We give meaning to differential equations with a rough path term and a Brownian noise term and study their regularity, that is we are interested in equations of the type Stη=S0+∫0ta(Srη)dr+∫0tb(Srη)∘dBr+∫0tc(Srη)dηr where η is a deterministic geometric, step-2 rough path and B is a...

This paper concerns the homogenization problem of a parabolic equation with large, time-dependent, random potentials in high dimensions d≥3. Depending on the competition between temporal and spatial mixing of the randomness, the homogenization procedure turns to be different. We characterize...

We show a method of uniform approximation of the value functions of uniformly nondegenerate stochastic differential games in smooth domains up to a constant over K with the ones having second-order derivatives bounded by a constant times K for any K≥1.

We study a zero-sum game where the evolution of a spectrally one-sided Lévy process is modified by a singular controller and is terminated by the stopper. The singular controller minimizes the expected values of running, controlling and terminal costs while the stopper maximizes them. Using...

Given a Brownian Motion W, in this paper we study the asymptotic behavior, as ε→0, of the quadratic covariation between f(εW) and W in the case in which f is not smooth. Among the main features discovered is that the speed of the decay in the case f∈Cα is at least polynomial in ε and not...

We consider an X-valued Markov chain X1,X2,…,Xn belonging to a class of iterated random functions, which is “one-step contracting” with respect to some distance d on X. If f is any separately Lipschitz function with respect to d, we use a well known decomposition of...

We consider a system of real-valued spins interacting with each other through a mean-field Hamiltonian that depends on the empirical magnetisation of the spins. The system is subjected to a stochastic dynamics where the spins perform independent Brownian motions. Using large deviation theory we...

Estimation of the service time distribution in the discrete-time GI/G/∞-queue based solely on information on the arrival and departure processes is considered. The focus is put on the estimation approach via the so called “sequence of differences”. Existing results for this approach are...

We study the minimal/endogenous solution R to the maximum recursion on weighted branching trees given by R=D(⋁i=1NCiRi)∨Q, where (Q,N,C1,C2,…) is a random vector with N∈N∪{∞}, P(|Q|0)0 and nonnegative weights {Ci}, and {Ri}i∈N is a sequence of i.i.d. copies of R independent of...

We consider the classical Wright–Fisher model with mutation and selection. Mutations occur independently in each locus, and selection is performed according to the sharp peak landscape. In the asymptotic regime studied in Cerf (2014), a quasispecies is formed. We find explicitly the...