Galerkin Approximation of Mean Correction in a Mean-Reverting SDE

Summary

The paper studies a Markovian stochastic differential equation with mean-reverting dynamics and a time-dependent correction term. Under the assumption that the density process from a Girsanov transformation is path independent, it derives a nonlinear partial differential equation for that correction function: the viscous Burgers equation.

It then proposes a Galerkin approximation scheme, applying a truncated discretized Fourier transform to approximate solutions of the equation. This connects a condition on a stochastic model’s change of measure to a numerical approach for estimating its correction function. The supplied description gives no convergence analysis, numerical experiments, or trading application, so it establishes the method’s outline but not its practical accuracy or market usefulness.

Key ideas

• The model is a mean-reverting stochastic differential equation with a correction function.
• Path independence of the Girsanov density leads to a viscous Burgers equation for that function.
• A Galerkin scheme approximates the solution using a truncated discretized Fourier transform.
• The description does not report numerical validation or a trading use case.

Tags

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


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









This paper is concerned with the following Markovian stochastic differential equation of mean-reversion type \[ dR_t= (θ+σα(R_t, t))R_t dt +σR_t dB_t \] with an initial value $R_0=r_0\in\mathbb{R}$, where $θ\in\mathbb{R}$ and $σ>0$ are constants, and the mean correction function $α:\mathbb{R}\times[0,\infty)\to α(x,t)\in\mathbb{R}$ is twice continuously differentiable in $x$ and continuously differentiable in $t$. We first derive that under the assumption of path independence of the density process of Girsanov transformation for the above stochastic differential equation, the mean correction function $α$ satisfies a non-linear partial differential equation which is known as the viscous Burgers equation. We then develop a Galerkin type approximation scheme for the function $α$ by utilizing truncation of discretised Fourier transformation to the viscous Burgers equation.

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