computation and analysis of the asymptotic expansions of the compound means we derive algorithms for computing asymptotic expansion of the composite mean of two arbitrary means m and n then we analyse asymptotic behaviour of the compound mean mn examples and application to some classical means are also presented

improved convergence towards generalized euler mascheroni constant we propose new simple sequences approximating the euler mascheroni constant and its generalization which converge faster towards their limits than those considered by detemple d w detemple a quicker convergence to euler s constant am math monthly 100 5 1998 468 470 sintamarian a sintamarian a generalization of euler s constant numer algorithms 46 2 2007 141 151 and vernescu a vernescu a new accelerate convergence to the constant of euler gazeta matem ser a bucharest xvii xcvi 4 1999 273 278

entropy solutions to a strongly degenerate anisotropic convection diffusion equation with application to utility theory we study the deterministic counterpart of a backward forward stochastic differential utility which has recently been characterized as the solution to the cauchy problem related to a pde of degenerate parabolic type with a conservative first order term we first establish a local existence result for strong solutions and a continuation principle and we produce a counterexample showing that in general strong solutions fail to be globally smooth afterward we deal with discontinuous entropy solutions and obtain the global well posedness of the cauchy problem in this class eventually we select a sufficient condition of geometric type which guarantees the continuity of entropy solutions for special initial data as a byproduct we establish the existence of an utility process which is a solution to a backward forward stochastic differential equation for a given class of final utilities which is relevant for financial applications

on weak solutions of forward backward sdes in this paper we continue exploring the notion of weak solution of forward backward stochastic differential equations fbsdes and associated forward backward martingale problems fbmps the main purpose of this work is to remove the constraints on the martingale integrands in the uniqueness proofs in our previous work ma et al in ann probab 36 6 2092 2125 2008 we consider a general class of non degenerate fbsdes in which all the coefficients are assumed to be essentially only bounded and uniformly continuous and the uniqueness is proved in the space of all the square integrable adapted solutions the standard solution space in the fbsde literature a new notion of semi strong solution is introduced to clarify the relations among different definitions of weak solution in the literature and it is in fact instrumental in our uniqueness proof as a by product we also establish some a priori estimates of the second derivatives of the solution to the decoupling quasilinear pde

a combinatorial identity involving gamma function and pochhammer symbol in this paper we establish an identity that relates pochhammer symbol and the ratio of gamma functions the identity is derived using the mellin series representation for the solution of a general algebraic equation

a class of integral approximations for the factorial function we introduce stirling s formula in a more general class of approximation formulas to extend the integral representation of liu z liu a new version of the stirling formula tamsui oxf j math sci 23 4 2007 389 392 finally an accurate approximation for the factorial function is established

a filtering problem with uncertainty in observation abstract this paper is concerned with a generalized kalman bucy filtering model and corresponding robust problem under model uncertainty we find that this robust problem is equivalent to considering an estimate problem under some sublinear operator therefore we turn to obtaining the minimum mean square estimator under a sublinear operator by girsanov theorem and minimax theorem we obtain the optimal estimator x t of the signal process x t for given time t 0 t

semimartingale integral representation we provide an integral representation for smooth functionals of continuous semimartingales the representation is related to an infinite dimensional nonautonomous parabolic equation semimartingale integral representations including a martingale representation are given in terms of a solution to this equation

complete characterizations of the gamma function abstract in this paper we obtain the complete monotonicities regarding both the functions x and x defined by x 1 2 x x e x e x 12 x 2 x x e x 1 x as direct consequences x is strictly increasing and strictly concave on 0 and x is strictly decreasing and strictly convex on 0 furthermore we show that x consists the properties lim x 0 x 0 and lim x x 1 as well as x holds the properties lim x 0 x and lim x x 0

on stirling s formula remainder let the sequence r n be defined by n 2 n n e n e r n we establish new estimates for stirling s formula remainder r n this improves some known results let x be defined by the relation x 1 2 x e x e x 12 x we prove that x is completely monotonic on 0 this implies the result given by mortici who proved that x 1 x is completely monotonic on 0

