In this paper, we characterize jump phenomena of the $n$-th eigenvalue of self-adjoint discrete Sturm–Liouville problems in any dimension. For a fixed Sturm–Liouville equation, we completely characterize jump phenomena of the $n$-th eigenvalue. For a fixed boundary condition, unlike in the continuous case, the $n$-th eigenvalue exhibits jump phenomena and we describe the singularity under a non-degenerate assumption. Compared with the continuous case in Hu et al. (2019, J. Differ. Equ. 266, 4106–4136) and Kong et al. (1999, J. Differ. Equ. 156, 328–354), the jump set here is involved with coefficients of the Sturm–Liouville equations. This, along with arbitrariness of the dimension, causes difficulty when dividing the jump areas. We study the singularity by partitioning and analysing the local coordinate systems, and provide a Hermitian matrix which can determine the areas’ division. To prove the asymptotic behaviour of the $n$-th eigenvalue, we generalize the method developed in Zhu and Shi (2016, J. Differ. Equ. 260, 5987–6016) to any dimension. As an application, by transforming the continuous Sturm–Liouville problem of Atkinson type to a discrete one, we determine the number of eigenvalues and obtain complete characterization of jump phenomena of the $n$-th eigenvalue for the Atkinson type.

In this study, we consider option pricing under a Markov regime-switching GARCH-jump (RS-GARCH-jump) model. More specifically, we derive the risk neutral dynamics and propose a lattice algorithm to price European and American options in this framework. We also provide a method of parameter estimation in our RS-GARCH-jump setting using historical data on the underlying time series. To measure the pricing performance of the proposed algorithm, we investigate the convergence of the tree-based results to the true option values and show that this algorithm exhibits good convergence. By comparing the pricing results of RS-GARCH-jump model with regime-switching GARCH (RS-GARCH) model, GARCH-jump model, GARCH model, Black–Scholes (BS) model, and Regime-Switching (RS) model, we show that accommodating jump effect and regime switching substantially changes the option prices. The empirical results also show that the RS-GARCH-jump model performs well in explaining option prices and confirm the importance of allowing for both jump components and regime switching.

In this paper, we study the stationary distributions for reflected diffusions with jumps in the positive orthant. Under the assumption that the stationary distribution possesses a density in R+n that satisfies certain finiteness conditions, we characterize the Fokker-Planck equation. We then provide necessary and sufficient conditions for the existence of a product-form distribution for diffusions with oblique boundary reflections and jumps. To do so, we exploit a recent characterization of the boundary properties of such reflected processes. In particular, we show that the conditions generalize those for semimartingale reflecting Brownian motions and reflected Lévy processes. We provide explicit results for some models of interest.

Jump performance was studied in seven flea species (Xenopsylla conformis mycerini, Xenopsylla ramesis, Xenopsylla dipodilli, Parapulex chephrenis, Synosternus cleopatrae pyramidis, Nosopsyllus iranus theodori and Stenoponia tripectinata medialis). Jump length was compared among individuals, between sexes and among species, and an examination made of whether: (1) there is a consistent difference in jump performance between male and female fleas; (2) sexual dimorphism in body size affects jump performance; (3) there is a size-specific difference in jump performance among species; (4) there are morphological correlates of jumping ability among individuals, between sexes and among species. Males were smaller than females in all species and size dimorphism did not increase with increasing body size. In all species, except S. t. medialis, both absolute and size-specific jumping abilities of the males were less than those of the females. Males and females of S. t. medialis demonstrated similar jumping ability despite their body size difference. The best jumpers were S. c. pyramidis, N. i. theodori and S. t. medialis, whereas X. dipodilli as well as female X. c. mycerini and X. ramesis demonstrated the poorest ability to jump. Female:male performance ratio differed among species, being the highest in X. c. mycerini and X. ramesis, and the lowest in S. t. medialis. The proportion of the overall dimorphism in jumping ability that was owing to size differences between males and females was low, ranging from 0.10 to 0.34. No correlation between jumping ability and pleural height or length of hind coxa, femur and tibia was found in any species, except in X. dipodilli. Interspecific comparisons using size-corrected and phylogeny-controlled values did not show any relationship between jumping ability and any morphological trait. Fleas occupying habitats with a soft sandy substrate were better jumpers than those living in habitats with a hard stony substrate.