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Approximation methods for piecewise deterministic Markov processes and their costs. (English) Zbl 1411.91294

Summary: In this paper, we analyse piecewise deterministic Markov processes (PDMPs), as introduced in Davis (1984). Many models in insurance mathematics can be formulated in terms of the general concept of PDMPs. There one is interested in computing certain quantities of interest such as the probability of ruin or the value of an insurance company. Instead of explicitly solving the related integro-(partial) differential equation (an approach which can only be used in few special cases), we adapt the problem in a manner that allows us to apply deterministic numerical integration algorithms such as quasi-Monte Carlo rules; this is in contrast to applying random integration algorithms such as Monte Carlo. To this end, we reformulate a general cost functional as a fixed point of a particular integral operator, which allows for iterative approximation of the functional. Furthermore, we introduce a smoothing technique which is applied to the integrands involved, in order to use error bounds for deterministic cubature rules. We prove a convergence result for our PDMPs approximation, which is of independent interest as it justifies phase-type approximations on the process level. We illustrate the smoothing technique for a risk-theoretic example, and compare deterministic and Monte Carlo integration.

MSC:

91B30 Risk theory, insurance (MSC2010)
60J25 Continuous-time Markov processes on general state spaces
91G60 Numerical methods (including Monte Carlo methods)
65C05 Monte Carlo methods

Software:

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References:

