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The transition density of the diffusion process plays an important role in calculating the dynamic characteristics of the underlying variables as well as the model estimation. In this article, we combine the idea of conditional probability density function with non parametric kernel regression, and introduce the kernel estimation of joint density function and marginal density function, then construct the non parametric kernel estimator of the transition density of second-order diffusion process based on discrete observational samples. In order to obtain the asymptotic properties of the new kernel estimator, we analyze the asymptotic expectation and asymptotic variance of the proposed estimator under some mild conditions. Finally, the consistency and asymptotic normality of the new proposed non parametric estimator of the transition density function are proved. [ABSTRACT FROM AUTHOR]

The integrals of diffusion processes are of significant importance in the field of finance, particularly in relation to stochastic volatility models, which are frequently employed to represent the temporal variability of stock prices. In this paper, we consider the strong consistency of the nonparametric kernel estimator of the transition density for second-order diffusion processes, using the moment inequalities of ρ-mixing sequences to demonstrate the strong consistency under some regularity conditions. Furthermore, the asymptotic mean square error is provided based on the deviation and variance of the transition density kernel estimator. The optimal bandwidth is found and thus the convergence rate of the kernel estimator is obtained. At the same time, our results are compared with the conclusions of the univariate density function. [ABSTRACT FROM AUTHOR]

The integrals of diffusion processes are of significant importance in the field of finance, particularly in relation to stochastic volatility models, which are frequently employed to represent the temporal variability of stock prices. In this paper, we consider the strong consistency of the nonparametric kernel estimator of the transition density for second-order diffusion processes, using the moment inequalities of $ \rho $-mixing sequences to demonstrate the strong consistency under some regularity conditions. Furthermore, the asymptotic mean square error is provided based on the deviation and variance of the transition density kernel estimator. The optimal bandwidth is found and thus the convergence rate of the kernel estimator is obtained. At the same time, our results are compared with the conclusions of the univariate density function.

We study the first-exit-time problem for the two-dimensional Wiener and Ornstein–Uhlenbeck processes through time-varying ellipses which run according to specific rules depending on the processes. We obtain the Laplace Transform of the first-exit-time probability density function and the corresponding moments. For both processes, some computational results on the first-exit-time densities are provided by means of the numerical inversion of the relevant Laplace Transforms. Moreover, we also investigate the asymptotic behaviour of the first-exit-time moments when the ellipse grows. In particular, an asymptotic exponential trend holds for the first-exit-time density of the mean-reverting Ornstein–Uhlenbeck process. [ABSTRACT FROM AUTHOR]

This paper studies the transition density and exponential ergodicity for affine processes on the canonical state space R≥0m×Rn$\mathbb {R}_{\ge 0}^{m}\times \mathbb {R}^{n}$. Under a Hörmander‐type condition for diffusion components as well as a boundary nonattainment condition, we derive the existence and regularity of the transition densities and then prove the strong Feller property of the associated semigroup. Moreover, we also show that, under an additional subcriticality condition on the drift, the corresponding affine process is exponentially ergodic in the total variation distance. [ABSTRACT FROM AUTHOR]

We consider classical solutions to the kinetic Fokker–Planck equation on a bounded domain O ⊂ R d in position, and we obtain a probabilistic representation of the solutions using the Langevin diffusion process with absorbing boundary conditions on the boundary of the phase-space cylindrical domain D = O × R d . Furthermore, a Harnack inequality, as well as a maximum principle, are provided on D for solutions to this kinetic Fokker–Planck equation, together with the existence of a smooth transition density for the associated absorbed Langevin process. This transition density is shown to satisfy an explicit Gaussian upper-bound. Finally, the continuity and positivity of this transition density at the boundary of D are also studied. All these results are in particular crucial to study the behavior of the Langevin diffusion process when it is trapped in a metastable state defined in terms of positions. [ABSTRACT FROM AUTHOR]

In this paper, we establish sharp two-sided estimates for transition densities of a large class of subordinate Markov processes. As applications, we show that the parabolic Harnack inequality and Hölder regularity hold for parabolic functions of such processes, and derive sharp two-sided Green function estimates. [ABSTRACT FROM AUTHOR]

