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AN IMPROVEMENT OF MARKOVIAN INTEGRATION BY PARTS FORMULA AND APPLICATION TO SENSITIVITY COMPUTATION

Published online by Cambridge University Press:  05 January 2021

Yue Liu Open the ORCID record for Yue Liu [Opens in a new window] ,
Zhiyan Shi Open the ORCID record for Zhiyan Shi [Opens in a new window] ,
Ying Tang ,
Jingjing Yao  and
Xincheng Zhu
Yue Liu
Affiliation:
School of Geography, Nanjing Normal University, Nanjing, Jiangsu, China School of Finance and Economics, Jiangsu University, Zhenjiang, Jiangsu, China
Zhiyan Shi
Affiliation:
School of Mathematical Science, Jiangsu University, Zhenjiang, Jiangsu, China E-mail: shizhiyan1984@126.com
Ying Tang
Affiliation:
School of Mathematical Science, Jiangsu University, Zhenjiang, Jiangsu, China E-mail: shizhiyan1984@126.com
Jingjing Yao
Affiliation:
School of Finance and Economics, Jiangsu University, Zhenjiang, Jiangsu, China
Xincheng Zhu
Affiliation:
Department of Computer and Mathematics, Arcadia University, Glenside, Pennsylvania19038, USA School of Mathematical Science, Jiangsu University, Zhenjiang, Jiangsu, China
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Abstract

This paper establishes a new version of integration by parts formula of Markov chains for sensitivity computation, under much lower restrictions than the existing researches. Our approach is more fundamental and applicable without using Girsanov theorem or Malliavin calculus as did by past papers. Numerically, we apply this formula to compute sensitivity regarding the transition rate matrix and compare with a recent research by an IPA (infinitesimal perturbation analysis) method and other approaches.

Keywords

integration by parts formulaMarkov chainssensitivity computation
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 2 , April 2022 , pp. 357 - 377
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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References

Biane, Ph. (1989). Chaotic representations for finite Markov chains. Stochastics and Stochastic Reports 30: 61–68.CrossRefGoogle Scholar
Bismut, J.M. (1983). Calcul des variations stochastique et processus de sauts. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 63: 147–235.CrossRefGoogle Scholar
Cao, X.R. (2007). Stochastic learning and optimization: a sensitivity-based approach. Hong Kong, China: Springer.CrossRefGoogle Scholar
Cao, X.R. & Wan, Y.W. (1998). Algorithms for sensitivity analysis of Markov systems through potentials and perturbation realization. IEEE Transactions on Control Systems Technology 6(4): 482–494.Google Scholar
Cohen, S.N. (2012). Chaos representations for marked point processes. Communications on Stochastic Analysis 6(2): 263–279.CrossRefGoogle Scholar
Davis, M.H.A. & Johansson, M.P. (2006). Malliavin Monte Carlo Greeks for jump diffusions. Stochastic Processes and Their Applications 116(1): 101–129.CrossRefGoogle Scholar
Debelley, V. & Privault, N. (2004). Sensitivity analysis of European options in jump-diffusion models via the Malliavin calculus on the Wiener space. Preprint.Google Scholar
Denis, L. & Nguyen, T.M. (2016). Malliavin calculus for Markov chains using perturbations of time. Stochastics 88(6): 813–840.CrossRefGoogle Scholar
Di Nunno, G. & Sjursen, S. (2013). On chaos representation and orthogonal polynomials for the doubly stochastic Poisson process. In Seminar on stochastic analysis, random fields and applications VII. Progr. Probab. 6. Basel: Springer, pp. 23–54.CrossRefGoogle Scholar
Fournié, E., Lasry, J.M., Lebuchoux, J., Lions, P.L., & Touzi, N. (1999). Applications of Malliavin calculus to Monte Carlo methods in finance. Finance and Stochastics 3(4): 391–412.Google Scholar
Fu, M.C. & Hu, J.Q. (1995). Sensitivity analysis for Monte Carlo simulation of option pricing. Probability in the Engineering and Informational Sciences 9(3): 417–446.CrossRefGoogle Scholar
Fu, M.C. & Hu, J.Q. (1999). Efficient design and sensitivity analysis of control charts using Monte Carlo simulation. Management Science 45(3): 395–413.CrossRefGoogle Scholar
Glasserman, P. (1991). Gradient estimation via perturbation analysis. Boston: Kluwer Academic Publishers.Google Scholar
Heidergott, B., Leahu, H., Lopker, A., & Pflug, G. (2016). Perturbation analysis of inhomogeneous finite Markov chains. Advances in Applied Probability 48(1): 255–273.CrossRefGoogle Scholar
Ho, Y.C. & Cao, X.R. (1991). Discrete event dynamic systems and perturbation analysis. Boston, MA: Kluwer Academic Publishers.CrossRefGoogle Scholar
Kawai, R. & Takeuchi, A. (2011). Greeks formulas for an asset price model with gamma processes. Mathematical Finance 24(4): 723–742.Google Scholar
Kroeker, J.P. (1980). Wiener analysis of functionals of a Markov chain: application to neural transformations of random signals. Biological Cybernetics 36(4): 243–248.CrossRefGoogle Scholar
Liu, Y. & Privault, N. (2017). An integration by parts formula in a Markovian regime switching model and application to sensitivity analysis. Stochastic Analysis and Applications 35(5): 919–940.CrossRefGoogle Scholar
Privault, N. (2013). Understanding Markov chains - examples and applications. Springer Undergraduate Mathematics Series, x+354 pp.CrossRefGoogle Scholar
Privault, N. & Schoutens, W. (2002). Discrete chaotic calculus and covariance identities. Stochastics and Stochastic Reports 72: 289–315.CrossRefGoogle Scholar
Rubinstein, R.Y. & Shapiro, A. (1993). Discrete event systems: sensitivity analysis and stochastic optimization by the score function method. New York: Wiley.Google Scholar
Siu, T.K. (2014). Integration by parts and martingale representation for a Markov chain. Abstract and Applied Analysis 2014(1): 189–221.CrossRefGoogle Scholar