A Galerkin approximation scheme for the mean correction in a mean-reversion stochastic differential equation

JiangLun Wu, Wei Yang

Research output: Working paper

Abstract

This paper is concerned with the following Markovian stochastic
dierential equation of mean-reversion type
dRt = ( + (Rt; t))Rtdt + RtdBt
with an initial value R0 = r0 2 R, where 2 R and > 0 are constants, and the mean correction function : R [0; 1) 7! (x; t) 2 R
is twice continuously dierentiable in x and continuously dierentiable
in t. We rst derive that under the assumption of path indepen-
dence of the density process of Girsanov transformation for the above
stochastic dierential equation, the mean correction function sat-
ises a non-linear partial dierential equation which is known as the
viscous Burgers equation. We then develop a Galerkin type approxi-
mation scheme for the function by utilizing truncation of discretised
Fourier transformation to the viscous Burgers equation.
LanguageEnglish
Number of pages16
Publication statusPublished - 2013

Fingerprint

Mean Reversion
Galerkin Approximation
Approximation Scheme
Stochastic Equations
Differential equation
Burgers Equation
Girsanov Transformation
Truncation
Galerkin
Partial
Path

Keywords

  • galerkin approximation scheme
  • mean correction
  • stochastic differential equation
  • mean-reversion
  • markovian stochastic differential equation of mean-revision type
  • viscous burgers equation
  • truncation of (discretised) fourier transformation
  • numerical approximation scheme

Cite this

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title = "A Galerkin approximation scheme for the mean correction in a mean-reversion stochastic differential equation",
abstract = "This paper is concerned with the following Markovian stochasticdierential equation of mean-reversion typedRt = ( + (Rt; t))Rtdt + RtdBtwith an initial value R0 = r0 2 R, where 2 R and > 0 are constants, and the mean correction function : R [0; 1) 7! (x; t) 2 Ris twice continuously dierentiable in x and continuously dierentiablein t. We rst derive that under the assumption of path indepen-dence of the density process of Girsanov transformation for the abovestochastic dierential equation, the mean correction function sat-ises a non-linear partial dierential equation which is known as theviscous Burgers equation. We then develop a Galerkin type approxi-mation scheme for the function by utilizing truncation of discretisedFourier transformation to the viscous Burgers equation.",
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T1 - A Galerkin approximation scheme for the mean correction in a mean-reversion stochastic differential equation

AU - Wu, JiangLun

AU - Yang, Wei

PY - 2013

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N2 - This paper is concerned with the following Markovian stochasticdierential equation of mean-reversion typedRt = ( + (Rt; t))Rtdt + RtdBtwith an initial value R0 = r0 2 R, where 2 R and > 0 are constants, and the mean correction function : R [0; 1) 7! (x; t) 2 Ris twice continuously dierentiable in x and continuously dierentiablein t. We rst derive that under the assumption of path indepen-dence of the density process of Girsanov transformation for the abovestochastic dierential equation, the mean correction function sat-ises a non-linear partial dierential equation which is known as theviscous Burgers equation. We then develop a Galerkin type approxi-mation scheme for the function by utilizing truncation of discretisedFourier transformation to the viscous Burgers equation.

AB - This paper is concerned with the following Markovian stochasticdierential equation of mean-reversion typedRt = ( + (Rt; t))Rtdt + RtdBtwith an initial value R0 = r0 2 R, where 2 R and > 0 are constants, and the mean correction function : R [0; 1) 7! (x; t) 2 Ris twice continuously dierentiable in x and continuously dierentiablein t. We rst derive that under the assumption of path indepen-dence of the density process of Girsanov transformation for the abovestochastic dierential equation, the mean correction function sat-ises a non-linear partial dierential equation which is known as theviscous Burgers equation. We then develop a Galerkin type approxi-mation scheme for the function by utilizing truncation of discretisedFourier transformation to the viscous Burgers equation.

KW - galerkin approximation scheme

KW - mean correction

KW - stochastic differential equation

KW - mean-reversion

KW - markovian stochastic differential equation of mean-revision type

KW - viscous burgers equation

KW - truncation of (discretised) fourier transformation

KW - numerical approximation scheme

M3 - Working paper

BT - A Galerkin approximation scheme for the mean correction in a mean-reversion stochastic differential equation

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