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This paper deals with the period function of the reversible quadratic centers X ν = − y (1 − x) ∂ x + (x + D x 2 + F y 2) ∂ y , where ν = (D , F) ∈ R 2. Compactifying the vector field to S 2 , the boundary of the period annulus has two connected components, the center itself and a polycycle. We call them the inner and outer boundary of the period annulus, respectively. We are interested in the bifurcation of critical periodic orbits from the polycycle Π ν at the outer boundary. A critical period is an isolated critical point of the period function. The criticality of the period function at the outer boundary is the maximal number of critical periodic orbits of X ν that tend to Π ν 0 in the Hausdorff sense as ν → ν 0. This notion is akin to the cyclicity in Hilbert's 16th Problem. Our main result (Theorem A) shows that the criticality at the outer boundary is at most 2 for all ν = (D , F) ∈ R 2 outside the segments { − 1 } × [ 0 , 1 ] and { 0 } × [ 0 , 2 ]. With regard to the bifurcation from the inner boundary, Chicone and Jacobs proved in their seminal paper on the issue that the upper bound is 2 for all ν ∈ R 2. In this paper the techniques are different because, while the period function extends analytically to the center, it has no smooth extension to the polycycle. We show that the period function has an asymptotic expansion near the polycycle with the remainder being uniformly flat with respect to ν and where the principal part is given in a monomial scale containing a deformation of the logarithm, the so-called Écalle-Roussarie compensator. More precisely, Theorem A follows by obtaining the asymptotic expansion to fourth order and computing its coefficients, which are not polynomial in ν but transcendental. Theorem A covers two of the four quadratic isochrones, which are the most delicate parameters to study because its period function is constant. The criticality at the inner boundary in the isochronous case is bounded by the number of generators of the ideal of all the period constants but there is no such approach for the criticality at the outer boundary. A crucial point to study it in the isochronous case is that the flatness of the remainder in the asymptotic expansion is preserved after the derivation with respect to parameters. We think that this constitutes a novelty that is of particular interest also in the study of similar problems for limit cycles in the context of Hilbert's 16th Problem. Theorem A also reinforces the validity of a long standing conjecture by Chicone claiming that the quadratic centers have at most two critical periodic orbits. A less ambitious goal is to prove the existence of a uniform upper bound for the number of critical periodic orbits in the family of quadratic centers. By a compactness argument this would follow if one can prove that the criticality of the period function at the outer boundary of any quadratic center is finite. Theorem A leaves us very close to this existential result. [ABSTRACT FROM AUTHOR]

Assume Ω ⊂ ℝ 2 is a planar domain, and u is a locally bounded distributional solution to the elliptic equation − Δ u = | x | 2 β h (x) f (u) in  Ω , where β > − 1 is a constant, h and f are real analytic functions defined on Ω and the real line ℝ , respectively. We establish asymptotic expansions of u (x) to arbitrary orders near 0 , which complements the recent results of Han–Li–Li on the Yamabe equation, Guo–Li–Wanon the weighted Yamabe equation, and partly extends that of Guo–Wan–Yang on the Liouville equation in a punctured disc. Our method is a combination of a priori estimate and mathematical induction. [ABSTRACT FROM AUTHOR]

• The notion of an asymptotic continued fraction expansion of a complex-valued function is introduced. • Some general results about asymptotic continued fraction expansions are established. • Two continued fraction expansions of the exponential integral E n (z) are shown to yield two asymptotic continued fraction expansions of the prime counting function π (x). • All of the best rational function approximations of the function π (e x) / e x are determined. • Our results on π (x) are generalized to any arithmetic semigroup satisfying Axiom A, hence to any number field. We provide several asymptotic expansions of the prime counting function π (x) and related functions. We define an asymptotic continued fraction expansion of a complex-valued function of a real or complex variable to be a possibly divergent continued fraction whose approximants provide an asymptotic expansion of the given function. We show that, for each positive integer n , two well-known continued fraction expansions of the exponential integral function E n (z) correspondingly yield two asymptotic continued fraction expansions of π (x) / x. We prove this by first establishing some general results about asymptotic continued fraction expansions. We show, for instance, that the "best" rational function approximations of a function possessing an asymptotic Jacobi continued fraction expansion are precisely the approximants of the continued fraction, and as a corollary we determine all of the best rational function approximations of the function π (e x) / e x. Finally, we generalize our results on π (x) to any arithmetic semigroup satisfying Axiom A, and thus to any number field. [ABSTRACT FROM AUTHOR]

