Dependence properties of dynamic credit risk models

We give a unified mathematical framework for reduced-form models for portfolio credit risk and identify properties which lead to positive dependence of default times. Dependence in the default hazard rates is modeled by common macroeconomic factors as well as by inter-obligor links. It is shown that popular models produce positive dependence between defaults in terms of association. Implications of these results are discussed, in particular when we turn to pricing of credit derivatives. In mathematical terms our paper contains results about association of a class of non-Markovian processes

Scaled penalization of Brownian motion with drift and the Brownian ascent

We study a scaled version of a two-parameter Brownian penalization model introduced by Roynette-Vallois-Yor in arXiv:math/0511102. The original model penalizes Brownian motion with drift $h\in\mathbb{R}$ by the weight process ${\big(\exp(\nu S_t):t\geq 0\big)}$ where $\nu\in\mathbb{R}$ and $\big(S_t:t\geq 0\big)$ is the running maximum of the Brownian motion. It was shown there that the resulting penalized process exhibits three distinct phases corresponding to different regions of the $(\nu,h)$-plane. In this paper, we investigate the effect of penalizing the Brownian motion concurrently with scaling and identify the limit process. This extends a result of Roynette-Yor for the ${\nu<0,~h=0}$ case to the whole parameter plane and reveals two additional "critical" phases occurring at the boundaries between the parameter regions. One of these novel phases is Brownian motion conditioned to end at its maximum, a process we call the Brownian ascent. We then relate the Brownian ascent to some well-known Brownian path fragments and to a random scaling transformation of Brownian motion recently studied by Rosenbaum-Yor.Comment: 32 pages; made additions to Section

On self-attracting dd-dimensional random walks

Let $\{X_t\}_{t \geq 0}$ be a symmetric, nearest-neighbor random walk on $\mathbb{Z}^d$ with exponential holding times of expectation $1/d$, starting at the origin. For a potential $V: \mathbb{Z}^d \to [0, \infty)$ with finite and nonempty support, define transformed path measures by $d \hat{\mathbb{P}}_T \equiv \exp (T^{-1} \int_0^T \int_0^T V(X_s - X_t) ds dt) d \mathbb{P}/Z_T$ for $T > 0$, where $Z_T$ is the normalizing constant. If $d = 1$ or if the self-attraction is sufficiently strong, then $||X_t||_{\infty}$ has an exponential moment under $\mathbb{P}_T$ which is uniformly bounded for $T > 0$ and $t \in [0, T]$. We also prove that ${X_t}_{t \geq 0}$ under suitable subsequences of ${\hat{\mathbb{P}}_T}_{T > 0}$ behaves for large $T$ asymptotically like a mixture of space-inhomogeneous ergodic random walks. For special cases like a sufficiently strong Dirac-type interaction, we even prove convergence of the transformed path measures and the law of $X_T$ as well as of the law of the empirical measure $L_T$ under ${\hat{\mathbb{P}}_T}_{T > 0}$

On self-attracting random walks

In this survey paper we mainly discuss the results contained in two of our recent articles [2] and [5]. Let {Xt}t≥0 be a continuous-time, symmetric, nearest-neighbour random walk on Zd. For every T > 0 we define the transformed path measure dPT = (1/ZT ) exp(HT ) dP, where P is the original one and ZT is the appropriate normalizing constant. The Hamiltonian HT imparts the self-attracting interaction of the paths up to time T. We consider the case where HT is given by a potential function V on Zd with finite support, and the case HT = −NT , where NT denotes the number of points visited by the random walk up to time T. In both cases the typical paths under PT as T →∞ clump together much more than those of the free random walk and give rise to localization phenomena

Convergence of path measures arising from a mean field or polaron type interaction

We discuss the limiting path measures of Markov processes with either a mean field or a polaron type interaction of the paths. In the polaron type situation the strength is decaying at large distances on the time axis, and so the interaction is of short range in time. In contrast, in the mean field model, the interaction is weak, but of long range in time. Donsker and Varadhan proved that for the partition functions, there is a transition from the polaron type to the mean field interaction when passing to a limit by letting the strength tend to zero while increasing the range. The discussion of the path measures is more subtle. We treat the mean field case as an example of a differentiable interaction and discuss the transition from the polaron type to the mean field interaction for two instructive examples

On the maximum entropy principle for uniformly ergodic Markov chains

For strongly ergodic discrete time Markov chains we discuss the possible limits as n→∞ of probability measures on the path space of the form exp(nH(Ln)) dP/Zn· Ln is the empirical measure (or sojourn measure) of the process, H is a real-valued function (possibly attaining −∞) on the space of probability measures on the state space of the chain, and Zn is the appropriate norming constant. The class of these transformations also includes conditional laws given Ln belongs to some set. The possible limit laws are mixtures of Markov chains minimizing a certain free energy. The method of proof strongly relies on large deviation techniques