COMPLETE MONOTONICITY FOR A NEW RATIO OF FINITELY MANY GAMMA FUNCTIONS

In this paper,by deriving an inequality involving the generating function of the Bernoulli numbers,the author introduces a new ratio of finitely many gamma functions,finds complete monotonicity of the second logarithmic derivative of the ratio,and simply reviews the complete monotonicity of several linear combinations of finitely many digamma or trigamma functions.

Complete Monotonicity of Functions Related to Trigamma and Tetragamma Functions

In this paper,we study the completely monotonic property of two functions involving the functionΔ(x)=[ψ′(x)]2+ψ″(x)and deduce the double inequality x^(2)+3x+3/3x^(4)(2x+1)^(2)<Δ(x)<625x^(2)+2275x+5043/3x^(4)(50x+41)^(2),x>0which improve some recent results,whereψ(x)is the logarithmic derivative of the Gamma function.Also,we deduce the completely monotonic degree of a function involvingψ′(x).

COMPLETE MONOTONICITY OF THE PROBABILITY OF RUIN AND DE FINETTI'S DIVIDEND PROBLEM

This paper studies the complete monotonicity of the probability of ruin in the the classical risk model and the classical risk model that is perturbed by a diffusion. As a byproduct, the authors give an alternative proof to a result on the optimal dividend problem due to Loeffen （2008）.

Complete monotonicity of a function involving the divided diference of digamma functions

In the paper, necessary and sufficient conditions are provided for a function involving the divided difference of two psi functions to be completely monotonic. Consequently, a class of inequalities for sums are presented, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions are derived, and two double inequalities for bounding the ratio of two gamma functions are discovered.

SOME COMPLETELY MONOTONIC FUNCTIONS ASSOCIATED WITH THE q-GAMMA AND THE q-POLYGAMMA FUNCTIONS

In this paper, the q-analogue of the Stirling formula for the q-gamma function （Moak formula） is exploited to prove the complete monotonicity properties of some functions involving the q-gamma and the q-polygamma functions for all real number q 〉 0. The monotonicity of these functions is used to establish sharp inequalities for the q-gamma and the q-polygamma functions and the q-Harmonic number. Our results are shown to be a generalization of results which were obtained by Selvi and Batir [23].

Predictable forward performance processes in complete markets

We establish existence of Predictable Forward Performance Processes(PFPPs)in conditionally complete markets,which has been previously shown only in the binomial setting.Our market model can be a discrete-time or a continuous-time model,and the investment horizon can be finite or infinite.We show that the main step in construction of PFPPs is solving a one-period problem involving an integral equation,which is the counterpart of the functional equation found in the binomial case.Although this integral equation has been partially studied in the existing literature,we provide a new solution method using the Fourier transform for tempered distributions.We also provide closedform solutions for PFPPs with inverse marginal functions that are completely monotonic and establish uniqueness of PFPPs within this class.We apply our results to two special cases.The first one is the binomial market and is included to relate our work to the existing literature.The second example considers a generalized Black–Scholes model which,to the best of our knowledge,is a new result.

Rank-dependent ppredictableforward performance processes

Predictable forward performance processes(PFPPs)are stochastic optimal control frameworks for an agent who controls a randomly evolving system but can only prescribe the system dynamics for a short period ahead.This is a common scenario in which a controlling agent frequently re-calibrates her model.We introduce a new class of PFPPs based on rank-dependent utility,generalizing existing models that are based on expected utility theory(EUT).We establish existence of rank-dependent PFPPs under a conditionally complete market and exogenous probability distortion functions which are updated periodically.We show that their construction reduces to solving an integral equation that generalizes the integral equation obtained under EUT in previous studies.We then propose a new approach for solving the integral equation via theory of Volterra equations.We illustrate our result in the special case of conditionally complete Black-Scholes model.

The time discretization in classes of integro-differential equations with completely monotonic kernels:Weighted asymptotic stability

We study discretization in classes of integro-differential equationswhere the functions aj（t）, 1 ≤ j ≤n, are completely monotonic on （0, ∞） and locally integrable, but not constant. The equations are discretized using the backward Euler method in combination with order one convolution quadrature for the memory term. The stability properties of the discretization are derived in the weighted 11 （p; 0, ∞） norm, where p is a given weight function. Applications to the weighted l^1 stability of the numerical solutions of a related equation in Hilbert space are given.