Lanpeng Ji

On the supremum of γ-reflected processes with fractional Brownian motion as input

Abstract Let { X H ( t ) , t ≥ 0 } be a fractional Brownian motion with Hurst index H ∈ ( 0 , 1 ] and define a γ -reflected process W γ ( t ) = X H ( t ) − c t − γ inf s ∈ [ 0 , t ] ( X H ( s ) − c s ) , t ≥ 0 with c > 0 , γ ∈ [ 0 , 1 ] two given constants. In this paper we establish the exact tail asymptotic behaviour of M γ ( T ) = sup t ∈ [ 0 , T ] W γ ( t ) for any T ∈ ( 0 , ∞ ] . Furthermore, we derive the exact tail asymptotic behaviour of the supremum of certain non-homogeneous mean-zero Gaussian random fields.

Extremes of a class of nonhomogeneous Gaussian random fields

This contribution establishes exact tail asymptotics of sup(s,t)∈E X(s,t) for a large class of nonhomogeneous Gaussian random fields X on a bounded convex set E⊂R2, with variance function that attains its maximum on a segment on E. These findings extend the classical results for homogeneous Gaussian random fields and Gaussian random fields with unique maximum point of the variance. Applications of our result include the derivation of the exact tail asymptotics of the Shepp statistics for stationary Gaussian processes, Brownian bridge and fractional Brownian motion as well as the exact tail asymptotic expansion for the maximum loss and span of stationary Gaussian processes.

The Gerber-Shiu penalty functions for two classes of renewal risk processes

In this paper, we study the Gerber-Shiu functions for a risk model with two independent classes of risks. We suppose that both of the two claim number processes are renewal processes with phase-type inter-claim times. By re-composing and analyzing the Markov chains associated with two given phase-type distributions, we obtain systems of integro-differential equations for two types of Gerber-Shiu functions. Explicit expressions for the Laplace transforms of the two types of Gerber-Shiu functions are established, respectively. And explicit results for the Gerber-Shiu functions are derived when the initial surplus is zero and when the two claim amount distributions are both from the rational family. Finally, an example is considered to illustrate the applicability of our main results.

Gaussian risk models with financial constraints

In this paper, we investigate Gaussian risk models which include financial elements, such as inflation and interest rates. For some general models for inflation and interest rates, we obtain an asymptotic expansion of the finite-time ruin probability for Gaussian risk models. Furthermore, we derive an approximation of the conditional ruin time by an exponential random variable as the initial capital tends to infinity.

Extremes of ()-locally stationary Gaussian random fields

This contribution derives the exact asymptotic behaviour of the supremum of alpha(t)-locally stationary Gaussian random fields over a finite hypercube. We present two applications of our result; the first one deals with extremes of ggregate multifractional Brownian motions, whereas the second application establishes the exact asymptotics of the supremum of chi-processes generated by multifractional Brownian motions.

On the probability of conjunctions of stationary Gaussian processes

Let {Xi(t),t≥0},1≤i≤n be independent centered stationary Gaussian processes with unit variance and almost surely continuous sample paths. For given positive constants u,T, define the set of conjunctions C[0,T],u≔{t∈[0,T]:min1≤i≤nXi(t)≥u}. Motivated by some applications in brain mapping and digital communication systems, we obtain exact asymptotic expansion of P{C[0,T],u≠ϕ}, as u→∞. Moreover, we establish the Berman sojourn limit theorem for the random process {min1≤i≤nXi(t),t≥0} and derive the tail asymptotics of the supremum of each order statistics process.

Extremes and First Passage Times of Correlated Fractional Brownian Motions

Let {Xi(t), t ⩾ 0}, i = 1, 2 be two standard fractional Brownian motions being jointly Gaussian with constant cross-correlation. In this paper, we derive the exact asymptotics of the joint survival function as u → ∞. A novel finding of this contribution is the exponential approximation of the joint conditional first passage times of X1, X2. As a by-product, we obtain generalizations of the Borell-TIS inequality and the Piterbarg inequality for 2-dimensional Gaussian random fields.

Extremes of locally stationary chi-square processes with trend

Abstract Chi-square processes with trend appear naturally as limiting processes in various statistical models. In this paper we are concerned with the exact tail asymptotics of the supremum taken over ( 0 , 1 ) of a class of locally stationary chi-square processes with particular admissible trends. An important tool for establishing our results is a weak version of Slepian’s lemma for chi-square processes. Some special cases including squared Brownian bridge and Bessel process are discussed.

Random shifting and scaling of insurance risks

Random shifting typically appears in credibility models whereas random scaling is often encountered in stochastic models for claim sizes reflecting the time-value property of money. In this article we discuss some aspects of random shifting and random scaling of insurance risks focusing in particular on credibility models, dependence structure of claim sizes in collective risk models, and extreme value models for the joint dependence of large losses. We show that specifying certain actuarial models using random shifting or scaling has some advantages for both theoretical treatments and practical applications.

Asymptotics of the Finite-time Ruin Probability for the Sparre Andersen Risk Model Perturbed by an Inflated Stationary Chi-process

In this article, we consider the Sparre Andersen risk model that is perturbed by an inflated chi-process with non-negative random inflator R. Under some conditions on the perturbation and the random inflator, which allow for both small and large fluctuations, exact asymptotic behaviour of the finite-time ruin probability is obtained when initial reserve tends to infinity.