A NOTE ON REAL-WORLD AND RISK-NEUTRAL DYNAMICS FOR HEATH–JARROW–MORTON FRAMEWORKS

We show that for time-inhomogeneous Markovian Heath–Jarrow–Morton models driven by an infinite-dimensional Brownian motion and a Poisson random measure an equivalent change of measure exists whenever the real-world and the risk-neutral dynamics can be defined uniquely and are related via a drift and a jump condition.

A Semimartingale Bellman Equation and the Variance-Optimal Martingale Measure

We consider a financial market model, where the dynamics of asset prices is given by an Rm -valued continuous semimartingale. Using the dynamic programming approach we obtain an explicit description of the variance optimal martingale measure in terms of the value process of a suitable problem of an optimal equivalent change of measure and show that this value process uniquely solves the corresponding semimartingale backward equation. This result is applied to prove the existence of a unique generalized solution of Bellman's equation for stochastic volatility models, which is used to determine the variance-optimal martingale measure.