The Picard boundary value problem for a third order stochastic difference equation
Summary: It is considered the multidimensional third-order stochastic difference equation \[ \Delta^3X_{n-2}= f(X_n)+ \xi_n,\quad n\in\{2,\dots,N-1\},\quad N\geq 5, \] where \(X_i\in\mathbb{R}^d\), \(d\geq 1\), and \(\{\xi_i\}\) is a sequence of \(d\)-dimensional independent random vectors, with the Picard boundary condition \[ X_0= a_0,\quad X_1=a_1,\quad X_N= a_N,\quad a_i\in\mathbb{R}^d,\quad i=0,1,N. \] We first prove that the boundary value problem admits a unique solution if \(f\) is a monotone application. Moreover, we are able to compute the density of the law of the solution if the random vectors \(\{\xi_i\}\) are absolutely continuous. Thanks to this explicit computation, in the scalar case we prove that the process \(\{(X_i,\Delta X_i,\Delta^2X_i): i=0,\dots,N-2\}\) is a Markov chain if and only if \(f\) is affine and we provide a simple counterexample to show that a similar strong condition does not hold in the multidimensional case.
scientific article; zbMATH DE number 1021761