Abstract
In this paper we develop a class of applied probabilistic continuous time but discretized state space decompositions of the characterization of a multivariate generalized diffusion process. This decomposition is novel and, in particular, it allows one to construct families of mimicking classes of processes for such continuous state and continuous time diffusions in the form of a discrete state space but continuous time Markov chain representation. Furthermore, we present this novel decomposition and study its discretization properties from several perspectives. This class of decomposition both brings insight into understanding locally in the state space the induced dependence structures from the generalized diffusion process as well as admitting computationally efficient representations in order to evaluate functionals of generalized multivariate diffusion processes, which is based on a simple rank one tensor approximation of the exact representation. In particular, we investigate aspects of semimartingale decompositions, approximation and the martingale representation for multidimensional correlated Markov processes. A new interpretation of the dependence among processes is given using the martingale approach. We show that it is possible to represent, in both continuous and discrete space, that a multidimensional correlated generalized diffusion is a linear combination of processes originated from the decomposition of the starting multidimensional semimartingale. This result not only reconciles with the existing theory of diffusion approximations and decompositions, but defines the general representation of infinitesimal generators for both multidimensional generalized diffusions and, as we will demonstrate, also for the specification of copula density dependence structures. This new result provides immediate representation of the approximate weak solution for correlated stochastic differential equations. Finally, we demonstrate desirable convergence results for the proposed multidimensional semimartingales decomposition approximations.
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References
Billingsley P (1999) Convergence of Probability Measures, 2nd Edn. Wiley Series in Probability and Statistics
Brunick G, Shreve S (2013) Mimicking an Ito process by a solution of a stochastic differential equation. Ann Appl Probab 23(4):1584–1628
Brunick G (2013) Uniqueness in law for a class of degenerate diffusions with continuous covariance. Probab Theory Related Fields 155:265–302. MR3010399
Ekström E, Hobson D, Janson S, Tysk J (2013) Can time-homogeneous diffusions produce any distribution?. Probab Theory Related Fields 155:493–520. MR3034785
Ethier ST, Kurtz TG (1985) Markov Processes, Characterization and Convergence. Wiley
Gyöngy I (1986) Mimicking the one-dimensional marginal distributions of processes having an ito differential. Probab Theory Relat Fields 71(4):501–516
Hackbusch W (2012) Tensor Spaces and Numerical Tensor Calculus. Springer Series in Computational Mathematics
Karatzas I, Shreve S (2000) Brownian Motion and Stochastic Calculus - 2nd Edn. Springer
Kushner HJ, Dupuis P (2001) Numerical Methods for Stochastic Control Problems in Continuous Time - 2nd Edn, Springer Science & Business Media
Lukacs E (1958) Characteristic Functions. Charles Griffin & Company Limited London
McNeil AJ, Frey R, Embrechts P (2005) Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton Series in Finance
Nelsen RB (1999) An Introduction to Copulas. Springer, New York. ISBN 0-387-98623-5
Portenko NI (1982) Generalized Diffusion Processes. American Mathematical Society
Rogers LCG, Williams D (2000) Diffusions, Markov Processes and Martingales. Cambridge University Press
Scarsini M (1984) On measures of concordance. Stochastica 8(3):201-218
Sklar A (1996) Random Variables, Distribution Functions, and Copulas: a personal look backward and forward. Lecture notes-monograph series, JSTOR
Stroock DW, Srinivasa Varadhan SR (1997) Multidimensional Diffusion Processes. Springer
Tavella D, Randall C (2000) Pricing Financial Instrument with the Finite Difference Method. Whiley Financial Engneering
Zhang H, Ding F (2013) On theKronecker Products and Their Applications. J Appl Math 2013:8. Article ID 296185
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Dalessandro, A., Peters, G.W. Tensor Approximation of Generalized Correlated Diffusions and Functional Copula Operators. Methodol Comput Appl Probab 20, 237–271 (2018). https://doi.org/10.1007/s11009-017-9545-8
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DOI: https://doi.org/10.1007/s11009-017-9545-8