[Paper Review] Space-Distribution PDEs for Path Independent Additive Functionals of McKean-Vlasov SDEs

Panpan Ren, Feng‐Yu Wang|arXiv (Cornell University)|May 28, 2018

Stochastic processes and financial applications13 references4 citations

TL;DR

This paper establishes a class of space-distribution PDEs on $\mathbb{R}^d \times \mathscr{P}_2(\mathbb{R}^d)$ involving standard and Lions' $L$-derivatives to characterize path independence of additive functionals for McKean-Vlasov SDEs. It proves that such path independence is equivalent to solving a nonlinear PDE, with solutions constructed probabilistically, recovering classical Girsanov transformation results as special cases.

ABSTRACT

Let P2(Rd) be the space of probability measures on Rd with finite second moment. The path independence of additive functionals of McKean-Vlasov SDEs is characterized by PDEs on the product space Rd*P2(Rd) equipped with the usual derivative in space variable and Lions derivative in distribution. These PDEs are solved by using probabilis- tic arguments developed from [2]. In particular, the path independence of the Girsanov transformation killing the drift term is identified with a nonlinear PDE on Rd*P2(Rd), which includes corresponding results derived earlier for the classical SDEs as special situations.

Motivation & Objective

• To characterize the path independence of additive functionals in McKean-Vlasov SDEs driven by both drift and diffusion terms.
• To identify conditions under which Girsanov transformations for these SDEs yield path-independent densities.
• To extend classical path independence results for standard SDEs to the distribution-dependent setting using PDEs on product space $\mathbb{R}^d \times \mathscr{P}_2(\mathbb{R}^d)$.
• To establish a connection between path independence and solutions of nonlinear PDEs involving both spatial and distributional derivatives.

Proposed method

• Formulates the path independence condition as a PDE on $\mathbb{R}^d \times \mathscr{P}_2(\mathbb{R}^d)$ with standard derivative in space and Lions' $L$-derivative in distribution.
• Uses probabilistic arguments, including time-changed processes and expectations over conditional distributions, to construct solutions to the PDEs.
• Applies the Feynman-Kac formula in a distributional setting to represent solutions as conditional expectations of terminal functions.
• Relies on the existence and uniqueness of solutions to McKean-Vlasov SDEs under Lipschitz-type conditions on drift and diffusion coefficients.
• Establishes equivalence between path independence and the solvability of a nonlinear PDE involving $\partial_t + \mathbf{L}_{\sigma,b}$ and quadratic terms in the gradient.
• Applies transformation techniques, such as $V = -\beta \log \tilde{V}$, to relate solutions of linear and nonlinear PDEs in the context of Girsanov changes of measure.

Experimental results

Research questions

• RQ1 Under what conditions is the additive functional $A_{s,t}^{\mathbf{f},\mathbf{g}}$ of a McKean-Vlasov SDE path independent?
• RQ2 How can path independence be characterized via a PDE on the product space $\mathbb{R}^d \times \mathscr{P}_2(\mathbb{R}^d)$ involving both spatial and distributional derivatives?
• RQ3 What is the connection between path independence of Girsanov transformations and solutions of nonlinear PDEs in the McKean-Vlasov framework?
• RQ4 How do classical results on path independence in standard SDEs emerge as special cases of the proposed framework?

Key findings

• Path independence of additive functionals $A_{s,t}^{\mathbf{f},\mathbf{g}}$ is equivalent to the existence of a function $V \in C^{1,2,(1,1)}([0,T] \times \mathbb{R}^d \times \mathscr{P}_2(\mathbb{R}^d))$ solving the PDE $(\partial_t + \mathbf{L}_{\sigma,b})V = -\frac{1}{2}|\sigma^* \partial_x V|^2 + \mathbf{f}$.
• Solutions to the PDE are constructed probabilistically via $V(t,x,\mu) = \mathbb{E}[\Phi(X_{t,T}^{x,\mu}, P_{t,T}^*\mu)] - \mathbb{E}\left[\int_t^T \mathbf{f}(r,X_{t,r}^{x,\mu}, P_{t,r}^*\mu) dr\right]$.
• The Girsanov transformation killing the drift term is path independent if and only if $b(t,x,\mu) = \sigma(t,x,\mu)\sigma^*(t,x,\mu) \partial_x V(t,x,\mu)$, which corresponds to solving a nonlinear PDE with $\mathbf{f} = \frac{1}{2}|\sigma^* \partial_x V|^2$.
• The classical path independence result for standard SDEs is recovered as a special case when the distribution dependence vanishes, i.e., $\mu = \delta_x$.
• For the case $\mathbf{f} = \frac{1}{2}|\mathbf{g}|^2$, the path independence of the functional $A_{s,t}^{\mathbf{g}}$ is equivalent to solving the PDE $\partial_t V + \mathbf{L}_{\sigma,b} V = \frac{1}{2}|\sigma^* \partial_x V|^2$.
• The transformation $V = -\beta \log \tilde{V}$ links the nonlinear PDE to a linear PDE $\partial_t \tilde{V} + \mathbf{L}_{\sigma,b} \tilde{V} = 0$, enabling probabilistic solution representation.