In this paper, enlightened by the asymptotic expansion methodology developed by Li [(2013). Maximum-likelihood estimation for diffusion processes via closed-form density expansions. Annals of Statistics 41: 1350–1380] and Li and Chen [(2016). Estimating jump-diffusions using closed-form likelihood expansions. Journal of Econometrics 195(1): 51–70], we propose a Taylor-type approximation for the transition densities of the stochastic differential equations (SDEs) driven by the gamma processes, a special type of Lévy processes. After representing the transition density as a conditional expectation of Dirac delta function acting on the solution of the related SDE, the key technical method for calculating the expectation of multiple stochastic integrals conditional on the gamma process is presented. To numerically test the efficiency of our method, we examine the pure jump Ornstein–Uhlenbeck model and its extensions to two jump-diffusion models. For each model, the maximum relative error between our approximated transition density and the benchmark density obtained by the inverse Fourier transform of the characteristic function is sufficiently small, which shows the efficiency of our approximated method. [ABSTRACT FROM AUTHOR]

The core crust transition density and core crust transition pressure of neutron stars are analyzed with the thermodynamical approach using finite range effective interaction for two different splittings of exchange strength parameters. The role of density derivatives of symmetry energy on and are studied and it is found that reducing the effect of slope, the curvature of symmetry energy dominates the variation of transition density. The results are compared with various models quoted in the literature. [ABSTRACT FROM AUTHOR]

We introduce Sim.DiffProc, an R package for symbolic and numerical computations on scalar and multivariate systems of stochastic differential equations (SDEs). It provides users with a wide range of tools to simulate, estimate, analyze, and visualize the dynamics of these systems in both forms, Itô and Stratonovich. One of Sim.DiffProc key features is to implement the Monte Carlo method for the iterative evaluation and approximation of an interesting quantity at a fixed time on SDEs with parallel computing, on multiple processors on a single machine or a cluster of computers, which is an important tool to improve capacity and speed-up calculations. We also provide an easy-to-use interface for symbolic calculation and numerical approximation of the first and central second-order moments of SDEs (i.e., mean, variance and covariance), by solving a system of ordinary differential equations, which yields insights into the dynamics of stochastic systems. The final result object of Monte Carlo and moment equations can be derived and presented in terms of LATEX math expressions and visualized in terms of LATEX tables. Furthermore, we illustrate various features of the package by proposing a general bivariate nonlinear dynamic system of Haken-Zwanzig, driven by additive, linear and nonlinear multiplicative noises. In addition, we consider the particular case of a scalar SDE driven by three independent Wiener processes. The Monte Carlo simulation thereof is obtained through a transformation to a system of three equations. We also study some important applications of SDEs in different fields.

This paper revitalizes the investigation of the classical cusp catastrophe model in catastrophe theory and tackles the unsolved statistical inference problem concerning stochastic cusp differential equation. This model is challenging because its associated transition density hence the likelihood function is analytically intractable. We propose a novel Bayesian approach combining Hamiltonian Monte Carlo with two likelihood approximation methods, namely, Euler approximation and Hermite expansion. We validate this novel approach through a series of simulation studies. We further demonstrate potential application of this novel approach using the real USD/EUR exchange rate. [ABSTRACT FROM AUTHOR]

Many phenotypic traits have a polygenic genetic basis, making it challenging to learn their genetic architectures and predict individual phenotypes. One promising avenue to resolve the genetic basis of complex traits is through evolve-and-resequence experiments, in which laboratory populations are exposed to some selective pressure and trait-contributing loci are identified by extreme frequency changes over the course of the experiment. However, small laboratory populations will experience substantial random genetic drift, and it is difficult to determine whether selection played a roll in a given allele frequency change. Predicting how much allele frequencies change under drift and selection had remained an open problem well into the 21 st century, even those contributing to simple, monogenic traits. Recently, there have been efforts to apply the path integral, a method borrowed from physics, to solve this problem. So far, this approach has been limited to genic selection, and is therefore inadequate to capture the complexity of quantitative, highly polygenic traits that are commonly studied. Here we extend one of these path integral methods, the perturbation approximation, to selection scenarios that are of interest to quantitative genetics. In particular, we derive analytic expressions for the transition probability (i.e., the probability that an allele will change in frequency from x , to y in time t ) of an allele contributing to a trait subject to stabilizing selection, as well as that of an allele contributing to a trait rapidly adapting to a new phenotypic optimum. We use these expressions to characterize the use of allele frequency change to test for selection, as well as explore optimal design choices for evolve-and-resequence experiments to uncover the genetic architecture of polygenic traits under selection.

