Stochastic differential equations and their applications

Book•

Stochastic differential equations and their applications

01 Jan 1997-

About: The article was published on 1997-01-01 and is currently open access. It has received 1170 citations till now. The article focuses on the topics: Stochastic partial differential equation & Numerical partial differential equations.

...read moreread less

Citations

PDF

Open Access

More filters

Journal Article•DOI•

Environmental Brownian noise suppresses explosions in population dynamics

[...]

Xuerong Mao, Glenn Marion¹, Eric Renshaw²•Institutions (2)

University of Edinburgh¹, University of Strathclyde²

01 Jan 2002-Stochastic Processes and their Applications

TL;DR: In this article, the authors show that the presence of even a tiny amount of environmental noise can suppress a potential population explosion, with probability one that the associated stochastic differential equation does not.

...read moreread less

764 citations

Additional excerpts

...When there are no interspecific interactions, a bounded system can be described by the purely logistic scheme ẋ1(t) = x1(t)[b1 − a11x1(t)] ẋ2(t) = x2(t)[b2 − a22x2(t)] , (4) for positive parameters b1, b2, a11 and a22....
[...]

Journal Article•DOI•

Stability of stochastic differential equations with Markovian switching

[...]

Xuerong Mao

01 Jan 1999-Stochastic Processes and their Applications

TL;DR: In this paper, the authors discuss the exponential stability of nonlinear stochastic differential equations with Markovian switching and show that the stability can be improved by using Markovians.

...read moreread less

714 citations

Book Chapter•DOI•

The Bayesian Approach to Inverse Problems

[...]

Masoumeh Dashti¹, Andrew M. Stuart¹•Institutions (1)

University of Warwick¹

29 Jun 2017-arXiv: Probability

TL;DR: In this paper, the authors highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations, and describe measure-preserving dynamics on the underlying infinite dimensional space.

...read moreread less

Abstract: These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions.

...read moreread less

520 citations

Journal Article•DOI•

Stability analysis for stochastic Cohen-Grossberg neural networks with mixed time delays

[...]

Zidong Wang¹, Yurong Liu², Maozhen Li, Xiaohui Liu¹•Institutions (2)

Brunel University London¹, Yangzhou University²

01 May 2006-IEEE Transactions on Neural Networks

TL;DR: It is shown that the addressed stochastic Cohen-Grossberg neural networks with mixed delays are globally asymptotically stable in the mean square if two LMIs are feasible, where the feasibility of LMIs can be readily checked by the Matlab LMI toolbox.

...read moreread less

Abstract: In this letter, the global asymptotic stability analysis problem is considered for a class of stochastic Cohen-Grossberg neural networks with mixed time delays, which consist of both the discrete and distributed time delays. Based on an Lyapunov-Krasovskii functional and the stochastic stability analysis theory, a linear matrix inequality (LMI) approach is developed to derive several sufficient conditions guaranteeing the global asymptotic convergence of the equilibrium point in the mean square. It is shown that the addressed stochastic Cohen-Grossberg neural networks with mixed delays are globally asymptotically stable in the mean square if two LMIs are feasible, where the feasibility of LMIs can be readily checked by the Matlab LMI toolbox. It is also pointed out that the main results comprise some existing results as special cases. A numerical example is given to demonstrate the usefulness of the proposed global stability criteria

...read moreread less

433 citations

Additional excerpts

...…to (19), and Q 1 0 and Q 2 0 are defined by Q 1 := " 2 G T 2 G 2 + 6 T 2 6 2 ; Q 2 := " 3 G T 3 G 3 : (22) By Itô's differential formula (see, e.g., [8]), the stochastic derivative of V (t; x(t)) along (10) can be obtained as follows: dV (t; x(t)) = 02x T (t)P (x(t)) 2 (x(t)) 0 Al1 (x(t)) 0 Bl2…...
[...]

Journal Article•DOI•

Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients

[...]

Martin Hutzenthaler, Arnulf Jentzen, Peter E. Kloeden

01 Aug 2012-Annals of Applied Probability

TL;DR: In this article, an explicit and easily implementable numerical method for such an SDE was proposed, which converges strongly with the standard order one-half to the exact solution of the SDE.

...read moreread less

Abstract: On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.

...read moreread less

386 citations

Cites background from "Stochastic differential equations a..."

...We refer to Theorem 2 in Alyushina [1], Theorem 1 in Krylov [21] and Theorem 2.4.1 in Mao [24] for existence and uniqueness results for SDEs of the form (1)....
[...]
...1 in Mao [24] for existence and uniqueness results for SDEs of the form (1)....
[...]
...In the same setting as in this article, Higham, Mao and Stuart showed in Theorem 5.3 in [12] (see also [11, 14, 22, 25, 34–37] and the references therein for more approximation results on implicit numerical methods for SDEs of the form (1)) that the implicit Euler scheme (6) converges with order 12 to the exact solution of the SDE (1) in the root mean square sense, that is, they established the existence of a real number C ∈ [0,∞) such that (E[‖XT − ˜̃Y NN‖2])1/2 ≤C ·N−1/2(7) for all N ∈N....
[...]
...In the literature (see, e.g., Theorem 10.2.2 in Kloeden and Platen [20], Theorem 1.1 in Milstein [27] or Theorem 3.1 in Yuan and Mao [38]) the convergence results for the explicit Euler scheme require the drift coefficient µ of the SDE (1) to be globally Lipschitz continuous or to grow at most linearly, which we have not assumed in our setting....
[...]
...In particular, in 2002, Higham, Mao and Stuart formulated in [12], page 1060, the following open problem: “In general, it is not clear when such moment bounds can be expected to hold for explicit methods with f, g ∈C1” (drift and diffusion coefficients are denoted by f and g in [12] instead of µ and σ here)....
[...]

Collapse

Stochastic differential equations and their applications

Citations

Additional excerpts

Additional excerpts

Cites background from "Stochastic differential equations a..."

Related Papers (5)