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

Covariance 

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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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