Stochastic Processes and their Applications最新文献

We study a general contracting problem between the principal and a finite set of competitive agents, who perform equivalent changes of measure by controlling the drift of the output process and the compensator of its associated jump measure. In this setting, we generalize the dynamic programming approach developed by Cvitanić et al. (2018) and we also relax their assumptions. We prove that the problem of the principal can be reformulated as a standard stochastic control problem in which she controls the continuation utility (or certainty equivalent) processes of the agents. Our assumptions and conditions on the admissible contracts are minimal to make our approach work. We review part of the literature and give examples on how they are usually satisfied. We also present a smoothness result for the value function of a risk–neutral principal when the agents have exponential utility functions. This leads, under some additional assumptions, to the existence of an optimal contract.

The model of directed polymer in a random environment is a fundamental model of interaction between a simple random walk and ambient disorder. This interaction gives rise to complex phenomena and transitions from a central limit theory to novel statistical behaviours. Despite its intense study, there are still many aspects and phases which have not yet been identified. In this review we focus on the current status of our understanding of the transition between weak and strong disorder phases, give an account of some of the methods that the study of the model has motivated and highlight some open questions.

A birth and death process is a continuous-time Markov chain with minimal state space $N$, whose transition matrix is standard and whose density matrix is a birth–death matrix. Birth and death process is unique if and only if $\infty$ is an entrance or natural. When $\infty$ is neither an entrance nor natural, there are two ways in the literature to obtain all birth and death processes. The first one is an analytic treatment proposed by Feller in 1959, and the second one is a probabilistic construction completed by Wang in 1958.

In this paper we will give another way to study birth and death processes using the Ray–Knight compactification. This way has the advantage of both the analytic and probabilistic treatments above. By applying the Ray–Knight compactification, every birth and death process can be modified into a càdlàg Ray process on $N\cup \left\{\infty \right\}\cup \left\{\partial \right\}$, which is either a Doob processes or a Feller $Q$-process. Every birth and death process in the second class has a modification that is a Feller process on $N\cup \left\{\infty \right\}\cup \left\{\partial \right\}$. We will derive the expression of its infinitesimal generator, which explains its boundary behaviours at $\infty$. Furthermore, by using the killing transform and the Ikeda–Nagasawa–Watanabe piecing out procedure, we will also provide a probabilistic construction for birth and death processes. This construction relies on a triple determining the resolvent matrix introduced by Wang and Yang in their work (Wang and Yang, 1992).

We consider the stochastic heat equation on the 1-dimensional torus $T≔\left(-1,1\right)$ with periodic boundary conditions: ${\partial }_{t}u(t,x)={\partial }_x^{2}u(t,x)+\sigma (t,x,u)\dot{F}(t,x),x\in T,t\in R_{+},$where $\dot{F}(t,x)$ is a generalized Gaussian noise, which is white in time but colored in space. Assuming that $\sigma$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,x)\equiv 0$.

In this paper, we propose a new definition of catastrophes and present our results on large deviations for Poisson processes with catastrophes that satisfy this definition. Our earlier work focused on (almost) uniformly distributed catastrophes, but the current paper extends the results to a larger class of catastrophes. We show that the rate function remains the same regardless of the distribution of catastrophic events. Additionally, we extend and generalize our previous results on the limiting behavior of the supremum of the considered processes.

The main objective of the present paper is to construct a new class of space–time discretizations for the stochastic $p$-Stokes system and analyze its stability and convergence properties. We derive regularity results for the approximation that are similar to the natural regularity of solutions. One of the key arguments relies on discrete extrapolation that allows us to relate lower moments of discrete maximal processes. We show that, if the generic spatial discretization is constraint conforming, then the velocity approximation satisfies a best-approximation property in the natural distance. Moreover, we present an example such that the resulting velocity approximation converges with rate $1/2$ in time and 1 in space towards the (unknown) target velocity with respect to the natural distance. The theory is corroborated by numerical experiments.

This paper presents a finite-dimensional approximation for a class of partial differential equations on the space of probability measures. These equations are satisfied in the sense of viscosity solutions. The main result states the convergence of the viscosity solutions of the finite-dimensional PDE to the viscosity solutions of the PDE on Wasserstein space, provided that uniqueness holds for the latter, and heavily relies on an adaptation of the Barles & Souganidis monotone scheme (Barles and Souganidis, 1991) to our context, as well as on a key precompactness result for semimartingale measures. We illustrate our convergence result with the example of the Hamilton–Jacobi–Bellman and Bellman–Isaacs equations arising in stochastic control and differential games, and propose an extension to the case of path-dependent PDEs.

We derive a formula for the quasi-potential of the one-dimensional symmetric exclusion process in weak contact with reservoirs. The interaction with the boundary is so weak that, in the diffusive scale, the density profile evolves as the one of the exclusion process with reflecting boundary conditions. In order to observe an evolution of the total mass, the process has to be observed in a longer time-scale, in which the density profile becomes immediately constant.

We study the stochastic Camassa–Holm equation with pure jump noise. We establish the existence of the global martingale solution by the regularization method, the tightness criterion, the generalization of the Skorokhod theorem for nonmetric spaces and the stochastic renormalized formulations.

In this paper, we formulate and investigate the notion of causal feedback strategies arising in linear–quadratic control problems for stochastic Volterra integral equations (SVIEs) with singular and non-convolution-type coefficients. We show that there exists a unique solution, which we call the causal feedback solution, to the closed-loop system of a controlled SVIE associated with a causal feedback strategy. Furthermore, introducing two novel equations named a Type-II extended backward stochastic Volterra integral equation and a Lyapunov–Volterra equation, we prove a duality principle and a representation formula for a quadratic functional of controlled SVIEs in the framework of causal feedback strategies.