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 Levy 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 Levy measure of the subordinator. When T is a d-dimensional additive subordinator, Y is a time-inhomogeneous Markov process. When Tⁱ= ∫₀^tV_sⁱ ds with Vⁱ forming a multi-variate 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-dependent 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 multi-name unified credit-equity models and correlated commodity models