In this paper, we discuss estimates on the transition densities of subordinators, which are global in time. We establish sharp two-sided estimates on the transition densities of subordinators whose Lévy measures are absolutely continuous and decaying in mixed polynomial orders. Under a weaker assumption on Lévy measures, we also obtain precise asymptotic behaviors of the transition densities at infinity. Our results cover geometric stable subordinators, Gamma subordinators and much more. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a generalization of Brownian motion, called sticky skew Brownian motion, which has two interesting characteristics: stickiness and skewness. This kind of processes spends a lot more time at its sticky points so that the time they spend at the sticky points has positive Lebesgue measure. By using time change, we obtain an SDE for the sticky skew Brownian motion. Then, we present the explicit relationship between symmetric local time and occupation time. Some basic probability properties, such as transition density, are studied and we derive the explicit expression of Laplace transform of transition density for the sticky skew Brownian motion. We also consider the first hitting time problems over a constant boundary and a random jump boundary, respectively, and give some corollaries based on the results above. [ABSTRACT FROM AUTHOR]

We investigate densities of vaguely continuous convolution semigroups of probability measures on ℝd. First, we provide results that give upper estimates in a situation when the corresponding jump measure is allowed to be highly non-symmetric. Further, we prove upper estimates of the density and its derivatives if the jump measure compares with an isotropic unimodal measure and the characteristic exponent satisfies a certain scaling condition. Lower estimates are discussed in view of a recent development in that direction, and in such a way to complement upper estimates. We apply all those results to establish precise estimates of densities of non-symmetric Lévy processes. [ABSTRACT FROM AUTHOR]

In this paper, we study doubly skewed CIR processes, which are extensions of skew Brownian motion. We use modified spectral expansion to obtain some properties, including the transition densities and first hitting time distributions, of doubly skewed CIR processes. [ABSTRACT FROM AUTHOR]

Advances in empirical population genetics have made apparent the need for models that simultaneously account for selection and demography. To address this need, we here study the Wright-Fisher diffusion under selection and variable effective population size. In the case of genic selection and piecewise-constant effective population sizes, we obtain the transition density by extending a recently developed method for computing an accurate spectral representation for a constant population size. Utilizing this extension, we show how to compute the sample frequency spectrum in the presence of genic selection and an arbitrary number of instantaneous changes in the effective population size. We also develop an alternate, efficient algorithm for computing the sample frequency spectrum using a moment-based approach. We apply these methods to answer the following questions: If neutrality is incorrectly assumed when there is selection, what effects does it have on demographic parameter estimation? Can the impact of negative selection be observed in populations that undergo strong exponential growth?

We consider isotropic Lévy processes on a compact Riemannian manifold, obtained from an R d -valued Lévy process through rolling without slipping. We prove that the Feller semigroups associated with these processes extend to strongly continuous contraction semigroups on L p , for 1 ≤ p < ∞ . When p = 2 , we show that these semigroups are self-adjoint. If, in addition, the motion has a non-trivial Brownian part, we prove that the generator has a discrete spectrum of eigenvalues and that the semigroup is trace-class. [ABSTRACT FROM AUTHOR]

Properties of the ground states and transitions in 40Ca and 48Ca nuclei are studied using the self-consistent Hartree - Fock and random phase approximation calculations with Skyrme-type interactions: KDE0, SLy4, LNS, RAPT and T6. The purpose of the paper is to obtain the best Skyrme-force parameterizations for description of the experimental data. All the calculated values were compared with the available data. The calculated binding energy per nucleon, charge root mean square, ground charge density distribution and transition strength distribution agree very well with the experimental data. The overall behavior of the calculated transition densities demonstrated the reliability of the method.

