Before revision:  In the portfolio optimization context, we obtain explicit constructions of value functions and investment strategies for investors with Constant Relative Risk Aversion (CRRA) utilities, built on the Hamilton-Jacobi-Bellman (HJB) framework developed in Capponi and Figueroa-Lopez (2011). We give a precise characterization of the investment strategies in terms of corporate bond returns, forward rates, and expected recovery at default, and illustrate the dependence of the optimal strategies on time, losses given default, and risk aversion level of the investor through a detailed economical analysis .
After revision:  We provide a detailed error analysis and illustrate the accuracy and computational complexity of our method on several market traded instruments, such as defaultable bond prices, barrier options, and vulnerable call options. Using again our Poisson series representation, we show differentiability in time of the pre-default price function of European vulnerable claims, which enables us to rigorously deduce Feynman-Kac representations for the pre-default pricing function and new semimartingale representations for the price process of the vulnerable claim under both risk-neutral and objective probability measures .