Recent zbMATH articles in MSC 60J74https://zbmath.org/atom/cc/60J742024-03-13T18:33:02.981707ZWerkzeugLarge deviation principle and thermodynamic limit of chemical master equation via nonlinear semigrouphttps://zbmath.org/1528.490262024-03-13T18:33:02.981707Z"Gao, Yuan"https://zbmath.org/authors/?q=ai:gao.yuan|gao.yuan.1"Liu, Jian-Guo"https://zbmath.org/authors/?q=ai:liu.jian-guoSummary: Chemical reactions can be modeled by a random time-changed Poisson process on countable states. The macroscopic behaviors, such as large fluctuations, can be studied via the WKB reformulation. The WKB reformulation for the backward equation is Varadhan's discrete nonlinear semigroup and is also a monotone scheme that approximates the limiting first-order Hamilton-Jacobi equations (HJE). The discrete Hamiltonian is an \(m\)-accretive operator, which generates a nonlinear semigroup on countable grids and justifies the well-posedness of the chemical master equation and the backward equation with ``no reaction'' boundary conditions. The convergence from the monotone schemes to the viscosity solution of HJE is proved by constructing barriers to overcome the polynomial growth coefficients in the Hamiltonian. This implies the convergence of Varadhan's discrete nonlinear semigroup to the continuous Lax-Oleinik semigroup and leads to the large deviation principle for the chemical reaction process at any single time. Consequently, the macroscopic mean-field limit reaction rate equation is recovered with a concentration rate estimate. Furthermore, we establish the convergence from a reversible invariant measure to an upper semicontinuous viscosity solution of the stationary HJE.On Feller and strong Feller properties and irreducibility of regime-switching jump diffusion processes with countable regimeshttps://zbmath.org/1528.600782024-03-13T18:33:02.981707Z"Kunwai, Khwanchai"https://zbmath.org/authors/?q=ai:kunwai.khwanchai"Zhu, Chao"https://zbmath.org/authors/?q=ai:zhu.chaoThe Feller property and irreducibility are fundamental desirable properties of Markov processes. Motivated by control and optimization problems, this paper presents weak local non-Lipschitz conditions for Feller and strong Feller properties, as well as the irreducibility of regime-switching jump diffusion processes with countable regimes. The approach used for obtaining sufficient conditions for Feller and strong Feller properties is motivated by a gradient estimate for diffusion semigroups in the literature. While the technique is obtaining irreducibility under certain assumptions, it relies on an identity concerning the transition probability of the process. As an application, these conditions are used to obtain the existence of a unique invariant measure based on classical results of Markov processes.
Reviewer: Chuang Xu (Honolulu)A contagion process with self-exciting jumps in credit risk applicationshttps://zbmath.org/1528.910772024-03-13T18:33:02.981707Z"Pasricha, Puneet"https://zbmath.org/authors/?q=ai:pasricha.puneet"Selvamuthu, Dharmaraja"https://zbmath.org/authors/?q=ai:dharmaraja.selvamuthu|selvamuthu.dharmaraja"Natarajan, Selvaraju"https://zbmath.org/authors/?q=ai:natarajan.selvarajuSummary: The modeling of the probability of joint default or total number of defaults among the firms is one of the crucial problems to mitigate the credit risk since the default correlations significantly affect the portfolio loss distribution and hence play a significant role in allocating capital for solvency purposes. In this article, we derive a closed-form expression for the default probability of a single firm and probability of the total number of defaults by time \(t\) in a homogeneous portfolio. We use a contagion process to model the arrival of credit events causing the default and develop a framework that allows firms to have resistance against default unlike the standard intensity-based models. We assume the point process driving the credit events is composed of a systematic and an idiosyncratic component, whose intensities are independently specified by a mean-reverting affine jump-diffusion process with self-exciting jumps. The proposed framework is competent of capturing the feedback effect. We further demonstrate how the proposed framework can be used to price synthetic collateralized debt obligation (CDO). Finally, we present the sensitivity analysis to demonstrate the effect of different parameters governing the contagion effect on the spread of tranches and the expected loss of the CDO.