The inverse problem of reconstructing a source term from boundary measurements, for the wave equation, is revisited. We propose a novel approach to recover the unknown source through measuring the wave fields after injecting small particles, enjoying a high contrast, into the medium. For this purpose, we first derive the asymptotic expansion of the wave field, based on the time-domain Lippmann--Schwinger equation. The dominant term in the asymptotic expansion is expressed as an infinite series in terms of the eigenvalues \{ \lambda n\} n\in N of the Newtonian operator (for the pure Laplacian). Such expansions are useful under a certain scale between the size of the particles and their contrast. Second, we observe that the relevant eigenvalues appearing in the expansion have nonzero averaged eigenfunctions. We prove that the family \{ sin(\surd c1 \lambda n t), cos(\surd c1 \lambda n t)\}, for those relevant eigenvalues, with c1 as the contrast of the small particle, defines a Riesz basis (contrary to the family corresponding to the whole sequence of eigenvalues). Then, using the Riesz theory, we reconstruct the wave field, generated before injecting the particles, on the center of the particles. Finally, from internal values of the last field, we reconstruct the source term by numerical differentiation, for instance. A significant advantage of our approach is that we only need the measurements on \{ x\} \times (0, T) for a single point x away from \Omega, i.e., the support of the source, and large enough T. [ABSTRACT FROM AUTHOR]

In the present paper we prove the validity of the Camassa-Holm equation as a long wave limit to the double dispersion equation which describes the propagation of bidirectional weakly nonlinear and dispersive waves in an infinite elastic medium. First we show formally that the right-going wave solutions of the double dispersion equation can be approximated by the solutions of the Camassa-Holm equation in the long wave limit. Then we rigorously prove that the solutions of the double dispersion and the Camassa-Holm equations remain close over a long time interval, determined by two small parameters measuring the effects of nonlinearity and dispersion. [ABSTRACT FROM AUTHOR]

In this paper, we investigate so-called slowly traveling wave solutions for a quasilinear parabolic equation in detail. Over the past three decades, the motion of the plane curve by the power of its curvature with positive exponent α has been intensively investigated. For this motion, blow-up phenomena of curvature on cusp singularity in the plane curve with self-crossing points have been studied by several authors. In their analysis, particularly in estimating the blow-up rate, the slowly traveling wave solutions played a significantly important role. In this paper, aiming to clarify the blow-up phenomena, we derive an asymptotic expansion of the slowly traveling wave solutions with respect to the parameter κ , which is proportional to the maximum of the curvature of the curve, as κ goes to infinity. We discovered that the result depends discontinuously on the parameter δ = 1 + 1 / α . It suggests that the blow-up phenomenon may also drastically change according to parameter δ . [ABSTRACT FROM AUTHOR]

An accurate evaluation of three-dimensional (3D) Time-Domain Green Function (TDGF) in infinite water depth is essential for ship’s hydrodynamic analysis. Various numerical algorithms based on the TDGF properties are considered, including the ascending series expansion at small time parameter, the asymptotic expansion at large time parameter and the Taylor series expansion combines with ordinary differential equation for the time domain analysis. An efficient method (referred as “Present Method”) for a better accuracy evaluation of TDGF has been proposed. The numerical results generated from precise integration method and analytical solution of Shan et al. (2019) revealed that the “Present method” provides a better solution in the computational domain. The comparison of the heave hydrodynamic coefficients in solving the radiation problem of a hemisphere at zero speed between the “Present method” and the analytical solutions proposed by Hulme (1982) showed that the difference of result is small, less than 3%.

The boundary value problem for the steady Navier–Stokes system is considered in a 2D bounded domain with the boundary having a power cusp singularity at the point O. The case of a boundary value with a nonzero flow rate is studied. In this case there is a source/sink in O and the solution necessarily has an infinite Dirichlet integral. The formal asymptotic expansion of the solution near the singular point is constructed and the existence of a solution having this asymptotic decomposition is proved.

