Power Forward Performance in Semimartingale Markets with Stochastic Integrated Factors

We study the forward investment performance process (FIPP) in an incomplete semimartingale market model with closed and convex portfolio constraints, when the investor’s risk preferences are of the power form. We provide necessary and sufficient conditions for the existence of such a FIPP. In a semimartingale factor model, we show that the FIPP can be recovered as a triplet of processes that admit an integral representation with respect to semimartingales. Using an integrated stochastic factor model, we relate the factor representation of the triplet of processes to the smooth solution of an ill-posed partial integro-differential Hamilton–Jacobi–Bellman equation. We develop explicit constructions for the class of time-monotone FIPPs, generalizing existing results from Brownian to semimartingale market models. Funding: L. Bo was supported by the National Natural Science Foundation of China (NSFC) [Grant 11971368] and National Center for Applied Mathematics in Shaanxi (NCAMS). A. Capponi was supported in part by the National Science Foundation [Grant DMS-1716145]. C. Zhou was supported by the Singapore Ministry of Education Academic Research Fund [Grant R-146-000-271-112] and NSFC [Grant 11871364].