# zbMATH — the first resource for mathematics

On logarithmically optimal exact simulation of max-stable and related random fields on a compact set. (English) Zbl 1428.62426
Summary: We consider the random field $M(t)=\sup_{n\geq1}\{-\log A_n+X_n(t)\},\qquad t\in T,$ for a set $$T\subset\mathbb{R}^m$$, where $$(X_n)$$ is an i.i.d. sequence of centered Gaussian random fields on $$T$$ and $$0<A_1<A_2<\cdots$$ are the arrivals of a general renewal process on $$(0,\infty)$$, independent of $$(X_n)$$. In particular, a large class of max-stable random fields with Gumbel marginals have such a representation. Assume that one needs $$c(d)=c(\{t_1,\ldots,t_d\})$$ function evaluations to sample $$X_n$$ at $$d$$ locations $$t_1,\ldots,t_d\in T$$. We provide an algorithm which samples $$M(t_1),\ldots,M(t_d)$$ with complexity $$O(c(d)^{1+o(1)})$$ as measured in the $$L_p$$ norm sense for any $$p\ge1$$. Moreover, if $$X_n$$ has an a.s. converging series representation, then $$M$$ can be a.s. approximated with error $$\delta$$ uniformly over $$T$$ and with complexity $$O(1/(\delta\log(1/\delta))^{1/\alpha})$$, where $$\alpha$$ relates to the Hölder continuity exponent of the process $$X_n$$ (so, if $$X_n$$ is Brownian motion, $$\alpha=1/2)$$.

##### MSC:
 60G70 Extreme value theory; extremal stochastic processes 60G60 Random fields 60-08 Computational methods for problems pertaining to probability theory
Full Text:
##### References:
 [1] Adler, R.J. and Taylor, J.E. (2007). Random Fields and Geometry. Springer Monographs in Mathematics. New York: Springer. · Zbl 1149.60003 [2] Asmussen, S. (2003). Applied Probability and Queues: Stochastic Modelling and Applied Probability, 2nd ed. Applications of Mathematics (New York) 51. New York: Springer. · Zbl 1029.60001 [3] Asmussen, S., Glynn, P. and Pitman, J. (1995). Discretization error in simulation of one-dimensional reflecting Brownian motion. Ann. Appl. Probab.5 875-896. · Zbl 0853.65147 [4] Asmussen, S. and Glynn, P.W. (2007). Stochastic Simulation: Algorithms and Analysis. Stochastic Modelling and Applied Probability57. New York: Springer. · Zbl 1126.65001 [5] Ayache, A. and Taqqu, M.S. (2003). Rate optimality of wavelet series approximations of fractional Brownian motion. J. Fourier Anal. Appl.9 451-471. · Zbl 1050.60043 [6] Blanchet, J. and Chen, X. (2015). Steady-state simulation of reflected Brownian motion and related stochastic networks. Ann. Appl. Probab.25 3209-3250. · Zbl 1332.60120 [7] Blanchet, J., Chen, X. and Dong, J. (2017). $$\varepsilon$$ -strong simulation for multidimensional stochastic differential equations via rough path analysis. Ann. Appl. Probab.27 275-336. · Zbl 1436.65012 [8] Blanchet, J. and Wallwater, A. (2015). Exact sampling of stationary and time-reversed queues. ACM Trans. Model. Comput. Simul.25 Art. 26. · Zbl 1384.90025 [9] Blanchet, J.H. and Sigman, K. (2011). On exact sampling of stochastic perpetuities. J. Appl. Probab.48A 165-182. · Zbl 1230.65012 [10] Brown, B.M. and Resnick, S.I. (1977). Extreme values of independent stochastic processes. J. Appl. Probab.14 732-739. · Zbl 0384.60055 [11] de Haan, L. (1984). A spectral representation for max-stable processes. Ann. Probab.12 1194-1204. · Zbl 0597.60050 [12] de Haan, L. and Zhou, C. (2008). On extreme value analysis of a spatial process. REVSTAT6 71-81. · Zbl 1153.62074 [13] Dieker, A.B. and Mikosch, T. (2015). Exact simulation of Brown-Resnick random fields at a finite number of locations. Extremes18 301-314. · Zbl 1319.60108 [14] Dombry, C., Engelke, S. and Oesting, M. (2016). Exact simulation of max-stable processes. Biometrika103 303-317. Available at arXiv:1506.04430. · Zbl 07072113 [15] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Berlin: Springer. · Zbl 0873.62116 [16] Gut, A. (2009). Stopped Random Walks: Limit Theorems and Applications, 2nd ed. Springer Series in Operations Research and Financial Engineering. New York: Springer. · Zbl 1166.60001 [17] Kabluchko, Z., Schlather, M. and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. Ann. Probab.37 2042-2065. · Zbl 1208.60051 [18] Kenealy, B. (2013, August 11). New York’s MTA buys $$200 million cat bond to avoid storm surge losses. Bus. Insur. Available at https://www.businessinsurance.com$$ [19] Oesting, M., Schlather, M. and Zhou, C. (2018). Exact and fast simulation of max-stable processes on a compact set using the normalized spectral representation. Bernoulli24 1497-1530. · Zbl 1431.60042 [20] Pollock, M., Johansen, A.M. and Roberts, G.O. (2016). On the exact and $$\varepsilon$$ -strong simulation of (jump) diffusions. Bernoulli22 794-856. · Zbl 1343.60099 [21] Schilling, R.L. and Partzsch, L. (2012). Brownian Motion: An Introduction to Stochastic Processes. Berlin: de Gruyter. With a chapter on simulation by Björn Böttcher. · Zbl 1258.60002 [22] Schlather, M. (2002). Models for stationary max-stable random fields. Extremes5 33-44. · Zbl 1035.60054 [23] Steele, J.M. (2001). Stochastic Calculus and Financial Applications. Applications of Mathematics (New York) 45. New York: Springer. · Zbl 0962.60001 [24] Thibaud, E., Aalto, J., Cooley, D.S., Davison, A.C. and Heikkinen, J. (2016). Bayesian inference for the Brown-Resnick process, with an application to extreme low temperatures. Ann. Appl. Stat.10 2303-2324. Available at arXiv:1506.07836. · Zbl 1454.62462
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.