The Kelvin–Voigt model for a thin stratified two-dimensional visco-elastic strip is analyzed both in the quasistatic and in the dynamic cases. The Neumann boundary conditions on the upper and the lower parts of the boundary and periodicity conditions with respect to the longitudinal variable are stated. A complete asymptotic expansion of the solution is constructed in both cases, by using the dimension reduction combined with a homogenization technique. The error between the exact solution and the asymptotic one is evaluated in each case and the obtained results fully justify the asymptotic construction. The results were partially (quasistatic case) announced in the short note in C.R. Acad. Sci. Paris; the present article contains the complete proofs and generalizations in the dynamic case.

We consider a Urysohn integral operator K with the kernel of the type of Green's function. For r ≥ 1 , a space of piecewise polynomials of degree ≤ r − 1 with respect to a uniform partition is chosen to be the approximating space, and the projection is chosen to be the orthogonal projection. The iterated Galerkin method is applied to the integral equation x − K (x) = f. It is known that the order of convergence of the iterated Galerkin solution is r + 2 and is 2 r at the partition points. We obtain an asymptotic expansion of the iterated Galerkin solution at the partition points of the above Urysohn integral equation. Richardson extrapolation is used to improve the order of convergence. A numerical example is considered to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

We study the long-time behaviour of axisymmetric solutions without swirl for the three-dimensional Navier-Stokes equations in the whole space. Assuming that the initial vorticity is sufficiently localised, we compute explicitly the leading terms in the asymptotic expansion of the solution, both for the vorticity and the velocity field. In particular, we identify optimal temporal decay rates depending on the spatial localisation of the initial data. Our approach relies on accurate L p - L q estimates for the linearised evolution equation and its Taylor expansion in self-similar variables. [ABSTRACT FROM AUTHOR]

We consider weak boundary layer solutions to the singularly perturbed ODE systems of the type ε 2 (A (x , u (x) , ε) u ′ (x)) ′ = f (x , u (x) , ε) ε 2 (A (x , u (x) , ε) u ′ (x)) ′ = f (x , u (x) , ε). The new features are that we do not consider one scalar equation, but systems, that the systems are allowed to be quasilinear, and that the systems are spatially non-smooth. Although the results about existence, asymptotic behavior, local uniqueness and stability of boundary layer solutions are similar to those known for semilinear, scalar and smooth problems, there are at least three essential differences. First, the asymptotic convergence rates valid for smooth problems are not true anymore, in general, in the non-smooth case. Second, a specific local uniqueness condition from the scalar case is not sufficient anymore in the vectorial case. And third, the monotonicity condition, which is sufficient for stability of boundary layers in the scalar case, must be adjusted to the vectorial case. [ABSTRACT FROM AUTHOR]

Instantaneous volatility of logarithmic return in the lognormal fractional SABR model is driven by the exponentiation of a correlated fractional Brownian motion. Due to the mixed nature of driving Brownian and fractional Brownian motions, probability density for such a model is less studied in the literature. We show in this paper a bridge representation for the joint density of the lognormal fractional SABR model in a Fourier space. Evaluating the bridge representation along a properly chosen deterministic path yields a small time asymptotic expansion to the leading order for the probability density of the fractional SABR model. A direct generalization of the representation of joint density often leads to a heuristic derivation of the large deviations principle for joint density in a small time. Approximation of implied volatility is readily obtained by applying the Laplace asymptotic formula to the call or put prices and comparing coefficients. [ABSTRACT FROM AUTHOR]

In this article, we continue the investigation on the problem of the simultaneous confidence estimation of the vector of mean values for normal distributions when the confidence set is centered at the positive part James-Stein estimator. The coverage probability of these confidence sets for the vector , the same as the quadratic risk of the Stein estimator, depends on its components only through the so-called non-centrality parameter . We provide a survey of already known asymptotics ( and ) for , obtained with the help of the integral representation of the coverage probability established in our previous publications, and propose a new approximation of the second order that possesses the globality property approximating, with some accuracy, the coverage probability for all values of . We focus on the ability of the considered expansions to approximate the coverage probability simultaneously for all values of : the investigation of their global accuracy properties. With the help of an accurate global approximation, we construct confidence sets with the asymptotically constant coverage probability. [ABSTRACT FROM AUTHOR]

