Stochastic differential equations with boundary conditions driven by a {P}oisson noise (Q1767518)

The paper is devoted to the study of the stochastic flow (initial condition problem) associated with the stochastic differential equation \[ X_t= X_0+ \int_0^t f(r,X_{r^-})\,dr+\int_0^t F(r,X_{r^-})\,dN_r,\quad X_0=\psi(X_1), \quad t\in [0,1], \tag{1} \] where \(f\), \(F : [0, 1]\times \mathbb R\to\mathbb R\) and \(\psi :\mathbb R\to\mathbb R\) are measurable functions satisfying certain hypotheses, and \(N = \{N_t,t\geq 0\}\) is a Poisson process with intensity 1. Due to the boundary condition, the solution will anticipate any filtration to which \(N\) is adapted, and therefore the stochastic integral appearing in the equation is, strictly speaking, an anticipating integral. However, the bounded variation character of the Poisson process permits to avoid most of the technical difficulties of the anticipating stochastic integrals with respect to the Wiener process. The authors study existence, uniqueness, regularity and absolute continuity of the solution to the problem (1). The case of linear equations is studied as a special example. The authors find some sufficient conditions for the solution of the linear equation to enjoy the reciprocal property. It is established the relation of the forward equation (1) with the backward equation \[ X_t= X_0+ \int_0^t f(r,X_{r})\,dr+\int_0^t F(r,X_{r})\,dN_r,\quad X_0=\psi(X_1), \quad t\in[0,1], \tag{2} \] and the Skorokhod-type equation \[ X_t= X_0+ \int_0^t b(r,X_{r})\,dr+\int_0^t B(r,X_{r})\,\delta\tilde N_r, \quad X_0=\psi(X_1),\quad t\in [0,1], \tag{3} \] where \(\delta\tilde N_r\) denotes the Skorokhod integral with respect to the compensated Poisson process. While the stochastic integrals in (1) and (2) are no more than Stieltjes integrals, the Skorokhod integral operator is defined by means of the chaos decomposition on the canonical Poisson space. The linear backward and Skorokhod equations are again considered with particular attention, and the chaos decompositions of the solutions are computed.