Statement of Scope and Mission Reflected hyperexponential jump diffusion processes (RHEPs) have broad applications in operations research, financial engineering, insurance, economics, and queueing. Nonetheless, the extant literature lacks efficiently computable formulae for quantities related to first passage problems of RHEPs. This paper provides a new approach to derive such expressions. As a by-product, we obtain a formula for the Laplace transform of the RHEP transition density. Our approach relies on solving specific OIDEs numerically, applying martingale methods, conducting expansions, and performing matrix analysis. Numerical results show that, for cases where rounding error matters, our approach can provide effective approximation that is substantially more efficient than using "multiprecision computing." The extant literature on first passage problems of reflected hyperexponential jump diffusion processes (RHEPs) lacks efficiently computable formulae for the Laplace transform of the joint distribution of the RHEP and its first passage time, cumulative distribution function of the overshoot, expected cumulative value of the discounted increments of the local time up to the first passage time, expected cumulative discounted value of the RHEP up to the first passage time, and expectation of the first passage time. We combine numerical solutions to ordinary integro-differential equations and martingale methods in a novel manner to derive such expressions. For some of these quantities, our approach can deal with the subtle case in which both the RHEP's overall drift and the discount rate equal zero. As a by-product, we obtain a formula for the Laplace transform of the RHEP transition density. We illustrate the numerical performance of our methodology through a few examples. We observe that, when the RHEP's overall drift and the discount rate are very close to zero, rounding errors can make the evaluation of some of our formulae unreliable. In these situations our exact expression for the case in which the RHEP's overall drift and discount rate are both zero can be an effective approximation for the quantities in question that is substantially more efficient than reliably calculating them using their exact expressions and "multiprecision computing." Our research has applications in financial engineering, insurance, economics, and queueing. [ABSTRACT FROM AUTHOR]

In this paper we discuss non-local operators with killing potentials, which may not be in the standard Kato class. We first discuss factorization of their Dirichlet heat kernels in metric measure spaces. Then we establish explicit estimates of the Dirichlet heat kernels under critical killings in C 1 , 1 open subsets of R d or in R d ∖ { 0 }. The decay rates of our explicit estimates come from the values of the multiplicative constants in the killing potentials. Our method also provides an alternative and unified proof of the main results of [18–20]. [ABSTRACT FROM AUTHOR]

We consider a multi-factor Cox-Ingersoll-Ross (CIR) model of the term structure of interest rates with weak mean-reversion effect. We use perturbation theory to analyze its conditional characteristic function illustrated by a system of Riccati equations and derive the error bounds for the perturbation approximations. Using the Fourier inversion theorem, we clarify that the perturbation approximation of the conditional characteristic function can be applied to estimate the transition density and likelihood function. We provide their error bounds and accuracy orders. Finally, we discuss the performance of the perturbation approximation in estimating the transition density via simulation. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop a data augmentation technique for statistical inference concerning stochastic cusp catastrophe model subject to missing data and partially observed observations. We propose a Bayesian inference solution that naturally treats missing observations as parameters and we validate this novel approach by conducting a series of Monte Carlo simulation studies assuming the cusp catastrophe model as the underlying model. We demonstrate that this Bayesian data augmentation technique can recover and estimate the underlying parameters from the stochastic cusp catastrophe model.

The Laplace transforms of the transition probability density and distribution functions of the Feller process contain products of a Kummer and a Tricomi confluent hypergeometric function. The intricacies caused by the singularity at 0 of the Feller process imply that ultimately seven new inverse Laplace transforms can be derived of which four contain the Marcum Q function. The results of this paper together with a scarcely used link between the Marcum and Nuttall Q functions also provide two alternative proofs for an existing identity involving two Marcum Q functions with reversed arguments. The paper also expands the existing expression for the Marcum Q function with identical arguments and order 1. In particular, the new formula applies to all integer and fractional values of the order and is expressed in terms of the generalized hypergeometric function. [ABSTRACT FROM AUTHOR]

The paper deals with a three-dimensional family of diffusion processes on an infinite-dimensional simplex. These processes were constructed by Borodin and Olshanski in 2009 and 2010, and they include, as limit objects, the infinitely-many-neutral-allels diffusion model constructed by Ethier and Kurtz in 1981 and its extension found by Petrov in 2009. Each process X in our family possesses a unique symmetrizing measure M, called the z-measure. Our main result is that the transition function of X has a continuous density with respect to M. This is a generalization of earlier results due to Ethier (1992) and to Feng, Sun, Wang, and Xu (2011). Our proof substantially uses a special basis in the algebra of symmetric functions, which is related to the Laguerre polynomials. [ABSTRACT FROM AUTHOR]

We consider a subordinate Brownian motion X with Gaussian components when the scaling order of purely discontinuous part is between 0 and 2 including 2. In this paper we establish sharp two-sided bounds for transition density of X in ℝ d and C1,1-open sets. As a corollary, we obtain a sharp Green function estimates. [ABSTRACT FROM AUTHOR]

