  In this paper we investigate classical solution of a semi-linear system of
backward stochastic integral partial differential equations driven by a
Brownian motion and a Poisson point process. By proving an It\^{o}-Wentzell
formula for jump diffusions as well as an abstract result of stochastic
evolution equations, we obtain the stochastic integral partial differential
equation for the inverse of the stochastic flow generated by a stochastic
differential equation driven by a Brownian motion and a Poisson point process.
By composing the random field generated by the solution of a backward
stochastic differential equation with the inverse of the stochastic flow, we
construct the classical solution of the system of backward stochastic integral
partial differential equations. As a result, we establish a stochastic
Feynman-Kac formula.