[1] Albrecher, H.; Kainhofer, R., Risk theory with a nonlinear dividend barrier, Computing, 68, 4, 289-311 (2002) · Zbl 1076.91521 · doi:10.1007/s00607-001-1447-4
[2] Albrecher, H.; Lautscham, V., Dividends and the time of ruin under barrier strategies with a capital-exchange agreement, Anales del Instituo de Actuarios Espanoles, 21, 3, 1-30 (2015)
[3] Almudevar, A., A dynamic programming algorithm for the optimal control of piecewise deterministic Markov processes, SIAM Journal on Control and Optimization, 40, 2, 525-539 (2001) · Zbl 1061.93096 · doi:10.1137/S0363012999364474
[4] Asmussen, S.; Albrecher, H., Ruin probabilities (2010), Hackensack, NJ: World Scientific · Zbl 1247.91080
[5] Bakhvalov, N. S., On the approximate calculation of multiple integrals, Vestnik MGU, Series Mathematical, Mechanics & Astronomy Physical Chemistry, 4, 3-18 (1959)
[6] Bäuerle, N.; Rieder, U., Optimal control of piecewise deterministic Markov processes with finite time horizon, Modern Trends in Controlled Stochastic Processes: Theory and Applications, 123, 143 (2010)
[7] Bäuerle, N.; Rieder, U., Markov decision processes with applications to finance (2011), Heidelberg: Universitext, Springer · Zbl 1236.90004
[8] Cai, J.; Feng, R.; Willmot, G. E., On the expectation of total discounted operating costs up to default and its applications, Advances in Applied Probability, 41, 2, 495-522 (2009) · Zbl 1173.91023 · doi:10.1239/aap/1246886621
[9] Colaneri, K., Eksi, Z., Frey, R. & Szölgyenyi, M. (2017). Optimal liquidation under partial information with price impact. arXiv:1606.05079 · Zbl 1444.91196
[10] Costa, O. L.; Davis, M. H. A., Impulse control of piecewise-deterministic processes, Mathematics of Control, Signals, and Systems (MCSS), 2, 3, 187-206 (1989) · Zbl 0675.93077 · doi:10.1007/BF02551384
[11] Costa, O. L.; Dufour, F., Continuous average control of piecewise deterministic Markov processes (2013), New York: Springer, New York · Zbl 1213.60124
[12] Coulibaly, I.; Lefèvre, C., On a simple quasi-Monte Carlo approach for classical ultimate ruin probabilities, Insurance: Mathematics and Economics, 42, 3, 935-942 (2008) · Zbl 1141.91497
[13] Dassios, A.; Embrechts, P., Martingales and insurance risk, Communications in Statistics. Stochastic Models, 5, 2, 181-217 (1989) · Zbl 0676.62083 · doi:10.1080/15326348908807105
[14] Davis, M. H. A., Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models, Journal of the Royal Statistical Society Series B, 46, 3, 353-388 (1984) · Zbl 0565.60070
[15] Davis, M. H. A., Markov models and optimization (1993), London: Chapman & Hall, London · Zbl 0780.60002
[16] Davis, M. H. A. & Farid, M. (1999). Piecewise-deterministic processes and viscosity solutions. In W. M. McEneaney, G. George Yin, and Q. Zhang, eds., Stochastic Analysis, Control, Optimization and Applications. Boston: Springer. P. 249-268. · Zbl 0917.93071
[17] De Saporta, B.; Dufour, F.; Zhang, H., Numerical methods for simulation and optimization of piecewise deterministic markov processes (2016) · Zbl 1338.65021
[18] De Saporta, B.; Dufour, F.; Zhang, H.; Elegbede, C., Optimal stopping for the predictive maintenance of a structure subject to corrosion, Journal of Risk and Reliability, 226, 2, 169-181 (2012)
[19] Dempster, M. A. H.; Ye, J. J., Necessary and sufficient optimality conditions for control of piecewise deterministic Markov processes, Stochastics: An International Journal of Probability and Stochastic Processes, 40, 3-4, 125-145 (1992) · Zbl 0762.93080
[20] Eichler, A.; Leobacher, G.; Szölgyenyi, M., Utility indifference pricing of insurance catastrophe derivatives, European Actuarial Journal, 7, 2, 515-534 (2017) · Zbl 1405.91256 · doi:10.1007/s13385-017-0154-2
[21] Embrechts, P.; Schmidli, H., Ruin estimation for a general insurance risk model, Advances in Applied Probability, 26, 2, 404-422 (1994) · Zbl 0811.62096 · doi:10.2307/1427443
[22] Ethier, S. N.; Kurtz, T. G., Markov processes (1986), New York: John Wiley & Sons, Inc., New York · Zbl 0592.60049
[23] Forwick, L.; Schäl, M.; Schmitz, M., Piecewise deterministic Markov control processes with feedback controls and unbounded costs, Acta Applicandae Mathematica, 82, 3, 239-267 (2004) · Zbl 1084.49027 · doi:10.1023/B:ACAP.0000031200.76583.75
[24] Grigorian, A. (2009). Ordinary differential equation. Lecture notes.
[25] Hinrichs, A.; Novak, E.; Ullrich, M.; Woźniakowski, H., Product rules are optimal for numerical integration in classical smoothness spaces, Journal of Complexity, 38, 39-49 (2017) · Zbl 1354.65043 · doi:10.1016/j.jco.2016.09.001
[26] Jacobsen, M., Point process theory and applications (2006), Boston, MA: Birkhäuser Boston, Inc, Boston, MA
[27] Joe, S.; Kuo, F. Y., Constructing Sobol’ sequences with better two-dimensional projections, SIAM Journal of Scientific Computation, 30, 2635-2654 (2008) · Zbl 1171.65364 · doi:10.1137/070709359
[28] Kallenberg, O., Foundations of modern probability (2002), New York: Springer-Verlag, New York · Zbl 0996.60001
[29] Kamke, E., Differentialgleichungen. I. Gewöhnliche differentialgleichungen (1964), Leipzig: Akademische Verlagsgesellschaft, Leipzig
[30] Kritzer, P.; Pillichshammer, F.; Wasilkowski, G. W., Very low truncation dimension for high dimensional integration under modest error demand, Journal of Complexity, 35, 63-85 (2016) · Zbl 1342.65098 · doi:10.1016/j.jco.2016.02.002
[31] Kuo, F. Y. (n.d.). F. Y. Kuo’s homepage. . Last visited 14/12/2017.
[32] Kuo, F. Y.; Sloan, I. H.; Wasilkowski, G. W.; Woźniakowski, H., Liberating the dimension, Journal of Complexity, 26, 422-454 (2010) · Zbl 1203.65057 · doi:10.1016/j.jco.2009.12.003
[33] Kurtz, T. G. & Protter, P. E. (1996). Weak convergence of stochastic integrals and differential equations I. In D. Talay and L. Tubaro, eds., Probabilistic Models for Nonlinear Partial Differential Equations. Berlin, Heidelberg: Springer. · Zbl 0862.60041
[34] Lenhart, S.; Liaot, Y., Integro-differential equations associated with optimal stopping time of a piecewise-deterministic process, Stochastics: An International Journal of Probability and Stochastic Processes, 15, 3, 183-207 (1985) · Zbl 0582.60053 · doi:10.1080/17442508508833356
[35] Leobacher, G.; Ngare, P., Utility indifference pricing of derivatives written on industrial loss indexes, Journal of Computational and Applied Mathematics, 300, 68-82 (2016) · Zbl 1331.91100 · doi:10.1016/j.cam.2015.11.028
[36] Niederreiter, H. (1992). Random number generation and quasi-Monte Carlo methods. CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia: SIAM. · Zbl 0761.65002
[37] Owen, A. B. (2000). Monte Carlo, quasi-Monte Carlo, and randomized quasi-Monte Carlo. In H. Niederreiter and J. Spanier, eds., Monte Carlo and Quasi- Monte Carlo Methods 1998. Springer. P. 86-97. · Zbl 0942.65024
[38] Pausinger, F.; Svane, A. M., A Koksma-Hlawka inequality for general discrepancy systems, Journal of Complexity, 31, 773-793 (2015) · Zbl 1377.11084 · doi:10.1016/j.jco.2015.06.002
[39] Preischl, M., Thonhauser, S. & Tichy, R. F. (2018). Integral equations, quasi-monte carlo methods and risk modeling. In J. Dick, F. Y. Kuo and H. Woźniakowski, eds., Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, Vol. 1, 2. Cham: Springer. P. 1051-1074. · Zbl 1405.65177
[40] Riedler, M. G., Almost sure convergence of numerical approximations for piecewise deterministic Markov processes, Journal of Computational and Applied Mathematics, 239, 50-71 (2013) · Zbl 1255.65028 · doi:10.1016/j.cam.2012.09.021
[41] Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J., Stochastic processes for insurance and finance (1999), New York: John Wiley & Sons, New York · Zbl 0940.60005
[42] Schäl, M., On piecewise deterministic Markov control processes: Control of jumps and of risk processes in insurance, Insurance: Mathematics and Economics, 22, 1, 75-91 (1998) · Zbl 0906.90170
[43] Siegl, T.; Tichy, R. F., Ruin theory with risk proportional to the free reserve and securitization, Insurance: Mathematics and Economics, 26, 1, 59-73 (2000) · Zbl 0946.91025
[44] Sloan, I. H.; Woźniakowski, H., When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?, Journal of Complexity, 14, 1-33 (1998) · Zbl 1032.65011 · doi:10.1006/jcom.1997.0463
[45] Tichy, R. F., Über eine zahlentheoretische Methode zur numerischen Integration und zur Behandlung von Integralgleichungen, Osterreichische Akademie der Wissenschaften Mathematisch-Naturwissenschaftliche Klasse. Sitzungsberichte. Abteilung II, 193, 4-7, 329-358 (1984) · Zbl 0549.65008
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