Existence and uniqueness of solutions for forward and backward nonlocal Fokker-Planck equations with time-dependent coefficients (Q6559408)

The authors consider the forward nonlocal Fokker-Planck equation: \( dX_{t}=f(X_{t},t)dt+g(X_{t},t)dB(t)+d\xi (t)\), with the initial value \( X_{0}=x_{0}\in \mathbb{R}^{d}\), where \(f:\mathbb{R}^{d}\times \lbrack 0,T]\rightarrow \mathbb{R}^{d}\), \(g:\mathbb{R}^{d}\times \lbrack 0,T]\rightarrow \mathbb{R}^{d}\times \mathbb{R}^{d}\), \(\xi(t)\) is a mutually independent \(d\)-dimensional Lévy process, which is independent of the \(d\)-dimensional Brownian motion \(B\). They start with the symmetric case, assuming that \(\xi (t)\) has the generating triplet \((0,0,\nu )\) where \(\nu \) is a symmetric Lévy measure, \(\nu (D)=\nu (-D)\) for each \( D\in B(\mathbb{R}^{d}\setminus \{0\})\), which is independent of the \(d\) -dimensional Brownian motion \(B\). Assuming that for each \(t\in \lbrack 0,T]\) , \(f(x,t)\in C^{1}(\mathbb{R}^{d};\mathbb{R}^{d})\) and \(g(x,t)\in C^{1}( \mathbb{R}^{d};\mathbb{R}^{d}\times \mathbb{R}^{d})\), the authors prove that the probability density function for the preceding stochastic system in the sense of Cauchy principle value satisfies: \(\frac{\partial p(x,t)}{\partial t }=\widehat{\mathcal{Q}}^{\ast }p(x,t)\), \(p(x,0)=p_{0}(x)\), for some operator \(\widehat{\mathcal{Q}}^{\ast }\) whose expression is explicitly given in terms of the data through functions \(\mathcal{M}_{i}(x,t)\) and \(\mathcal{G} (x,t)\) which involve the components of the matrix \(gg^{T}\). They here essentially use Itô's formula. They derive a backward nonlocal Fokker-Planck equation for \(\widetilde{p}(x,\widetilde{t})=-p(x,T-\widetilde{ t})\). Under the same regularity properties as above, the authors also prove that if \(\widehat{X}_{s}\) is an Itô process which is the solution to the equation \(d\widehat{X}_{s}=\widetilde{\mathcal{M}}(\widehat{X}_{s},s)ds+ \widetilde{g}(\widehat{X}_{s},s)dB(s)+d\xi (s)\), \(\widetilde{t}\leq s\leq T\) , with the initial condition \(\widehat{X}_{\widetilde{t}}=x\), where \( \widetilde{g}(\widehat{X}_{s},s)=g(\widehat{X}_{T-s},T-s)\), and if \( \widetilde{p}(x,\widetilde{t})\in C^{1}([0,T];C_{b}^{2}(\mathbb{R}^{d}; \mathbb{R}))\) solves the associated equation with the terminal value \( \widetilde{p}(x,T)=\chi (x)\), and \(\widetilde{\mathcal{G}},\widetilde{p}, \widetilde{g}\), together with the first-order derivatives of \(\widetilde{p}\) in \(x\), are bounded on \(\mathbb{R}^{d}\times \lbrack 0,T]\), then \(\widetilde{ p}(x,\widetilde{t})\) admits the probabilistic representation \(\widetilde{p} (x,\widetilde{t})=\mathbb{E}[\exp (\int_{\widetilde{t}}^{T}\widetilde{ \mathcal{G}}(\widetilde{X}_{r},r)dr)\chi (\widetilde{X}_{T})\mid \widehat{X} _{\widetilde{t}}=x]\). The authors again use Itô's formula and direct computations. They derive an integral representation for \(p(x,t)\). They prove the existence of a solution to the backward nonlocal Fokker-Planck equation, that of a solution to the forward nonlocal Fokker-Planck equation and uniqueness results for these two equations. Moving to the asymmetric case, the equation becomes: \(dX_{t}=f(X_{t},t)dt+g(X_{t},t)dB(t)+d\xi _{\beta }(t)\), where \(\xi _{\beta }(t)\) is the mutually independent \(d\) -dimensional asymmetric Lévy process with the generating triplet \( (0,0,\nu _{\beta })\) where \(\nu _{\beta }\) is an asymmetric Lévy measure and \(\beta \in \lbrack -1,1]\) is the skewness parameter, which is independent of \(d\)-dimensional Brownian motion \(B\). They prove existence and uniqueness results in this case. The paper ends with the presentation of numerical simulations in both cases.