Stochastics and Dynamics

We study the statistical properties of a cellular automata model of traffic flow with the look-ahead potential. The model defines stochastic rules for the movement of cars on a lattice. We analyze the underlying statistical assumptions needed for the derivation of the coarse-grained model and demonstrate that it is possible to relax some of them to obtain an improved coarse-grained ODE model. We also demonstrate that spatial correlations play a crucial role in the presence of the look-ahead potential and propose a simple empirical correction to account for the spatial dependence between neighboring cells.

An infection spreads in a binary tree of height n as follows: initially, each leaf is either infected by one of k states or it is not infected at all. The infection state of each leaf is independently distributed according to a probability vector p = (p₁, …, p_k+1). The remaining nodes become infected or not via annihilation and coalescence: nodes whose two children have the same state (infected or not) are infected (or not) by this state; nodes whose two children have different states are not infected; nodes such that only one of the children is infected are infected by this state. In this paper we characterize, for every p, the limiting distribution at the root node of as n goes to infinity.

We also consider a variant of the model when k = 2 and a mutation can happen, with a fixed probability q, at each infection step. We characterize, in terms of p and q, the limiting distribution at the root node of as n goes to infinity.

The distribution at the root node is driven by a dynamical system, and the proofs rely on the analysis of this dynamics.

Based on the observation that the elasticity of variance of risky assets is randomly varying around a constant, we take an underlying asset model in which the averaged constant elasticity of variance is perturbed by a small fast fluctuating process and study the Merton type portfolio optimization problem using dynamic programming as well as asymptotic expansions. The Hamilton–Jacobi–Bellman equation for each of the power and exponential utility functions leads to an optimal trading strategy as a perturbation around the well known one. We reveal the impact of both the constant elasticity of variance upon the Merton investment optimal control under the Black–Scholes model and the stochastic elasticity of variance upon the investment optimal control under the constant elasticity of variance model. The concavity of the investment policy with respect to the excess return is characteristic of a market economy with the constant or stochastic elasticity of variance.

A time and space inhomogeneous Markov process is a Feller evolution process, if the corresponding evolution system on the continuous functions vanishing at infinity is strongly continuous. We discuss generators of such systems and show that under mild conditions on the generators a Feller evolution can be approximated by Markov chains with Lévy increments.

The result is based on the approximation of the time homogeneous spacetime process corresponding to a Feller evolution process. In particular, we show that a d-dimensional Feller evolution corresponds to a (d + 1)-dimensional Feller process. It is remarkable that, in general, this Feller process has a generator with discontinuous symbol.

We consider the limiting behavior of fluctuations of small noise diffusions with multiple scales around their homogenized deterministic limit. We allow full dependence of the coefficients on the slow and fast motion. These processes arise naturally when one is interested in short time asymptotics of multiple scale diffusions. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. Depending on the order of interaction between the fast scale and the size of the noise, we get different behavior. In certain cases additional drift terms arise in the limiting process, which are explicitly characterized. These results provide a better approximation to the limiting behavior of such processes when compared to the law of large numbers homogenization limit.

We define the Abelian distribution and study its basic properties. Abelian distributions arise in the context of neural modeling and describe the size of neural avalanches in fully-connected integrate-and-fire models of self-organized criticality in neural systems.

In this paper, we study the asymptotic behaviors of parameter estimator in a diagonalizable stochastic evolution equation driven by additive fractional Brownian motion with Hurst parameter H ∈ [½, 1). The moderate deviation for this estimator can be obtained.

This paper provides two different strong BSDE representations for optimal switching problems in the case where the dynamics of the underlying diffusion process depends on the current value of the switching mode. These new representations are valid in a non-Markovian framework and make use of either one-dimensional constrained BSDEs with jumps or multidimensional BSDEs with oblique reflections, thus extending the framework considered by Hu and Tang [12]. In particular, the numerical resolution of the corresponding switching problem can therefore be treated via the entirely probabilistic schemes presented in [4] or [8].

In this paper, we consider coupled forward–backward stochastic differential equations (FBSDEs in short) with parameter ε > 0, of the following type