nonlinear Feynman-Kac formula

The nonlinear Feynman-Kac formula is the fundamental bridge between semilinear parabolic PDEs and backward stochastic differential equations (BSDEs). The classical (linear) Feynman-Kac formula gives a stochastic representation for solutions of linear Kolmogorov-type PDEs as expectations of functionals of forward diffusions. The nonlinear generalisation, developed by Pardoux and Peng (1992), extends this to semilinear PDEs by introducing the BSDE formulation.

Specifically, consider the semilinear parabolic PDE: u_t + mu . grad u + (1/2) Tr[sigma sigma^T Hess u] + f(t, x, u, sigma^T grad u) = 0 with terminal condition u(T,x) = g(x). If u is a solution of this PDE and X_t solves the forward SDE dX_t = mu(t,X_t) dt + sigma(t,X_t) dW_t, then defining Y_t = u(t,X_t) and Z_t = sigma(t,X_t)^T grad u(t,X_t), the triple (X_t, Y_t, Z_t) satisfies the forward-backward SDE system. Conversely, by the uniqueness of the BSDE solution (Pardoux-Peng 1992), solving the BSDE recovers the PDE solution. The formula is “nonlinear” because the PDE driver f can depend nonlinearly on u and grad u, unlike the classical case where the PDE is linear.

This equivalence is the theoretical foundation of the deep BSDE method: it converts the PDE problem into a stochastic control problem where one optimises over Y_0 and the control process Z_t to match the terminal condition. The nonlinearity of f allows the formula to cover important financial applications including nonlinear pricing with differential rates, XVA computation (see Gnoatto et al. 2023), and Hamilton-Jacobi-Bellman equations arising in optimal control.

Key Details

• Linear case: f = f(t,x) only, reduces to classical Feynman-Kac: u(0,x) = E[g(X_T)] (no BSDE needed)
• Semilinear case: f = f(t,x,y,z) depends on the solution and its gradient, requiring the full BSDE formulation
• Uniqueness: Pardoux and Peng (1992) showed existence and uniqueness of the BSDE solution (X_t, Y_t, Z_t) in R^d x R x R^d under appropriate regularity (878 citations)
• Interpretation: Y_t = u(t,X_t) is the PDE solution along the diffusion path; Z_t = sigma^T grad u encodes the hedging strategy in financial applications
• For XVA: the driver f encodes default intensities, funding spreads, and collateral terms, making the BSDE inherently nonlinear
• For Deep BSDE: instead of solving the high-dimensional PDE on a grid, one simulates paths of X forward and approximates Z via neural networks

Textbook References

Continuous-Time Stochastic Control and Optimization with Financial Applications (Pham, 2009)

• Proposition 6.3.2 (p. 145): If $v\in C^{1,2}$ is a classical solution to the semilinear PDE $-{\partial }_{t}v-\mathcal{L}v-f(t,x,v,{\sigma }^{\prime }{D}_{x}v)=0$, then $(Y_t,Z_t)=(v(t,X_t),{\sigma }^{\prime }(X_t)D_{x}v(t,X_t))$ solves the BSDE. Provides the classical-to-BSDE direction
• Theorem 6.3.3 (p. 145): The converse: $v(t,x):=Y_t^{t,x}$ is continuous on $[0,T]\times {\mathbb{R}}^n$ and is a viscosity solution to the semilinear PDE (6.13)-(6.14). Proof via Itô’s formula for the forward direction; viscosity subsolution/supersolution argument for the converse
• Section 6.7 (p. 169): Credits the connection between BSDEs and PDEs to Pardoux’s survey [Pa98]; extensions under relaxed conditions (quadratic growth in $z$) due to Kobylanski [Ko00]

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