We consider a large class of symmetric pure jump Markov processes dominated by isotropic unimodal Lévy processes with weak scaling conditions. First, we establish sharp two-sided heat kernel estimates for these processes in C 1 , 1 open sets. As corollaries of our main results, we obtain sharp two-sided Green function estimates and a scale invariant boundary Harnack inequality with explicit decay rates in C 1 , 1 open sets. [ABSTRACT FROM AUTHOR]

Early detection combined with effective treatments are the only ways to fight against cancer, and cancer screening is the primary technique for early detection. Although mass cancer screening has been carried out for decades, there are many unsolved problems, and the statistical theory of cancer screening is still under developed. Screening sensitivity, time duration in the preclinical state, and time duration in the disease free state are the three key parameters, which are critical in cancer screening, since all other estimates are functions of the three key parameters. Lead time is the diagnosis time advanced by screening, and it serves as a measurement of effectiveness of screening programs. In this article, we provide a review for major probability models and statistical methodologies that have been developed on the estimation of the three key parameters and the lead time distributions. These methods can be applied to screening of other chronic diseases after slight modifications.

Quenched and annealed heat kernel estimates are established for Fontes–Isopi–Newman (FIN) processes on spaces equipped with a resistance form. These results are new even in the case of the one-dimensional FIN diffusion, and also apply to fractals such as the Sierpinski gasket and carpet. [ABSTRACT FROM AUTHOR]

Assume that g (| ξ | 2) , ξ ∈ R k , is for every dimension k ∈ N the characteristic function of an infinitely divisible random variable X k. By a classical result of Schoenberg f : = − log ⁡ g is a Bernstein function. We give a simple probabilistic proof of this result starting from the observation that X k = X 1 k can be embedded into a Lévy process (X t k) t ≥ 0 and that Schoenberg's theorem says that (X t k) t ≥ 0 is subordinate to a Brownian motion. Key ingredients in our proof are concrete formulae which connect the transition densities, resp., Lévy measures of subordinated Brownian motions across different dimensions. As a by-product of our proof we obtain a gradient estimate for the transition semigroup of a subordinated Brownian motion. [ABSTRACT FROM AUTHOR]

We study a spatial asymptotic behavior at infinity of kernels pt(x) for convolution semigroups of nonlocal pseudo-differential operators. We give general and sharp sufficient conditions under which the limits lim r→∞ pt(rθ - y) t ν(rθ), t∈ T, θ ∈ E, y ∈ Rd exist and can be effectively computed. Here ν is the corresponding L'evy density, T ⊂ (0,∞) is a bounded time-set, and E is a subset of the unit sphere in Rd, d ≥ 1. Our results are local on the unit sphere. They apply to a wide class of convolution semigroups, including those corresponding to highly asymmetric (finite and infinite) L'evy measures. Key examples include fairly general families of stable, tempered stable, jump-diffusion, and compound Poisson semigroups. A main emphasis is put on the semigroups with L'evy measures that are exponentially localized at infinity, for which our assumptions and results are strongly related to the existence of the multidimensional exponential moments. Here a key example is the evolution semigroup corresponding to the so-called quasi-relativistic Hamiltonian √ -Δ+m2 -m, m > 0. As a byproduct, we also obtain sharp two-sided estimates of the kernels pt in generalized cones, away from the origin. [ABSTRACT FROM AUTHOR]

Properties of the ground states and transitions in 40Ca and 48Ca nuclei are studied using the self-consistent Hartree - Fock and random phase approximation calculations with Skyrme-type interactions: KDE0, SLy4, LNS, RAPT and T6. The purpose of the paper is to obtain the best Skyrme-force parameterizations for description of the experimental data. All the calculated values were compared with the available data. The calculated binding energy per nucleon, charge root mean square, ground charge density distribution and transition strength distribution agree very well with the experimental data. The overall behavior of the calculated transition densities demonstrated the reliability of the method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an efficient stochastic method for solving space-fractional diffusion equations in high dimensions based on the Feynman–Kac formula. The key idea is to approximate the trajectory of the process by using a series of balls. As an extension of our first work Sheng et al. (2022) for the Poisson equation, the new algorithm finds remarkably efficient in solving time-dependent linear problems with integral fractional Laplacian on the bounded and unbounded domains. We present some numerical examples to validate the robustness and efficiency of our methods. [ABSTRACT FROM AUTHOR]