Exact asymptotics of component-wise extrema of two-dimensional Brownian motion

We derive the exact asymptotics of ${\mathbb {P} \left \{ \underset {t\ge 0}{\sup } \left (X_{1}(t) - \mu _{1} t\right )> u, \ \underset {s\ge 0}{\sup } \left (X_{2}(s) - \mu _{2} s\right )> u \right \} },\ \ u\to \infty ,$ where (X1(t), X2(s))t, s≥ 0 is a correlated two-dimensional Brownian motion with correlation ρ ∈ [− 1,1] and μ1, μ2 > 0. It appears that the play between ρ and μ1, μ2 leads to several types of asymptotics. Although the exponent in the asymptotics as a function of ρ is continuous, one can observe different types of prefactor functions depending on the range of ρ, which constitute a phase-type transition phenomena

Quadratic Models for Portfolio Credit Risk with Shot-Noise Effects

We propose a reduced form model for default that allows us to derive closed-form solutions to all the key ingredients in credit risk modeling: risk-free bond prices, defaultable bond prices (with and without stochastic recovery) and probabilities of survival. We show that all these quantities can be represented in general exponential quadratic forms, despite the fact that the intensity is allowed to jump producing shot-noise effects. In addition, we show how to price defaultable digital puts, CDSs and options on defaultable bonds. Further on, we study a model for portfolio credit risk where we consider both firm specific and systematic risks. The model generalizes the attempt from Duffie and Garleanu (2001). We find that the model produces realistic default correlation and clustering of defaults. Then, we show how to price first-to-default swaps, CDOs, and draw the link to currently proposed credit indices

Stochastic ordering and thinning of point processes

We study the stochastic ordering of random measures and point processes generated by a partial order [mu]stochastic ordering thinning realizable thinning random measure point process renewal process Markov renewal process

Long-Range Dependence in a Cox Process Directed by a Markov Renewal Process

A Cox process NCox directed by a stationary random measure ξ has second moment var NCox(0,t]=E(ξ(0,t])+var ξ(0,t], where by stationarity E(ξ(0,t])=(const.)t=E(NCox(0,t]), so long-range dependence (LRD) properties of NCox coincide with LRD properties of the random measure ξ. When ξ(A)=∫AνJ(u)du is determined by a density that depends on rate parameters νi(i∈) and the current state J(⋅) of an -valued stationary irreducible Markov renewal process (MRP) for some countable state space (so J(t) is a stationary semi-Markov process on ), the random measure is LRD if and only if each (and then by irreducibility, every) generic return time Yjj(j∈X) of the process for entries to state j has infinite second moment, for which a necessary and sufficient condition when is finite is that at least one generic holding time Xj in state j, with distribution function (DF)\ Hj, say, has infinite second moment (a simple example shows that this condition is not necessary when is countably infinite). Then, NCox has the same Hurst index as the MRP NMRP that counts the jumps of J(⋅), while as t→∞, for finite , var NMRP(0,t]∼2λ2∫0t(u)du, var NCox(0,t]∼2∫0t∑i∈(νi−ν¯)2ϖiℋi(t)du, where ν¯=∑iϖiνi=E[ξ(0,1]], ϖj=Pr{J(t)=j},1/λ=∑jpˇjμj, μj=E(Xj), {pˇj} is the stationary distribution for the embedded jump process of the MRP, ℋj(t)=μi−1∫0∞min(u,t)[1−Hj(u)]du, and (t)∼∫0tmin(u,t)[1−Gjj(u)]du/mjj∼∑iϖiℋi(t) where Gjj is the DF and mjj the mean of the generic return time Yjj of the MRP between successive entries to the state j. These two variances are of similar order for t→∞ only when each ℋi(t)/(t) converges to some [0,∞]-valued constant, say, γi, for t→∞