A Girsanov transformed Clark-Ocone-Haussmann type formula for \(L^1\)-pure jump additive processes and its application to portfolio optimization

Abstract

We derive a Clark-Ocone-Haussmann (COH) type formula under a change of measure for \( L^1 \)-canonical additive processes, providing a tool for representing financial derivatives under a risk-neutral probability measure. COH formulas are fundamental in stochastic analysis, providing explicit martingale representations of random variables in terms of their Malliavin derivatives. In mathematical finance, the COH formula under a change of measure is crucial for representing financial derivatives under a risk-neutral probability measure. To prove our main results, we use the Malliavin-Skorohod calculus in \( L^0 \) and \( L^1 \) for additive processes, as developed by Di Nunno and Vives (2017). An application of our results is solving the local risk minimization (LRM) problem in financial markets driven by pure jump additive processes. LRM, a prominent hedging approach in incomplete markets, seeks strategies that minimize the conditional variance of the hedging error. By applying our COH formula, we obtain explicit expressions for locally risk-minimizing hedging strategies in terms of Malliavin derivatives under the market model underlying the additive process. These formulas provide practical tools for managing risks in financial market price fluctuations with \(L^1\)-additive processes.