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MULTIVARIATE SUBORDINATION OF MARKOV PROCESSES WITH FINANCIAL APPLICATIONS
Author(s) -
MendozaArriaga Rafael,
Linetsky Vadim
Publication year - 2016
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/mafi.12061
Subject(s) - subordinator , markov chain , markov process , multivariate statistics , econometrics , markov renewal process , mathematics , markov model , markov property , subordination (linguistics) , computer science , lévy process , statistics , linguistics , philosophy
This paper develops the procedure of multivariate subordination for a collection of independent Markov processes with killing. Starting from d independent Markov processes X i with killing and an independent d ‐dimensional time change T , we construct a new process by time, changing each of the Markov processes X i with a coordinateT i . When T is a d ‐dimensional Lévy subordinator, the time changed process ( Y i : = X i ( T i ( t ) )is a time‐homogeneous Markov process with state‐dependent jumps and killing in the product of the state spaces of X i . 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. WhenT i = ∫ 0 t V s i d s with V i forming a multivariate Markov process, ( Y i , V i ) is a Markov process, where each V i plays a role of stochastic volatility of Y i . 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.