We consider a continuous time Markov chain on a countable state space. We prove a joint large deviation principle (LDP) of the empirical measure and current in the limit of large time interval. The proof is based on results on the joint large deviations of the empirical measure and flow obtained...

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 a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power p, for p∈[0,2). The asymptotic behaviour of the right-most particle for this system is already known; in...

We study the asymptotics of the probabilities of extreme slowdown events for transient one-dimensional excited random walks. That is, if {Xn}n≥0 is a transient one-dimensional excited random walk and Tn=min{k:Xk=n}, we study the asymptotics of probabilities of the form P(Xn≤nγ) and...

We consider the effect of recovery rates on a pool of credit assets. We allow the recovery rate to depend on the defaults in a general way. Using the theory of large deviations, we study the structure of losses in a pool consisting of a continuum of types. We derive the corresponding rate...

In this article, we establish a large deviation principle for invariant measures of solutions of stochastic partial differential equations with two reflecting walls driven by a space–time white noise.

We establish large deviation estimates for the optimal filter where the observation process is corrupted by a fractional Brownian motion. The observation process is transformed to an equivalent model which is driven by a standard Brownian motion. The large deviations in turn are established by...

We consider the branching processes in random environment. In this paper, we deal with the case of environments which are chosen stationary and ergodic from the finite set of geometrical offspring distributions. We denote by Zn the population at the n-th generation. We show that the large...

Let X1,X2,… be independent identically distributed (i.i.d.) random variables with EXk=0, V arXk=1. Suppose that φ(t)≔logEetXk<∞ for all t>−σ0 and some σ00. Let Sk=X1+⋯+Xk and S0=0. We are interested in the limiting distribution of the multiscale scan statisticMn=max0≤i<j≤nSj−Sij−i. We prove that for an appropriate normalizing sequence an, the random variable Mn2−an converges to the Gumbel extreme-value law exp{−e−cx}. The behavior of Mn depends strongly on the distribution of the Xk’s. We distinguish between four cases. In the superlogarithmic case we assume that φ(t)<t2/2 for every t>0. In this case, we show...</j≤nsj−sij−i.></∞>

We give an analytic characterization of a large-time “downside risk” probability associated with an investor’s wealth. We assume that risky securities in our market model are affected by “hidden” economic factors, which evolve as a finite-state Markov chain. We formalize and prove a...