In 1981, Zagier conjectured that the Lambert series associated to the weight 12 cusp form Δ should have an asymptotic expansion in terms of the nontrivial zeros of the Riemann zeta function. This conjecture was proven by Hafner and Stopple. In 2017 and 2019, Chakraborty et al. established an asymptotic relation between Lambert series associated to any primitive cusp form (for full modular group, congruence subgroup and in Maass form case) and the nontrivial zeros of the Riemann zeta function. In this paper, we study Lambert series associated with primitive Hilbert modular form and establish similar kind of asymptotic expansion. [ABSTRACT FROM AUTHOR]

The symmetric patterns that inequalities contain are reflected in researchers' studies in many mathematical sciences. In this paper, we prove an asymptotic expansion for the generalized gamma function Γ μ (v) and study the completely monotonic (CM) property of a function involving Γ μ (v) and the generalized digamma function ψ μ (v) . As a consequence, we establish some bounds for Γ μ (v) , ψ μ (v) and polygamma functions ψ μ (r) (v) , r ≥ 1 . [ABSTRACT FROM AUTHOR]

In this paper, we obtain an asymptotic expansion for the determinant of the matrix [ Ω n − i + j ] 1 ⩽ i , j ⩽ r , as n → ∞ , where Ω n denotes the volume of the unit ball in R n . [ABSTRACT FROM AUTHOR]

Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: H n ∼ γ + log n − ∑ k = 1 ∞ B k k · n k , where B k is the Bernoulli numbers. In this paper, we rewrite Ramanujan's harmonic number expansion into a similar form of Euler's asymptotic expansion as n approaches infinity: H n ∼ γ + c 0 (h) log (q + h) − ∑ k = 1 ∞ c k (h) k · (q + h) k , where q = n (n + 1) is the nth pronic number, twice the nth triangular number, γ is the Euler–Mascheroni constant, and c k (x) = ∑ j = 0 k k j c j x k − j , with c k is the negative of the median Bernoulli numbers. Then, 2 c n = ∑ k = 0 n n k B n + k , where B n is the Bernoulli number. By using the result obtained, we present two general Ramanujan's asymptotic expansions for the nth harmonic number. For example, H n ∼ γ + 1 2 log (q + 1 3) − 1 180 (q + 1 3) 2 ∑ j = 0 ∞ b j (r) (q + 1 3) j 1 / r as n approaches infinity, where b j (r) can be determined. [ABSTRACT FROM AUTHOR]

Let Ω n = π n / 2 / Γ (n 2 + 1) (n ∈ N) denote the volume of the unit ball in R n . In this paper, we present asymptotic expansions and new bound related to the quantities: Ω n Ω n - 1 and Ω n 2 Ω n - 1 Ω n + 1 . Finally, a numerical example is provided to demonstrate the effectiveness and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

In the present paper, a characterization of the Favard classes for the sampling Kantorovich operators based upon bandlimited kernels has been established. In order to achieve the above result, a wide preliminary study has been necessary. First, suitable high order asymptotic type theorems in L p -setting, 1 ≤ p ≤ + ∞ , have been proved. Then, the regularization properties of the sampling Kantorovich operators have been investigated. Here, we show how the regularity of the kernel influences the operator itself; this has been shown for bandlimited kernels, or more in general for kernels in Sobolev spaces, satisfying a Strang-Fix type condition of order r ∈ N + . Further, for the order of approximation of the sampling Kantorovich operators, quantitative estimates based on the L p modulus of smoothness of order r have been established. As a consequence, the qualitative order of approximation is also derived assuming f in suitable Lipschitz and generalized Lipschitz classes. Moreover, an inverse theorem of approximation has been stated, allowing to obtain a full characterization of the Lipschitz and of the generalized Lipschitz classes in terms of convergence of the above sampling type series. These approximation results have been proved for not necessarily bandlimited kernels. From the above mentioned characterization, it remains uncovered the saturation case that, however, can be treated by a totally different approach assuming that the kernel is bandlimited. Indeed, since sampling Kantorovich (discrete) operators based upon bandlimited kernels can be viewed as double-singular integrals, exploiting the properties of the convolution in Fourier Analysis, we become able to get the desired result obtaining a complete overview of the approximation properties in L p (R) , 1 ≤ p ≤ + ∞ , for the sampling Kantorovich operators. [ABSTRACT FROM AUTHOR]

