Perfect Sampling of Hawkes Processes and Queues with Hawkes Arrivals

In this paper we develop to our best knowledge the first perfect sampling algorithm for queues with Hawkes input (i.e., single-server queues with Hawkes arrivals and independent and identically distributed service times of general distribution). In addition to the stability condition, we also assume the excitation function of the Hawkes process has a light tail and the service time has finite moment-generating function in the neighborhood of the origin. In this procedure, we also propose a new perfect sampling algorithm for Hawkes processes with improved computational efficiency compared with the existing algorithm. Theoretical analysis and numerical tests on the algorithms’ correctness and efficiency are also included.

Numerical Method for a System of PIDEs Arising in American Contingent Claims under FMLS Model with Jump Diffusion and Regime-Switching Process

This paper investigates a numerical method for solving fractional partial integro-differential equations (FPIDEs) arising in American Contingent Claims, which follow finite moment log-stable process (FMLS) with jump diffusion and regime switching. Mathematically, the prices of American Contingent Claims satisfy a system of d problems with free-boundary values, where d is the number of regimes of the market. In addition, an optimal exercise boundary is needed to setup with each regime. Therefore, a fully implicit scheme based on the penalty term is arranged. In the end, numerical examples are carried out to verify the obtained theoretical results, and the impacts of state variables in our model on the optimal exercise boundary of American Contingent Claims are analyzed.

Pricing of Margin Call Stock Loan Based on the FMLS

In common stock loan, lenders face the risk that their loans will not be repaid if the stock price falls below loan, which limits the issuance and circulation of stock loans. The empirical test suggests that the log-return series of stock price in the US market reject the normal distribution and admit instead a subclass of the asymmetric distribution. In this paper, we investigate the model of the margin call stock loan problem under the assumption that the return of stock follows the finite moment log-stable process (FMLS). In this case, the pricing model of the margin call stock loan can be described by a space-fractional partial differential equation with a time-varying free boundary condition. We transform the free boundary problem to a linear complementarity problem, and the fully-implicit finite difference method that we used is unconditionally stable in both the integer and fractional order. The numerical experiments are carried out to demonstrate differences of the margin call stock loan model under the FMLS and the standard normal distribution. Last, we analyze the impact of key parameters in our model on the margin call stock loan evaluation and give some reasonable explanation.

Spectral risk measure of holding stocks in the long run

We investigate how the spectral risk measure associated with holding stocks rather than a risk-free deposit, depends on the holding period. Previous papers have shown that within a limited class of spectral risk measures, and when the stock price follows specific processes, spectral risk becomes negative at long periods. We generalize this result for arbitrary exponential Lévy processes. We also prove the same behavior for all spectral risk measures (including the important special case of Expected Shortfall) when the stock price grows realistically fast and when it follows a geometric Brownian motion or a finite moment log stable process. This result would suggest that holding stocks for long periods has a vanishing downside risk. However, using realistic models, we find numerically that spectral risk initially increases for a significant amount of time and reaches zero level only after several decades. Therefore, we conclude that holding stocks has spectral risk for all practically relevant periods.

Magnetocaloric Effect, Magnetoresistance of Sc0.28Ti0.72Fe2, and Phase Diagrams of Sc0.28Ti0.72Fe2−xTx Alloys with T = Mn or Co

(Sc,Ti)Fe2 Laves phases present a relatively unique case of first-order ferro-ferromagnetic transition originating from an instability of the Fe moment. In addition to large magnetoelastic effects making them potential negative thermal expansion materials, here, we show that Sc0.28Ti0.72Fe2 and related alloys also present sizable magnetocaloric and magnetoresistance effects. Both effects are found substantially larger at the ferro-ferromagnetic transition (Tt1) than near the Curie temperature TC, yet they remain limited in comparison to other classes of giant magnetocaloric materials. We suggest a strategy to improve these properties by bringing the transition at Tt1 close to TC, and test its possible realization by Co or Mn for Fe substitutions. The structural and magnetic phase diagrams of Sc0.28Ti0.72Fe2−xTx alloys with T = Mn or Co are explored. Substitutions for Fe by adjacent Mn or Co elements give rise to a breakdown of the long-range ferromagnetic order, as well as a swift disappearance of finite moment magnetism.

Review and Improvement of the Finite Moment Problem

Background: This paper reviews the Particle Size Distribution (PSD) problem in detail. Mathematically, the problem faced while recovering a function from a finite number of its geometric moments has been discussed with the help of the Spline Theory. Undoubtedly, the splines play a major role in the theory of interpolation and approximation in many fields of pure and applied sciences. B-Splines form a practical basis for the piecewise polynomials of the desired degree. A high degree of accuracy has been obtained in recovering a function within the first ten to fifteen geometric moments. The proposed approximation formula has been tested on several types of synthetic functions. This work highlights some advantages, such as the use of a practical basis for the approximating space, the exactness of computing the moments of these basis functions and the reduction of the size along with an appropriate transformation of the resulting linear system for stability. Objective: The aim is to recover a function from a finite number of its geometric moments. Methods: The main tool is the Spline Theory. Undoubtedly, the role of splines in the theory of interpolation and approximation in many fields of pure and applied sciences has been well- established. B-Splines form a practical basis for the piecewise polynomials of the desired degree. Results: A high degree of accuracy has been obtained in recovering a function within the first ten to fifteen geometric moments. The proposed approximation formula is tested on several types of synthetic functions. Conclusion: This work highlights some advantages, such as the use of a practical basis for the approximating space, the exactness of computing the moments of these basis functions and the reduction of the size along with the data transformation of the resulting linear system for stability.

Gumbel and Fréchet convergence of the maxima of independent random walks

We consider point process convergence for sequences of independent and identically distributed random walks. The objective is to derive asymptotic theory for the largest extremes of these random walks. We show convergence of the maximum random walk to the Gumbel or the Fréchet distributions. The proofs depend heavily on precise large deviation results for sums of independent random variables with a finite moment generating function or with a subexponential distribution.