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MULTIVARIATE SUBORDINATION OF MARKOV PROCESSES WITH FINANCIAL APPLICATIONS

This paper develops the procedure of multivariate subordination for a collection of independent Markov processes with killing. Starting from d independent Markov processes Xi with killing and an independent d‐dimensional time change T , we construct a new process by time, changing each of the Markov processes Xi with a coordinate Ti . When T is a d‐dimensional Lévy subordinator, the time changed process (Yi:=Xi(Ti(t)) is a time‐homogeneous Markov process with state‐dependent jumps and killing in the product of the state spaces of Xi . The dependence among jumps of its components is governed by the d‐dimensional Lévy measure of the subordinator. When T is a d‐dimensional additive subordinator, Y is a time‐inhomogeneous Markov process. When Ti=∫0tVsids with Vi forming a multivariate Markov process, (Yi,Vi) is a Markov process, where each Vi plays a role of stochastic volatility of Yi . This construction provides a rich modeling architecture for building multivariate models in finance with time‐ and state‐dependent jumps, stochastic volatility, and killing (default). The semigroup theory provides powerful analytical and computational tools for securities pricing in this framework. To illustrate, the paper considers applications to multiname unified credit‐equity models and correlated commodity models.