Stochastic process

About: Stochastic process is a research topic. Over the lifetime, 31227 publications have been published within this topic receiving 898736 citations. The topic is also known as: random process & stochastic processes.

Quantum stochastic processes as models for state vector reduction

Abstract: An elementary introduction of quantum-state-valued Markovian stochastic processes (QSP) for N-state quantum systems is given. It is pointed out that a so-called master constraint must be fulfilled. For a given master equation a continuous and, as a new alternative possibility, a discontinuous QSP are derived. Both are discussed as possible models for state reduction during measurement.

Reactive boundary conditions for stochastic simulations of reaction-diffusion processes.

Abstract: Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species (e.g. enzymes). Such reaction-diffusion processes can be mathematically modelled using either deterministic partial-differential equations or stochastic simulation algorithms. The latter provide a more detailed and precise picture, and several stochastic simulation algorithms have been proposed in recent years. Such models typically give the same description of the reaction-diffusion processes far from the boundary of the simulated domain, but the behaviour close to a reactive boundary (e.g. a membrane with receptors) is unfortunately model-dependent. In this paper, we study four different approaches to stochastic modelling of reaction-diffusion problems and show the correct choice of the boundary condition for each model. The reactive boundary is treated as partially reflective, which means that some molecules hitting the boundary are adsorbed (e.g. bound to the receptor) and some molecules are reflected. The probability that the molecule is adsorbed rather than reflected depends on the reactivity of the boundary (e.g. on the rate constant of the adsorbing chemical reaction and on the number of available receptors), and on the stochastic model used. This dependence is derived for each model.

A top down approach to multi-name credit

Abstract: We examine multi-name credit models from the perspective of point processes. In this context, it is natural to pursue a top down approach: the economy as a whole is modeled flrst. The technique of random thinning consistently generates sub-models for individual flrms or portfolios. A candidate for the top down approach is a self-exciting process, whose intensity at any time depends on the sequence of events observed up to that time. A self-exciting process incorporates the contagion observed in credit markets and avoids an ad hoc choice of copula. The familiar doubly stochastic process is at the opposite end of the spectrum in the sense that it is constructed from the bottom up: individual flrm intensities are estimated and then aggregated. We rigorously analyze self-exciting and doubly stochastic processes with respect to their ability to capture contagion. Model fltness can be tested using a deep result of Meyer (1971), which shows that any point process with continuous compensator can be transformed into a standard Poisson process by a change of time. Meyer’s result allows us to extend the scope of the tests proposed by Das, Du‐e & Kapadia (2004) for a doubly stochastic model.

Stochastic resonance in models of neuronal ensembles

Abstract: Two recently suggested mechanisms for the neuronal encoding of sensory information involving the effect of stochastic resonance with aperiodic time-varying inputs are considered. It is shown, using theoretical arguments and numerical simulations, that the nonmonotonic behavior with increasing noise of the correlation measures used for the so-called aperiodic stochastic resonance ~ASR! scenario does not rely on the cooperative effect typical of stochastic resonance in bistable and excitable systems. Rather, ASR with slowly varying signals is more properly interpreted as linearization by noise. Consequently, the broadening of the ‘‘resonance curve’’ in the multineuron stochastic resonance without tuningscenario can also be explained by this linearization. Computation of the input-output correlation as a function of both signal frequency and noise for the model system further reveals conditions where noise-induced firing with aperiodic inputs will benefit from stochastic resonance rather than linearization by noise. Thus, our study clarifies the tuning requirements for the optimal transduction of subthreshold aperiodic signals. It also shows that a single deterministic neuron can perform as well as a network when biased into a suprathreshold regime. Finally, we show that the inclusion of a refractory period in the spike-detection scheme produces a better correlation between instantaneous firing rate and input signal. @S1063-651X~97!01102-1#

Simulation of random processes and rate-distortion theory

Abstract: We study the randomness necessary for the simulation of a random process with given distributions, on terms of the finite-precision resolvability of the process. Finite-precision resolvability is defined as the minimal random-bit rate required by the simulator as a function of the accuracy with which the distributions are replicated. The accuracy is quantified by means of various measures: variational distance, divergence, Orstein (1973), Prohorov (1956) and related measures of distance between the distributions of random process. In the case of Ornstein, Prohorov and other distances of the Kantorovich-Vasershtein type, we show that the finite-precision resolvability is equal to the rate-distortion function with a fidelity criterion derived from the accuracy measure. This connection leads to new results on nonstationary rate-distortion theory. In the case of variational distance, the resolvability of stationary ergodic processes is shown to equal entropy rate regardless of the allowed accuracy. In the case of normalized divergence, explicit expressions for finite-precision resolvability are obtained in many cases of interest; and connections with data compression with minimum probability of block error are shown.