Multiplicative stochastic differential equation

A further complication that has been much studied in the literature is that of multiplicative noise in which the random force in stochastic differential equations like Eq. (1) is modified by a modulating term, i.e.,... 

Haken, H., Wunderlin, A. (1982) Slaving principle for stochastic differential equations with additive and multiplicative noise and for discrete noisy maps. Z. Phys. 47, 179 Hastings, S. P. (1976) Periodic plane waves for the Oregonator. Stud. Appl. Math. 55, 293 Herschkowitz-Kaufman, M. (1975) Bifurcation analysis of nonlinear reaction-diffusion equations II. Steady state solutions and comparison with numerical simulations. Bull. Math. Biol. 37, 589 Howard, L. N., Kopell, N. (1977) Slowly varying waves and shock structures in reaction-diffusion equations. Stud. Appl. Math. 56, 95... 

Eq. (40) has to be interpreted as a stochastic differential equation with multiplicative noise, and its solution is a difficult task ... 

The CLE is a multivariate Ito stochastic differential equation with multiple, multiplicative noise. We define the CLE again and present methods to solve it in Chapter 18, where we discuss numerical simulations of stochastic reaction kinetics. 

In the fast-continuous region, species populations can be assumed to be continuous variables. Because the reactions are sufficiently fast in comparison to the rest of the system, it can be assumed that they have relaxed to a steady-state distribution. Furthermore, because of the frequency of reaction rates, and the population size, the population distributions can be assumed to have a Gaussian shape. The subset of fast reactions can then be approximated as a continuous time Markov process with chemical Langevin Equations (CLE). The CLE is an ltd stochastic differential equation with multiplicative noise, as discussed in Chapter 13. 

A CLE is an Ito stochastic differential equation (SDE) with multiplicative noise terms and represents one possible solution of the Eokker-Planck equation. From a multidimensional Fokker-Planck equation we end up with a system of CLEs ... 

Note that the subdivision refers to the form of the equation, not to the process described by it the term multiplicative noise is a misnomer. There are other categories, such as stochastic partial differential equations, eigenvalue problems 0, and random boundaries ), but they will not be treated here. 

This chapter will focus on practicable methods to perform both the model specification and model estimation tasks for systems/models that are static or dynamic and linear or nonlinear. Only the stationary case win be detailed here, although the potential use of nonstationary methods will be also discussed briefly when appropriate. In aU cases, the models will take deterministic form, except for the presence of additive error terms (model residuals). Note that stochastic experimental inputs (and, consequently, outputs) may stiU be used in connection with deterministic models. The cases of multiple inputs and/or outputs (including multidimensional inputs/outputs, e.g., spatio-temporal) as well as lumped or distributed systems, will not be addressed in the interest of brevity. It will also be assumed that the data (single input and single output) are in the form of evenly sampled time-series, and the employed models are in discretetime form (e.g., difference equations instead of differential equations, discrete summations instead of integrals). 

Two important challenges exist for multiscale systems. The first is multiple time scales, a problem that is familiar in chemical engineering where it is called stiffness, and we have good solutions to it. In the stochastic world there doesn t seem to be much knowledge of this phenomenon, but I believe that we recently have found a solution to this problem. The second challenge—one that is even more difficult—arises when an exceedingly large number of molecules must be accounted for in stochastic simulation. I think the solution will be multiscale simulation. We will need to treat some reactions at a deterministic scale, maybe even with differential equations, and treat other reactions by a discrete stochastic method. This is not an easy task in a simulation. 

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