Results 261 to 270 of about 43,061 (301)

Stochastic differential equations

Mathematical Proceedings of the Cambridge Philosophical Society, 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 ...
J. E. Moyal, D. A. Edwards
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Stochastic Differential Equations [PDF]

, 1988
We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of ...
Steven E. Shreve, Ioannis Karatzas
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Stochastic Differential Equations

1991
In previous chapters stochastic differential equations have been mentioned several times in an informal manner. For instance, if M is a continuous local martingale, its exponential e(M) satisfies the equality $$\mathcal{E}{(M)_t} = 1 + \int_0^t {\mathcal{E}{{(M)}_s}} d{M_s};$$ this can be stated: e(M) is a solution to the stochastic differential
Daniel Revuz, Marc Yor
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Stochastic differential equations

2011
In this chapter we present some basic results on stochastic differential equations, hereafter shortened to SDEs, and we examine the connection to the theory of parabolic partial differential equations.
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Stochastic differential equations

Physics Reports, 1976
Abstract In chapter I stochastic differential equations are defined and classified, and their occurrence in physics is reviewed. In chapter II it is shown for linear equation show a differential equation for the averaged solution is obtained by expanding in ατ c , where α measures the size of the fluctuations and τ c their autocorrelation time. This
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Stochastic Differential Equations

1985
We now return to the possible solutions X t (ω) of the stochastic differential equation (5.1) where W t is 1-dimensional “white noise”. As discussed in Chapter III the Ito interpretation of (5.1) is that X t satisfies the stochastic integral equation or in differential form (5.2) .
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Stochastic Differential Equations

2009
The chapter begins with Section 4.1 in which motivational examples of random walks and stochastic phenomena in nature are presented. In Section 4.2 the concept of random processes is introduced in a more precise way. In Section 4.3 the concept of a Gaussian and Markov random process is developed. In Section 4.4 the important special case of white noise
Alexander Kukush, Andrey Pilipenko, Dmytro Gusak, Alexey Kulik, Yuliya Mishura   +4 more
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Stochastic Integrals and Differential Equations

2004
This chapter provides the tools needed for option pricing. The field of stochastic processes in continuous time, which are defined as solutions of stochastic differential equations, has an important role to play. To illustrate these notions we use repeated approximations by stochastic processes in discrete time and refer to the results from Chapter 4.
Wolfgang Karl Härdle, Wolfgang Karl Härdle, Christian M. Hafner, Jürgen Franke   +3 more
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Stochastic Differential Equations

2019
In this chapter, we consider the stochastic differential equations and backward stochastic differential equations driven by G-Brownian motion. The conditions and proofs of existence and uniqueness of a stochastic differential equation is similar to the classical situation.
Radek Erban, S. Jonathan Chapman
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Stochastic Differential Equations

2016
Let \(\mathbf{W} = (W^{1},\ldots,W^{m})\) be an m-dimensional Brownian motion, and let $$\displaystyle{\boldsymbol{\sigma }= (\sigma _{ij})_{1\leq i\leq d,1\leq j\leq m}: [0,\infty ) \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d} \times \mathbb{R}^{m}}$$ and $$\displaystyle{\boldsymbol{\mu }= (\mu ^{1},\ldots,\mu ^{d}): [0,\infty ) \times \
Setsuo Taniguchi, Hiroyuki Matsumoto
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