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The Lie symmetry algebra of the Longstaff-Schwartz model

Ouknine Anas

TL;DR

This work computes the Lie point symmetry groups of the Kolmogorov backward equation associated with the Longstaff-Schwartz model, a 1+2 dimensional parabolic PDE parameterized by $a,b,d,e$. By applying the infinitesimal symmetry criterion and a second prolongation, the authors derive a determining system that splits into four main parameter regimes, yielding explicit finite-dimensional symmetry generators and identifying an infinite-dimensional ideal generated by solutions $\mu$ of the PDE itself. They classify the resulting finite-dimensional algebras into isomorphism classes (e.g., $\mathfrak{sl}_2\ltimes\mathfrak{h}_3$, $\mathfrak{iso}_2$, and related semidirect products) and describe how these algebras sit inside larger structures via inclusions and central extensions. The paper also provides isomorphisms between the finite-dimensional symmetry algebras across cases and discusses the implications for both the PDE and its associated SDEs, including W-symmetries. Overall, the results offer a comprehensive symmetry-based taxonomy that can aid exact solutions, reductions, and comparisons with computer algebra outputs in financial modeling contexts.

Abstract

This study uses Lie's theory of symmetries to compute the symmetry group of a class of partial differential equations parameterized by four constants: $u_{t}=-\left((a-bx)u_{x}+(d-ey)u_{y}+\frac{x}{2}u_{xx}+\frac{y}{2}u_{yy}\right)$; under the various conditions on the constants $a,b,d$ and $e$, we deduce the largest and smallest Lie algebra of symmetries, and we also determined the structure of these algebras.

The Lie symmetry algebra of the Longstaff-Schwartz model

TL;DR

This work computes the Lie point symmetry groups of the Kolmogorov backward equation associated with the Longstaff-Schwartz model, a 1+2 dimensional parabolic PDE parameterized by . By applying the infinitesimal symmetry criterion and a second prolongation, the authors derive a determining system that splits into four main parameter regimes, yielding explicit finite-dimensional symmetry generators and identifying an infinite-dimensional ideal generated by solutions of the PDE itself. They classify the resulting finite-dimensional algebras into isomorphism classes (e.g., , , and related semidirect products) and describe how these algebras sit inside larger structures via inclusions and central extensions. The paper also provides isomorphisms between the finite-dimensional symmetry algebras across cases and discusses the implications for both the PDE and its associated SDEs, including W-symmetries. Overall, the results offer a comprehensive symmetry-based taxonomy that can aid exact solutions, reductions, and comparisons with computer algebra outputs in financial modeling contexts.

Abstract

This study uses Lie's theory of symmetries to compute the symmetry group of a class of partial differential equations parameterized by four constants: ; under the various conditions on the constants and , we deduce the largest and smallest Lie algebra of symmetries, and we also determined the structure of these algebras.
Paper Structure (49 sections, 6 theorems, 132 equations)

This paper contains 49 sections, 6 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Prerequisites and preliminaries
  3. Symmetry groups
  4. A Lie alebgra
  5. Equations Defining Infinitesimal Symmetries.
  6. Computing the Infinitesimal Symmetries.
  7. Case 1 ($a=\frac{1}{4},d= \frac{1}{4}$)
  8. Subcase 1.1 ($b\ne\pm e\,,\,a=\frac{1}{4},d= \frac{1}{4}$)
  9. Subcase 1.2 ($b=e\,,\,b\ne 0\,,\,a=\frac{1}{4},d=\frac{1}{4}$)
  10. Subcase 1.3$(b=-e\,,\,b\ne 0\,,\,a=\frac{1}{4},d=\frac{1}{4}$)
  11. Subcase 1.4$(b=e=0\,,\,a=\frac{1}{4},d=\frac{1}{4})$
  12. Case 2 ($a\ne\frac{1}{4},d\ne\frac{1}{4}$)
  13. Subcase 2.1 ($b\ne\pm e\,,\,a\ne\frac{1}{4},d\ne \frac{1}{4}$)
  14. Subcase 2.2($b=e\,,b\ne 0\,,\,a\ne\frac{1}{4},d\ne\frac{1}{4}$)
  15. Subcase 2.3 $(b=-e\,,\,b\ne 0\,,\,a\ne\frac{1}{4},d\ne\frac{1}{4})$
  16. ...and 34 more sections

Key Result

Theorem 2.1

OlverOLV86 Let $\mathbf{v}$ be a vector field of the form (Vector) on an open subset $M\subset X\times U$. Then, the $n$-th prolongation of $\mathbf{v}$ is the vector field: defined on the corresponding jet space $M^{(n)} \subset X \times U^{(n)}$, the second summation is over all multi-indices $J=\left(j_1, \ldots, j_k\right)$, with $1 \leqslant j_k \leqslant p, 1 \leqslant k \leqslant n$. The c

Theorems & Definitions (18)

  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Remark 2.6
  • Example 2.7: Heat equation
  • Remark 3.1
  • Lemma 3.2
  • proof
  • ...and 8 more