Polynomial approximation of discounted moments

We introduce an approximation strategy for the discounted moments of a stochastic process that can approximate the true moments for a large class of problems. These moments appear in pricing formulas of financial products such as bonds and credit derivatives. The approximation relies on a high-order power series expansion of the infinitesimal generator and draws parallels with the theory of polynomial processes. We demonstrate applications to bond pricing and credit derivatives. In the special cases that allow an analytical solution, the approximation error decreases to around 10 to 100 times machine precision for higher orders. When no analytical solution exists, we numerically compare the approximation with existing numerical techniques.