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Symmetry classification and invariant solutions of the classical geometric mean reversion process

Jin Zhang, Dapeng Gao

TL;DR

This work applies Lie symmetry analysis to the Feynman-Kac PDE associated with the classical geometric mean reversion process, identifying a rich symmetry structure that depends on model parameters. It derives the full Lie algebra of infinitesimal symmetries, constructs an optimal system of one-dimensional subalgebras, and performs symmetry reductions to produce closed-form invariant solutions expressed through special functions. The resulting invariant solutions provide exact pricing- and value-related formulas for the GMR dynamics and offer insights for applications in discounted project values and commodity pricing. Overall, the paper delivers a systematic symmetry-based framework for solving and interpreting the GMR pricing PDE with nonnegative, mean-reverting dynamics.

Abstract

Based on the Lie symmetry method, we investigate a Feynman-Kac formula for the classical geometric mean reversion process, which effectively describing the dynamics of short-term interest rates. The Lie algebra of infinitesimal symmetries and the corresponding one-parameter symmetry groups of the equation are obtained. An optimal system of invariant solutions are constructed by a derived optimal system of one-dimensional subalgebras. Because of taking into account a supply response to price rises, this equation provides for a more realistic assumption than the geometric Brownian motion in many investment scenarios.

Symmetry classification and invariant solutions of the classical geometric mean reversion process

TL;DR

This work applies Lie symmetry analysis to the Feynman-Kac PDE associated with the classical geometric mean reversion process, identifying a rich symmetry structure that depends on model parameters. It derives the full Lie algebra of infinitesimal symmetries, constructs an optimal system of one-dimensional subalgebras, and performs symmetry reductions to produce closed-form invariant solutions expressed through special functions. The resulting invariant solutions provide exact pricing- and value-related formulas for the GMR dynamics and offer insights for applications in discounted project values and commodity pricing. Overall, the paper delivers a systematic symmetry-based framework for solving and interpreting the GMR pricing PDE with nonnegative, mean-reverting dynamics.

Abstract

Based on the Lie symmetry method, we investigate a Feynman-Kac formula for the classical geometric mean reversion process, which effectively describing the dynamics of short-term interest rates. The Lie algebra of infinitesimal symmetries and the corresponding one-parameter symmetry groups of the equation are obtained. An optimal system of invariant solutions are constructed by a derived optimal system of one-dimensional subalgebras. Because of taking into account a supply response to price rises, this equation provides for a more realistic assumption than the geometric Brownian motion in many investment scenarios.

Paper Structure

This paper contains 5 sections, 4 theorems, 47 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Symmetry analysis
  3. Optimal system of one-dimensional symmetry algebras
  4. Symmetry reductions and exact solutions
  5. Conclusion

Key Result

Proposition 2.1

For the case $\alpha=0$ and $\lambda=\frac{k^2}{2\sigma^2}$, the Lie algebra of infinitesimal symmetries of (eq1) is spanned by the six vector fields and the infinite-dimensional subalgebra $V_{\varphi}=\varphi(t,x)\partial_u$, where $\varphi(t,x)$ is an arbitrary solution of (eq1).

Figures (7)

  • Figure 1: Three different dynamical structures of (\ref{['inv2']}) for $k = 0.5$, $c_1 = 2$, $c_2 = -1$.
  • Figure 2: Three different dynamical structures of (\ref{['inv3']}) for $k = 1$, $c_1 = 2$, $c_2 = -1$.
  • Figure 3: Three different dynamical structures of (\ref{['inv4']}) for $k = 1$, $c = 1$.
  • Figure 4: Three different dynamical structures of (\ref{['inv5']}) for $a = 2$, $k = 1$, $c_1 = 2$, $c_2 = 1$.
  • Figure 5: Three different dynamical structures of (\ref{['inv7-1']}) for $c_1 =-1$, $c_2 = 1$, $k = 1$.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 3.1
  • Theorem 4.1