We are interested in problems related to forward starting options for Hull–White stochastic volatility model. Our objective is to obtain analytical, semi-analytical, or approximated expressions of its price for simulation. To obtain an analytical representation of the price, we use Yor's formula. However, the analytical formula is difficult to implement. Next we consider semi-analytical expressions for the price. In order to have them, we use the tower property for conditional expectations with a certain filtration and explicitly calculate it. Then, we consider an expansion expression for the price using the semi-analytical expression to have a simple expression. The semi-analytical expressions and the expansion expression can reduce computational costs and standard errors when the Monte Carlo method is used. Finally, some numerical results are given to show their accuracy and efficiency. [ABSTRACT FROM AUTHOR]

In this study, we consider a bias reduction of the conditional maximum likelihood estimators for the unknown parameters of a Gaussian second-order moving average (MA(2)) model. In many cases, we use the maximum likelihood estimator because the estimator is consistent. However, when the sample size n is small, the error is large because it has a bias of $O({n^{-1}})$. Furthermore, the exact form of the maximum likelihood estimator for moving average models is slightly complicated even for Gaussian models. We sometimes rely on simpler maximum likelihood estimation methods. As one of the methods, we focus on the conditional maximum likelihood estimator and examine the bias of the conditional maximum likelihood estimator for a Gaussian MA(2) model. Moreover, we propose new estimators for the unknown parameters of the Gaussian MA(2) model based on the bias of the conditional maximum likelihood estimators. By performing simulations, we investigate properties of this bias, as well as the asymptotic variance of the conditional maximum likelihood estimators for the unknown parameters. Finally, we confirm the validity of the new estimators through this simulation study.

We study a precise asymptotic behavior of the tail probability of the first hitting time of the Bessel process. We deduce the order of the third term and decide the explicit form of its coefficient.

In this study, we compare two methods for using neural networks to efficiently learn the price of derivatives. The first method, proposed by Funahashi (Quant Financ 21(4):575–592, 2021a), involves training the neural networks to learn the difference between the derivative price and its asymptotic expansion, rather than learning the derivative price directly. The target derivative price is then obtained by adding the approximate solution with the predicted value of neural networks. This method reduces the required amount of learning data, often by a factor of one hundred to one thousand, compared to the case where the derivative price is directly learned through neural networks. In the second method, established in this paper, the neural networks learn the difference between the derivative price written on the underlying asset price that follows the target complex stochastic process and the derivative price written on the underlying asset price that has a relatively simple stochastic process that has a closed-form solution for the target derivative prices. This method provides an alternative valuation method when no efficient approximate solution for the derivative value is observed and if one can arbitrarily determine the model parameters of the quasi-process that approximates the original process. We also propose a unified method to determine the model parameters of quasi-processes from underlying asset processes. These methods prove valuable in cases where general analytic solutions are absent, as seen in widely used financial models such as the stochastic volatility models. These cases involve time-consuming numerical calculations to generate learning data, highlighting the value of the two methods, which significantly compress calculation times. [ABSTRACT FROM AUTHOR]

Recent mathematical models of reliability computer systems and telecommunication networks are based on distributions with heavy tails. This paper falls into the category of exploring the classical models with heavy tails: Gnedenko–Weibull, Burr, Benktander distributions. The moment’s asymptotics for residual time have been derived especially for there heavy-tailed distributions. A simulation study is presented to give a practice characterization of the distributions with heavy tails.

We offer a new asymptotic expansion with an explicit remainder estimate in the central limit theorem. The results obtained are essentially based on new forms of asymptotic expansions in the central limit theorem. We also present a more accurate estimation of the CLT-expansions remainder which is rigorously proved and backed up numerically. It is shown that our approach can be used for further refinement of allied asymptotic expansions.

Both theoretical and applied mathematics depend heavily on inequalities, which are rich in symmetries. In numerous studies, estimations of various functions based on the characteristics of their symmetry have been provided through inequalities. In this paper, we study the monotonicity of certain functions that involve Gamma functions. We were able to obtain some of the bounds of Γ(v) that are more accurate than some recently published inequalities.

