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LOWER TAIL INDEPENDENCE OF HITTING TIMES OF TWO-DIMENSIONAL DIFFUSIONS

Published online by Cambridge University Press:  18 September 2017

David Saunders ,
Lung Kwan Tsui  and
Satish Iyengar
David Saunders
Affiliation:
Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada E-mail: [email protected]
Lung Kwan Tsui
Affiliation:
Independent Model Review, HSBC, Toronto, Canada E-mail: [email protected]
Satish Iyengar
Affiliation:
Department of Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania, USA E-mail: [email protected]
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Abstract

The coefficient of tail dependence is a quantity that measures how extreme events in one component of a bivariate distribution depend on extreme events in the other component. It is well known that the Gaussian copula has zero tail dependence, a shortcoming for its application in credit risk modeling and quantitative risk management in general. We show that this property is shared by the joint distributions of hitting times of bivariate (uniformly elliptic) diffusion processes.

Keywords

hitting timeslarge deviationsSchilder's theoremsmall time
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 32 , Issue 4 , October 2018 , pp. 522 - 535
Copyright
Copyright © Cambridge University Press 2017 

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References

1.Azencott, R. (1985). Petites perturbations aléatoires des systèmes dynamiques: Développements asymptotiques. Bulletin des Sciences Mathématiques. 2e Série 109(3): 253–308.Google Scholar
2.Baldi, P. & Chaleyat-Maurel, M. (1988). An extension of Ventsel-Freidlin estimates. In Korezlioglu, H. & Ustunel, A.S. (eds.), Stochastic analysis and related topics (Silivri, 1986), number 1316 in Lecture Notes in Mathematics. Berlin, Heidelberg, New York: Springer, pp. 305–327.Google Scholar
3.Balkema, G. & Embrechts, P. (2007). High risk scenarios and extremes: a geometric approach. Zurich: European Mathematical Society.Google Scholar
4.Bass, R.F. (1998). Diffusions and elliptic operators. New York: Springer.Google Scholar
5.Bielecki, T. & Rutkowski, M. (2002). Credit risk: modeling, valuation and hedging. Berlin, Heidelberg: Springer.Google Scholar
6.Black, F. & Cox, J. (1976). Valuing corporate securities: Some effects of bond indenture provisions. Journal of Finance 31(2): 351–367.Google Scholar
7.Clarke, F. (2013). Functional analysis, calculus of variations and optimal control. London: Springer.Google Scholar
8.Dębicki, K., Kosiński, K.M., Mandjes, M., & Rolski, T. (2010). Extremes of multidimensional Gaussian processes. Stochastic Processes and their Applications 120: 2289–2301.Google Scholar
9.do Carmo, M.P. (1992). Riemannian geometry. Boston: Birkhäuser.Google Scholar
10.Elliott, R.J., Siu, T.K., & Yang, H. (2012). A PDE approach to multivariate risk theory. In Zhang, T. & Zhou, X. (eds.), Stochastic analysis and applications to finance: essays in honour of Jia-an Yan. Singapore: World Scientific, pp. 111–124.Google Scholar
11.Evans, L.C. (2010). Partial differential equations, 2nd ed. USA: American Mathematical Society.Google Scholar
12.Figalli, A. & Villani, C. (2011). Optimal transport and curvature. In Ambrosio, L. & Savaré, G. (eds.), Nonlinear PDE's and applications, number 2028 in Lecture Notes in Mathematics. Heidelberg: Springer, pp. 171–217.Google Scholar
13.Fleming, W.H. & James, M.R. (1992). Asymptotic series and exit time probabilities. Annals of Probability 20(3): 1369–1384.Google Scholar
14.Freidlin, M.I. & Wentzell, A.D. (2012). Random perturbations of dynamical systems, 3rd ed. Heidelberg: Springer.Google Scholar
15.Friedman, A. (1964). Partial differential equations of parabolic type. Englewood Cliffs, NJ: Prentice–Hall.Google Scholar
16.Friedman, A. (2006). Stochastic differential equations and applications: two volumes bound as one. Mineral, NY: Dover.Google Scholar
17.Furman, E., Kuznetsov, A., Su, J., & Zitkis, R. (2016). Tail dependence of the Gaussian copula revisited. Insurance: Mathematics and Economics 69: 97–103.Google Scholar
18.Herrmann, S., Imkeller, P., & Peithmann, D. (2006). Transition times and stochastic resonance for multidimensional diffusions with time periodic drift: A large deviations approach. Annals of Applied Probability 16(4): 1851–1892.Google Scholar
19.Hua, L. & Joe, H. (2011). Tail order and intermediate tail dependence for multivariate copulas. Journal of Multivariate Analysis 102: 1454–1471.Google Scholar
20.Iyengar, S. (1985). Hitting lines with two-dimensional Brownian motion. SIAM Journal on Applied Mathematics 45(6): 983–989.Google Scholar
21.Ledford, A.W. & Tawn, J.A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83(1): 169–187.Google Scholar
22.Li, D. (2001). On default correlation: A copula function approach. Journal of Fixed Income 9: 43–54.Google Scholar
23.Li, H. (2009). Orthant tail dependence of multivariate extreme value distributions. Journal of Multivariate Analysis 100(1): 243–256.Google Scholar
24.McNeil, A.J., Frey, R., & Embrechts, P. (2005). Quantitative risk management. Princeton, NJ: Princeton University Press.Google Scholar
25.Metzler, A. (2008). Multivariate first-passage models in credit risk. PhD thesis, University of Waterloo.Google Scholar
26.Resnick, S.I. (2007). Heavy-tail phenomena: probabilistic and statistical modeling. New York: Springer.Google Scholar
27.Sacerdote, L., Tamborrino, M., & Zucca, C. (2014). First passage times of two-dimensional correlated diffusion processes: Analytical and numerical methods. Available at arxiv.org, arXiv:1212.5287.Google Scholar
28.Varadhan, S.R.S. (1967). Diffusion processes in a small time interval. Communications on Pure and Applied Mathematics 20: 659–685.Google Scholar