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Symmetry Analysis of Semi-Linear Partial Differential Equations and Forward Backward Stochastic Differential Equations

Anas Ouknine, Paul Lescot

TL;DR

This work analyzes Lie symmetries of a class of semi-linear PDEs that arise from forward-backward SDEs via a generalized Feynman-Kac representation. It employs Olver's prolongation and projectable time-dependent vector fields to derive symmetry-determining equations for the PDE and defines symmetry for FBSDEs using a time-change framework, linking the stochastic and deterministic sides. Under the assumption that the generator $g$ is independent of $z$, the FBSDE symmetry space is contained in the PDE symmetry algebra, with potential equality in special cases; when $g$ depends on $z$ in general, symmetries may not align, though reductions via transformations can restore z-independence. The heat equation example, along with Girsanov and quadratic-$z$ transformations, demonstrates how PDE and FBSDE symmetries coincide or broaden, and points to future work on jumps and integro-differential PDEs.

Abstract

We examine the Lie symmetries of a semi-linear partial differential equations and their connections to the analogous symmetries of the forward-backward stochastic differential equations (FBSDEs), established through the generalized Feynman-Kac formula.

Symmetry Analysis of Semi-Linear Partial Differential Equations and Forward Backward Stochastic Differential Equations

TL;DR

This work analyzes Lie symmetries of a class of semi-linear PDEs that arise from forward-backward SDEs via a generalized Feynman-Kac representation. It employs Olver's prolongation and projectable time-dependent vector fields to derive symmetry-determining equations for the PDE and defines symmetry for FBSDEs using a time-change framework, linking the stochastic and deterministic sides. Under the assumption that the generator is independent of , the FBSDE symmetry space is contained in the PDE symmetry algebra, with potential equality in special cases; when depends on in general, symmetries may not align, though reductions via transformations can restore z-independence. The heat equation example, along with Girsanov and quadratic- transformations, demonstrates how PDE and FBSDE symmetries coincide or broaden, and points to future work on jumps and integro-differential PDEs.

Abstract

We examine the Lie symmetries of a semi-linear partial differential equations and their connections to the analogous symmetries of the forward-backward stochastic differential equations (FBSDEs), established through the generalized Feynman-Kac formula.
Paper Structure (12 sections, 9 theorems, 68 equations)

This paper contains 12 sections, 9 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Some Facts about FBSDE and their Symmetries
  3. FBSDE
  4. Relation between FBSDE and semi-linear PDE
  5. Symmetries of FBSDE
  6. Symmetries of the semi-linear PDE
  7. Symmetry of FBSDE versus symmetry of semi linear PDE
  8. Application
  9. Heat equation
  10. Application of Girsanov transformation
  11. Quadratic case
  12. Conclusion

Key Result

Proposition 2.1

carmona2016lectures Let us assume that the function $(t, x) \longrightarrow u(t, x)$ is jointly continuous on $[0, T] \times \mathbb{R}^d$, continuously differentiable in $t$, and twice continuously differentiable in $x$, the first derivative $\partial_x u(t, x)$ being of polynomial growth in $x$. L Then $(\mathbf{Y}, \mathbf{Z})=\left(Y_t, Z_t\right)_{t \in[0, T]}$, defined by is the unique solu

Theorems & Definitions (20)

  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • Theorem 2.7
  • Remark 2.8
  • Theorem 2.9
  • ...and 10 more