### Abstract

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.

Original language | English |
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Number of pages | 16 |

Publication status | Published - 2013 |

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### 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

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

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**A Galerkin approximation scheme for the mean correction in a mean-reversion stochastic differential equation.** / Wu, JiangLun; Yang, Wei.

Research output: Working paper

TY - UNPB

T1 - A Galerkin approximation scheme for the mean correction in a mean-reversion stochastic differential equation

AU - Wu, JiangLun

AU - Yang, Wei

PY - 2013

Y1 - 2013

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

ER -