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Extremal behavior of hitting a cone by correlated Brownian motion with drift

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  • Dȩbicki, Krzysztof
  • Hashorva, Enkelejd
  • Ji, Lanpeng
  • Rolski, Tomasz

Abstract

This paper derives an exact asymptotic expression for Pxu{∃t≥0X(t)−μt∈U},asu→∞,where X(t)=(X1(t),…,Xd(t))⊤,t≥0 is a correlated d-dimensional Brownian motion starting at the point xu=−αu with α∈Rd, μ∈Rd and U=∏i=1d[0,∞). The derived asymptotics depends on the solution of an underlying multidimensional quadratic optimization problem with constraints, which leads in some cases to dimension-reduction of the considered problem. Complementary, we study asymptotic distribution of the conditional first passage time to U, which depends on the dimension-reduction phenomena.

Suggested Citation

  • Dȩbicki, Krzysztof & Hashorva, Enkelejd & Ji, Lanpeng & Rolski, Tomasz, 2018. "Extremal behavior of hitting a cone by correlated Brownian motion with drift," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4171-4206.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:12:p:4171-4206
    DOI: 10.1016/j.spa.2018.02.002
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    References listed on IDEAS

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    1. Hüsler, Jürg & Piterbarg, Vladimir, 2008. "A limit theorem for the time of ruin in a Gaussian ruin problem," Stochastic Processes and their Applications, Elsevier, vol. 118(11), pages 2014-2021, November.
    2. Dȩbicki, Krzysztof & Hashorva, Enkelejd & Ji, Lanpeng & Tabiś, Kamil, 2015. "Extremes of vector-valued Gaussian processes: Exact asymptotics," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4039-4065.
    3. Debicki, K. & Kosinski, K.M. & Mandjes, M. & Rolski, T., 2010. "Extremes of multidimensional Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2289-2301, December.
    4. Samorodnitsky, Gennady, 1991. "Probability tails of Gaussian extrema," Stochastic Processes and their Applications, Elsevier, vol. 38(1), pages 55-84, June.
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    Cited by:

    1. Krzysztof Dȩbicki & Lanpeng Ji & Tomasz Rolski, 2019. "Logarithmic Asymptotics for Probability of Component-Wise Ruin in a Two-Dimensional Brownian Model," Risks, MDPI, vol. 7(3), pages 1-21, August.
    2. Remco Hofstad & Harsha Honnappa, 2019. "Large deviations of bivariate Gaussian extrema," Queueing Systems: Theory and Applications, Springer, vol. 93(3), pages 333-349, December.
    3. Dȩbicki, Krzysztof & Hashorva, Enkelejd & Wang, Longmin, 2020. "Extremes of vector-valued Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5802-5837.
    4. Hashorva, Enkelejd, 2019. "Approximation of some multivariate risk measures for Gaussian risks," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 330-340.
    5. Ling, Chengxiu, 2019. "Asymptotics of multivariate conditional risk measures for Gaussian risks," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 205-215.

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