Diffusions In Random Environment Research Articles

Diffusions in random environment and ballistic behavior

In this article we investigate the ballistic behavior of diffusions in random environment. We introduce conditions in the spirit of ( T ) and ( T ′ ) of the discrete setting, cf. [A.-S. Sznitman, On a class of transient random walks in random environment, Ann. Probab. 29 (2) (2001) 723–764; A.-S. Sznitman, An effective criterion for ballistic behavior of random walks in random environment, Probab. Theory Related Fields 122 (4) (2002) 509–544], that imply, when d ⩾ 2 , a law of large numbers with non-vanishing limiting velocity (which we refer to as ‘ballistic behavior’) and a central limit theorem with non-degenerate covariance matrix. As an application of our results, we consider the class of diffusions where the diffusion matrix is the identity, and give a concrete criterion on the drift term under which the diffusion in random environment exhibits ballistic behavior. This criterion provides examples of diffusions in random environment with ballistic behavior, beyond what was previously known.

An invariance principle for isotropic diffusions in random environment

We investigate in this work the asymptotic behavior of isotropic diffusions in random environment that are small perturbations of Brownian motion. When the space dimension is three or more, we prove an invariance principle as well as transience. Our methods also apply to questions of homogenization in random media.

Monotonicity properties of multi-dimensional reflected diffusions in random environment and applications

We consider an N -dimensional reflected process, modeling an infinite capacity fluid queues network, of which service and input rates depend on the queue levels as well as on the state of an exterior ergodic stationary process. N is the number of queues in the network. We prove a monotonicity result for such a process, from which we deduce stability results for networks of queues. Particular attention is paid to the case N = 2 . Next, we give some applications of those stability results.

Diffusion in random environment and the renewal theorem

According to a theorem of Schumacher and Brox, for a diffusion X in a Brownian environment, it holds that (Xt−blogt)/log2t→0 in probability, as t→∞, where b is a stochastic process having an explicit description and depending only on the environment. We compute the distribution of the number of sign changes for b on an interval [1,x] and study some of the consequences of the computation; in particular, we get the probability of b keeping the same sign on that interval. These results have been announced in 1999 in a nonrigorous paper by Le Doussal, Monthus and Fisher [Phys. Rev. E 59 (1999) 4795–4840] and were treated with a Renormalization Group analysis. We prove that this analysis can be made rigorous using a path decomposition for the Brownian environment and renewal theory. Finally, we comment on the information these results give about the behavior of the diffusion.

On the diffusive behavior of isotropic diffusions in a random environment

We present here results concerning the asymptotic behavior of isotropic diffusions in random environment that are small perturbations of Brownian motion. When the space dimension is three or more we prove an invariance principle as well as transience. Our methods also apply to questions of homogenization in random media. To cite this article: A.-S. Sznitman, O. Zeitouni, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Bayesian inference for Markov processes with diffusion and discrete components

Data arising in certain radio‐tracking experiments consist of both a continuous spatial component and a discrete component related to behaviour. This leads naturally to stochastic models with a state space which is a product of continuous and discrete components. We consider a class of such models in continuous time, which can be thought of as diffusions in random environments. They are related to switching diffusion or hidden Markov models, but observations are made on both components at discrete time points, so that neither component is completely ‘hidden’. We describe and illustrate an approach to fully Bayesian inference for these general models. The algorithm used is a hybrid Markov chain Monte Carlo method. The diffusion parameters, the environment parameters and the sample path of the environment process itself are updated separately, in sequence, and the individual steps are a mixture of Gibbs and random walk Metropolis–Hastings types. Some implementation and model checking issues are discussed, and an example using data arising from a radio‐tracking experiment is described.

On ballistic diffusions in random environment

In this article we investigate diffusions in random environment. We provide a sufficient condition for a strong law of large numbers with non-vanishing limiting velocity and a functional central limit theorem. In the course of this work we introduce certain regeneration times and obtain a renewal structure. As an illustration, we apply our results to a class of anisotropic gradient-type diffusions in random environment, where the technique of the environment viewed from the particle does not apply well.

Mathematical Models in Finance.

Influence of Mathematical Models in Finance on Practice: Past, Present and Future Applied Mathematics and Finance Stock Price Fluctuations as a Diffusion in Random Environment A Note on Super-Replicating Strategies Worldwide Security Market Anomalies Making Money from Mathematical Models Path-Dependent Options and Transaction Costs Stochastic Equality Volatility and the Capital Structure of the Firm The General Mean-Variance Portfolio Section Problem On a Free Boundary Problem That Arises in Portfolio Management Interest Rate Volatility and the Shape of the Term Structure Multi-Factor Term Structure Models Dynamic Asset Allocation: Insights from Theory Index