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The Tikhonov Theory for SDEs

  • Yuri Kabanov
  • Sergei Pergamenshchikov
Chapter
Part of the Applications of Mathematics book series (SMAP, volume 49)

Abstract

Let us consider the following initial value problem for the system of ordinary differential equations
$$dx_t^\varepsilon = f\left( {t,x_t^\varepsilon ,y_t^\varepsilon } \right)dt,\;x_0^\varepsilon = {x^0}$$
(2.0.1)
$$\varepsilon dy_t^\varepsilon = F\left( {t,x_t^\varepsilon ,y_t^\varepsilon } \right)dt,\;y_0^\varepsilon = {y^0}$$
(2.0.2)
where the “slow” variable x takes values in R k and the “fast” variable y takes values in R n , ε ∈]0, 1] is a small parameter. The reduced problem corresponding to the formal substitution of the zero value of ε has the form
$$dx_t^0 = f\left( {t,x_t^0,y_t^0} \right)dt,\;x_0^0 = {x^0}$$
(2.0.3)
$$0 = F\left( {t,x_t^0,y_t^0} \right)$$
(2.0.4)

Keywords

Asymptotic Expansion Wiener Process Stochastic Approximation Fundamental Matrix Fast Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Yuri Kabanov
    • 1
  • Sergei Pergamenshchikov
    • 2
  1. 1.Département de MathématiquesUniversité de Franche-ComtéBesançon CedexFrance
  2. 2.LIFAR, UFR Sciences et TechniquesUniversité de RouenMont Saint Aignan CedexFrance

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