functions of continuous weak Dirichlet processes. It is motivated and illustrated by its applications to the hedging or superhedging problems of path-dependent options in mathematical finance, in particular in the case of model uncertainty
Read moreA K -rough path above the space-time fractional Brownian motion
We construct a K -rough path above either a space-time or a spatial fractional Brownian motion, in any space dimension d . This allows us to provide an interpretation and a unique solution for the corresponding parabolic Anderson model, understood in the renormalized sense. We also consider the case of a spatial fractional noise.
Read moreA CLT for degenerate diffusions with periodic coefficients, and application to homogenisation of linear PDEs
In this article, we obtain a functional CLT for a class of degenerate diffusion processes with periodic coefficients, thus generalizing the already classical results in the context of uniformly elliptic diffusions. As an application, we also discuss periodic homogenization of a class of linear degenerate elliptic and parabolic PDEs.
Read moreA McKean-Vlasov SDE and particle system with interaction from reflecting boundaries
We consider a one-dimensional McKean-Vlasov SDE on a domain and the associated mean-field interacting particle system. The peculiarity of this system is the combination of the interaction, which keeps the average position prescribed, and the reflection at the boundaries; these two factors make the effect of reflection non local. We show pathwise well-posedness for the McKean-Vlasov SDE and convergence for the particle system in the limit of large particle number.
Read moreA Resolvent Approach to Metastability
We provide a necessary and sufficient condition for the metastability of a Markov chain, expressed in terms of a property of the solutions of the resolvent equation. As an application of this result, we prove the metastability of reversible, critical zero-range processes starting from a configuration.
Read moreA Scaling Limit for Utility Indifference Prices in the Discretized Bachelier Model
We consider the discretized Bachelier model where hedging is done on an equidistant set of times. Exponential utility indifference prices are studied for path-dependent European options and we compute their non-trivial scaling limit for a large number of trading times n and when risk aversion is scaled like n??for some constant ??0 . Our analysis is purely probabilistic. We first use a duality argument to transform the problem into an optimal drift control problem with a penalty term. We further use martingale techniques and strong invariance principles and get that the limiting problem takes the form of a volatility control problem.
Read moreA Survey on Limit Theorems for Toeplitz Type Quadratic Functionals of Stationary Processes and Applications
This is a survey of recent results on central and non-central limit theorems for quadratic functionals of stationary processes. The underlying processes are Gaussian, linear or Lévy-driven linear processes with memory, and are defined either in discrete or continuous time. We focus on limit theorems for Toeplitz and tapered Toeplitz type quadratic functionals of stationary processes with applications in parametric and nonparametric statistical estimation theory. We discuss questions concerning Toeplitz matrices and operators, Fejér-type singular integrals, and Lévy-Itô-type and Stratonovich-type multiple stochastic integrals. These are the main tools for obtaining limit theorems.
Read moreA Topological Proof of Sklar's Theorem in Arbitrary Dimensions
We prove Sklar's theorem in infinite dimensions via a topological argument and the notion of inverse systems.
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