We study the convergence of the binomial, trinomial, and more generally m-nomial tree schemes when evaluating certain European path-independent options in the Black–Scholes setting. To our knowledge, the results here are the first for trinomial trees. Our main result provides formulae for the coefficients of 1/n and 1/n in the expansion of the error for digital and standard put and call options. This result is obtained from an Edgeworth series in the form of Kolassa–McCullagh, which we derive from a recently established Edgeworth series in the form of Esseen/Bhattacharya and Rao for triangular arrays of random variables. We apply our result to the most popular trinomial trees and provide numerical illustrations.

A steady-state convection-diffusion problem with a small diffusion of order O (ε) is considered in a thin three-dimensional graph-like junction consisting of thin cylinders connected through a domain (node) of diameter O (ε) , where ε is a small parameter. Using multiscale analysis, the asymptotic expansion for the solution is constructed and justified. The asymptotic estimates in the norm of Sobolev space H 1 as well as in the uniform norm are proved for the difference between the solution and proposed approximations with a predetermined accuracy with respect to the degree of ε. [ABSTRACT FROM AUTHOR]

In this paper, we generalize a sequence of positive linear operators introduced by Ismail and May and we study some of their approximation properties for different classes of continuous functions. First, we estimate the error of approximation in terms of the usual modulus of continuity and the second-order modulus of Ditzian and Totik. Then, we characterize the bounded functions that can be approximated uniformly by these new operators. In the last section, we obtain the most important results of the paper. We give the complete asymptotic expansion for the operators and we deduce a Voronovskaya-type theorem, results that hold true for smooth functions with exponential growth. [ABSTRACT FROM AUTHOR]

We consider an optimal control problem in the class of piecewise continuous controls with smooth geometric constraints for a linear system with constant coefficients and with an integral convex performance index containing a small parameter multiplying the integral term. Such problems are called cheap control problems. It is shown that the problem with a terminal performance index will be the limit one. Using an example of a control problem for a point on a plane without resistance, we show that the solution of the cheap control problem behaves more regularly than that of the time-optimal problem for the case in which the optimal control has a discontinuity in the limit problem and is continuous in the original problem. It is shown that the defining vector in the control problem for a point on the plane is expanded in a series in even powers of the small parameter. [ABSTRACT FROM AUTHOR]

One of the main problems of fracture mechanics is to describe the behavior of media with cracks. At the same time, more and more attention is paid to the study of large systems, including periodic structures. For several cracks, the coefficient of influence is considered to be an important parameter. It is the ratio of the stress intensity factor calculated in the problem of a system of cracks to the stress intensity factor for a single crack under the same load. The paper proposes a method that allows to accurately determine the coefficients of influence for large systems of cracks, including the case of periodic structures, in which the number of cracks is infinite. The method is based on a numerical algorithm developed by the authors that uses the method of discontinuous displacements of a high order of accuracy and an asymptotic representation of stress fields at the crack tip (M. Williams). To verify the method, the results of the implemented algorithm are compared with known analytical solutions both for single cracks and for an infinite doubly periodic system of cracks. • The suggested method makes it possible developing stresses in fractures interactions. • The method allows calculating infinite systems of cracks. • Numerically developed stress intensity factors are in good agreement with analytical solutions. • The interaction of periodic systems of cracks in space located in non-parallel planes was investigated. [ABSTRACT FROM AUTHOR]

The purpose of this paper is the study of the asymptotic modeling of a linearly thin elastic isotropic membrane shell which is subjected to frictionless contact with a rigid obstacle on part of the boundary. Using the formal asymptotic expansions method, we obtain a two dimensional Signorini membrane shell model, then we justify the result by convergence theorem. [ABSTRACT FROM AUTHOR]

We consider the asymptotic expansion of the one-parameter Mittag-Leffler function E a (- x) for x → + ∞ as the parameter a → 1 . The dominant expansion when 0 < a < 1 consists of an algebraic expansion of O (x - 1) (which vanishes when a = 1 ), together with an exponentially small contribution that approaches e - x as a → 1 . Here we concentrate on the form of this exponentially small expansion when a approaches the value 1. Numerical examples are presented to illustrate the accuracy of the expansion so obtained. [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]