laplace transform methods

This paper investigates the asymptotic behavior of the ruin probability under a risk model featuring stochastic premium income and investment in a risky asset governed by geometric Brownian motion, focusing on the limiting regime as initial capital tends to infinity. Assuming the premium density admits a rational Laplace transform, the authors employ Laplace transform techniques, stochastic process modeling, and asymptotic analysis to establish, for the first time within a risk-investment framework, a rigorous asymptotic ruin theory for premium processes with rational Laplace structure. They derive an explicit exponential decay rate for the ruin probability and its first-order asymptotic expansion. The results quantitatively characterize the joint impact of asset volatility and premium-structure parameters on ruin risk, thereby providing novel theoretical foundations and quantitative tools for insurance asset allocation and solvency assessment.

This paper addresses spatiotemporal epidemic dynamics modeling via a generalized multivariate Hawkes process featuring time-varying baseline intensities and nonparametric triggering kernels—inducing inherent non-Markovianity. Method: We derive, for the first time, a closed-form expression for the joint characteristic function at multiple time points. Leveraging functional Laplace transforms and integral equation theory, we establish a system of distributional equations for non-Markovian Hawkes processes and analytically decompose the temporal evolution of the first two moments and cross-time covariances—including both singular and absolutely continuous components. Combining infinite divisibility with numerical approximation techniques, we enable statistical inference and simulation for high-dimensional processes. Results: Empirical validation confirms the theoretical correctness of our formulations and demonstrates high simulation fidelity, substantially extending the applicability of Hawkes processes to complex, realistic epidemic propagation modeling.

This work addresses the lack of a unified framework for real-order moments—including positive, fractional, and negative orders—of arbitrary random variables on the real line, encompassing discrete, continuous, and mixed distributions. By extending the tail-integral identity known for non-negative variables, the authors develop a comprehensive representation valid for distributions supported on the entire real axis. The proposed framework expresses real-order moments via integrals of the cumulative distribution and survival functions, and yields series representations based on cumulative probabilities in the discrete case. It further uncovers a geometric–analytic connection between moment existence and tail decay, and establishes links between logarithmic moments, Laplace transforms, and Frullani’s identity. The approach provides concise existence criteria and is validated on Zeta and Skellam distributions, thereby deepening the theoretical interplay among tail probabilities, distribution functions, and moments.

This study addresses the computational challenges in evaluating Conditional Mean Risk Sharing (CMRS) within continuous multivariate risk models, where high-dimensional integrals often hinder efficient implementation. The authors propose a novel framework that systematically employs the joint Laplace–Stieltjes transform to derive CMRS allocations. By taking partial derivatives along the diagonal direction, they obtain a transformed representation of the allocation measure, which is then inverted numerically using one-dimensional Laplace inversion enhanced with exponential tilting to stabilize tail-event estimation. This approach yields closed-form or semi-analytical solutions across a range of distributions—including high-dimensional settings—significantly improving both computational efficiency and numerical stability. Consequently, the method overcomes the feasibility limitations of conventional techniques in complex, high-dimensional scenarios.

This paper addresses the challenge of analytically characterizing the conditional and unconditional density functions for affine jump-diffusion models with state-independent jump intensities. We propose a novel density approximation method based on closed-form moment recursion and generalized moment matching. For the first time, we derive explicit recursive formulas for all orders of conditional and unconditional moments of such processes and construct a density approximation that admits an analytical expression—up to the normalization constant. Compared to conventional Monte Carlo simulation, our method achieves comparable accuracy in option pricing and path simulation while improving computational efficiency by an order of magnitude. This advancement significantly extends the practical applicability of affine jump-diffusion models in financial derivative pricing and high-efficiency stochastic simulation.

This study addresses the first-passage time (FPT) distribution for a stochastic logistic growth model incorporating constant harvesting and multiplicative noise. By employing a power series expansion of the Laplace transform, the authors derive moments and cumulants for both upward and downward FPTs. They further introduce, for the first time, a high-accuracy analytical approximation of the FPT density based on a Laguerre orthogonal expansion weighted by a Gamma function. The proposed method exhibits numerical stability, controllable precision, and superior performance under moderate noise intensities, making it well-suited for modeling large-scale population dynamics. The approach is successfully applied to parameter estimation in fisheries management, demonstrating its practical utility.

This work addresses the absence of a unified theoretical framework for handling conditional quantities and transport functionals across continuous, discrete, and mixed distributions. By constructing a probabilistic framework grounded in distributional derivatives and Dirac delta functions—integrated with copula theory and Fréchet–Hoeffding extremal bounds—the study uncovers the localized mechanisms underlying conditional expectations, risk functions, and related concepts. It introduces, for the first time, Δ-antimonotonic functions to characterize dependence structures. This approach yields sharp bounds on absolute difference moments under fixed marginals and provides concise expressions for the Wasserstein distance, upper-bound transport functionals, and generalized absolute difference moments. Furthermore, it derives novel formulations of the bivariate Gini mean difference and Gini index, successfully applying the framework to normal approximations for Poisson, binomial, and negative binomial distributions.

This study addresses the computational complexity and insufficient accuracy in pricing credit derivatives under general intensity-based models by proposing a semi-analytical approximation framework rooted in path integral methods. The approach jointly models default intensity using the Black–Karasinski model and foreign exchange devaluation dynamics, achieving high accuracy and computational efficiency even in high-volatility and long-maturity regimes. Compared to conventional numerical techniques, the framework substantially enhances pricing performance for products such as Quanto CDS and is readily applicable to practical contexts like XVA calculations. The method thus offers both theoretical novelty and strong engineering utility.

This work investigates the non-asymptotic characterization of posterior contraction rates and finite-sample Bernstein–von Mises (BvM) behavior in nonparametric Bayesian models. By interpreting the posterior distribution as the invariant measure of a Langevin stochastic partial differential equation (SPDE) on a separable Hilbert space, the study extends diffusion-based analytical frameworks to infinite-dimensional settings for the first time. This approach enables precise control over posterior moments and yields sharp non-asymptotic concentration rates in Hilbert norm, along with a quantitative Laplace approximation of the posterior. In the context of nonparametric linear Gaussian inverse problems, the theory elucidates how likelihood curvature and prior regularity jointly govern posterior contraction rates and finite-sample BvM phenomena.

This study addresses the absence of a forward stochastic partial differential equation (SPDE) framework suitable for pricing path-dependent derivatives within existing local stochastic volatility models. By leveraging Wentzell theory, the authors introduce the Itô–Wentzell formula into the Dupire equation for the first time, deriving a conditional forward SPDE. They further integrate this with the Musiela parametrization to formulate a dynamic model for rolling-maturity vanilla options. Additionally, a density-weighted Rao–Blackwell estimator is proposed to calibrate the leverage function with high accuracy and computational efficiency. This work not only establishes a novel modeling framework for path-dependent products under local stochastic volatility but also significantly enhances both the precision and speed of leverage function estimation.