Abstract
We consider a surplus process of a drifted fractional Brownian motion with the Hurst index \(H>1/2\), which appears as a functional limit of drifted compound Poisson risk models with correlated claims. This is a kind of representation of a surplus with a long memory. Our interest is to construct confidence intervals of the ruin probability of the surplus when the volatility parameter is unknown. We obtain the derivative of the ruin probability w.r.t. the volatility parameter via the Malliavin calculus, and apply the delta method to identify the asymptotic distribution of an estimated ruin probability.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities. 2nd ed. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.
Billingsley, P. (1999). Convergence of Probability Measures. 2nd ed. John Wiley & Sons, New York.
Cai, C. and **ao, W. (2021). Simulation of an integro-differential equation and application in estimation of ruin probability with mixed fractional Brownian motion. J. Integral Equations Applications 33 1–17.
Corcuera, J. M., Nualart, D and Woerner, J.H.C. (2006). Power variation of some integral fractional processes. Bernoulli 12, 713–735.
Florit, C. and Nualart, D. (1995). A local criterion for smoothness of densities and application to the supremum of the Brownian sheet. Statistics and Probability Letters 22 25–31.
Gerber, H.U., Shiu, E.S.W. (1998). On the time value of ruin. N. Am. Actuar. 2 48–72.
Gobet, E. and Kohatsu-Higa, A. (2003). Computation of Greeks for barrier and look-back options using Malliavin calculus. Electron. Comm. Probab 8 51–62.
Ji, L. and Robert, R. (2018). Ruin problem of a two-dimensional fractional Brownian motion risk process. Stoch. Models 34 73–97.
Lundberg, F. (1903). Approximerad framställning av sannolikhetsftmktionen. Aterforsäknng av kollektivrisker. Akad Afhandling. Almqvist och Wiksell, Uppsala.
Michna, Z. (1998). Self-similar processes in collective risk theory, J. Appl. Math. Stochastic Anal. 11 429–448.
Norros, I., Valkeila, E. and Virtamo, J. (1999). An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motion. Bernoulli 5 571–587.
Nualart, D. (1995). Malliavin Calculus and Related Topics. (Probability and its Applications). Berlin Heidelberg New York Springer.
Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1994). Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science, Yverdon.
Shimizu, Y. (2021), Asymptotic Statistics in Insurance Risk Theory. Springer Briefs in Statistics.
van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. With Applications to Statistics. Springer Series in Statistics. Springer-Verlag, New York.
Zaïdi, N.L. and Nualart, D. (2003). Smoothness of the law of the supremum of the fractional Brownian motion. Electron. Comm. Probab. 8 102–111.
Acknowledgements
The authors express sincere thanks to Prof. A. Kohats-Higa for the valuable discussion related to the part of Malliavin calculus. This research was partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (C) #21K03358.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Nakamura, S., Shimizu, Y. (2023). Estimating Finite-Time Ruin Probability of Surplus with Long Memory via Malliavin Calculus. In: Liu, Y., Hirukawa, J., Kakizawa, Y. (eds) Research Papers in Statistical Inference for Time Series and Related Models. Springer, Singapore. https://doi.org/10.1007/978-981-99-0803-5_20
Download citation
DOI: https://doi.org/10.1007/978-981-99-0803-5_20
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-99-0802-8
Online ISBN: 978-981-99-0803-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)