Stochastic differential equations

By P. Kloeden. 1955

The work of which this paper is an account began as a study of differential equations for functions whose values are random variables of finite variance. It was intended that all questions of convergence should be treated from the standpoint of strong convergence in Hilbert space—familiar to probabilists from the writings of Karhunen(11) and Loève(13) as mean-square convergence. The more general Banach-space approach now adopted was made possible by the discovery of a theorem (Theorem 1 of this paper) which Mr D. G. Kendall, its apparent author, kindly communicated to us.

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Citations in the corpus

Neural Processes (2018, methodology): “As stated by the Kolmogorov Extension Theorem (Øksendal, 2003) these conditions are sufﬁcient to deﬁne a stochastic process.”
Stochastic Gradient Descent Performs Variational Inference, Converges to Limit Cycles for Deep Networks (2017, background): “However, the same PDE can be tied to different SDEs depending on the stochastic integration scheme, e.g., Ito, Stratonovich (Risken, 1996; Oksendal, 2003), Hanggi (Hänggi, 1978), α-type etc.”
MODELING, ANALYSIS AND DISCRETIZATION OF STOCHASTIC LOGISTIC EQUATIONS (2007, unknown): “Interestingly, the equation 2λ = σ(2) is well-known to exhibit a bifurcation point (threshold case) for stability of linear SDEs (see [3], [39]).”
Simulating Solute Transport in Porous or Fractured Formations Using Random Walk Particle Tracking: A Review (2005, background): “This standard equation is often referred to as the discrete form of the Itô (1951) stochastic differ-Applying the general Eq.”
Cooperative Advertising and Pricing in a Dynamic Stochastic Supply Chain: Feedback Stackelberg Strategies (2008, unknown): “Then, from Gihman and Skorohod (1972, p. 149, 157–158), 0 and 1 are the natural boundaries for the solutions of (1) with x ð 0 Þ ¼ x 2 ½ 0 ; 1 (cid:4) , i.e., x ð t Þ 2 ð 0 ; 1 Þ almost surely for t 4 0.”
A mild Itô formula for SPDEs (2010, background): “First, note that if α = β = 0 and if the mild solution process X of the SPDE (1) is also a D(A)-valued strong solution of the SPDE (1), then the standard Itô formula (see Itô [48]) in infinite dimensions can be applied to X .”
Quantitative estimates of propagation of chaos for stochastic systems with W-1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d (2017, methodology): “Gronwall-like estimates with Lipschitz force fields, but a fixed number of SDEs, were also at the basis of [54].”
General duality between optimal control and estimation (2008, background): “The second limit coincides with the notion of generalized derivative in the theory of stochastic processes and evaluates to z t + L [z], where the operator L is the infinitesimal generator [12] of the stochastic process with transition probability p (h) (x 0 |x).”
CONVERGENCE AND STABILITY OF IMPLICIT METHODS FOR JUMP-DIFFUSION SYSTEMS (2005, unknown): “Applications in economics, finance, and several areas of science and engineering, give rise to jump-diffusion Ito stochastic differential equations [2] [4] [24] of the form ))dN (t), t > 0, with X(0) given, where X(t − ) denotes lim s→t − X(s).”
The Theoretical Biologist's Toolbox: Quantitative Methods for Ecology and Evolutionary Biology (2006, background): “Two older but solid introductions to the material are Arnold (1973) and Gardiner (1983). Another good starting point is Karlin and Taylor (1981), who have a 240 page chapter on diffusion processes. A general discussion of numerical methods for stochastic differential equations is found in Higham (20”
INVARIANT MANIFOLDS FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS (2003, unknown): “The set of these ω's is {θ t } t∈R -invariant. ii) This can be proven as in Øksendal [18] Page 35.”
Dynamic response and reliability analysis of structures with uncertain parameters (2005, unknown): “Denote the joint PDF of (XT, ZT)T as pXZ(x, z, t), according to the principle of preservation of probability [20], the joint PDF is governed by”