symmetry algebra of the anisotropic harmonic oscillator with commensurate frequencies the symmetry algebra of the m dimensional quantum anisotropic harmonic oscillator hamiltonian h with commensurate frequencies is shown to be u m each eigenspace of h carries a single irreducible representation of u m however distinct eigenspaces can yield equivalent u m representations the dynamical algebra is the non compact symplectic algebra sp m r for m 3 the anisotropic hamiltonian is relevant to superdeformed nuclei

an inequality for ratios of gamma functions abstract let x denote euler s gamma function the following inequality is proved for y 0 and x 1 we have x y 1 y 1 1 x x y 2 y 1 1 x 1 x y x y 1 the inequality is reversed if 0 x 1 this resolves an open problem of guo and qi b n guo f qi inequalities and monotonicity for the ratio of gamma functions taiwanese j math 7 2003 239 247

noether s theorem and accidental degeneracy it is shown that the determination of the symmetry lie algebra of a quantum system with accidental degeneracy can be obtained in principle by using noether s theorem the procedure is illustrated by considering a quantum system which represents a generalization of the degeneracies present in an anisotropic two dimensional harmonic oscillator thus besides the u 2 algebra which gives the fundamental degeneracy of a two dimensional oscillator the studied system can have an infinite set of states with the same energy characterized by an u 1 1 lie algebra

schrodinger cat states of a non stationary generalized oscillator the two mode even and odd coherent schrodinger cat states are introduced for the generalized stationary and non stationary harmonic oscillator the expected values of positions and momenta and their dispersion matrices are calculated for the schrodinger cat states the quadrature squeezing and correlation phenomenon are studied and the wigner and husimi functions are constructed explicitly for these states the photon statistics of the two mode even and odd coherent states of the generalized oscillator is described in terms of the photon distribution function which is expressed in terms of hermite polynomials the photon number means and dispersions show oscillations which are characteristics of squeezing and correlation phenomenon

integral representations and complete monotonicity related to the remainder of burnside s formula for the gamma function abstract in the paper the authors establish integral representations of some functions related to the remainder of burnside s formula for the gamma function and find the logarithmically complete monotonicity of these and related functions these results extend and generalize some known conclusions

two new asymptotic expansions of the ratio of two gamma functions formal expansions giving as particular cases semiasymptotic expansions of the ratio of two gamma functions are obtained

a stable multistep scheme for solving backward stochastic differential equations in this paper we propose a stable multistep scheme on time space grids for solving backward stochastic differential equations in our scheme the integrands which are conditional mathematical expectations derived from the original equations are approximated by using lagrange interpolating polynomials with values of the integrands at multiple time levels they are then numerically evaluated using the gauss hermite quadrature rules and polynomial interpolations on the spatial grids error estimates are rigorously proved for the semidiscrete version of the proposed scheme for backward stochastic differential equations with certain types of simplified generator functions finally various numerical examples and comparisons with some other methods are presented to demonstrate high accuracy of the proposed multistep scheme

explicit form and robustness of martingale representations stochastic integral representation of martingales has been undergoing a renaissance due to questions motivated by stochastic finance theory in the brownian case one usually has formulas of differing degrees of exactness for the predictable integrands we extend some of these to markov cases where one does not necessarily have stochastic integral representation of all martingales moreover we study various convergence questions that arise naturally from for example approximations of price processes via euler schemes for solutions of stochastic differential equations we obtain general results of the following type let u u n be random variables with decompositions u alpha int 0 infty xi s dx s n infty u n alpha n int 0 infty xi n s dx n s n n infty where x n x n n n are martingales if x n to x and u n to u when and how does xi n rightarrow xi

on semi linear degenerate backward stochastic partial differential equations in this paper we study a class of one dimensional degenerate semilinear backward stochastic partial differential equations bspdes for short of parabolic type by establishing some new a priori estimates for both linear and semilinear bspdes we show that the regularity and uniform boundedness of the adapted solution to the semilinear bspde can be determined by those of the coefficients a special feature that one usually does not expect from a stochastic differential equation the proof follows the idea of the so called bootstrap method which enables us to analyze each of the derivatives of the solution under consideration some related results including some comparison theorems of the adapted solutions for semilinear bspdes as well as a nonlinear stochastic feynman kac formula are also given

noether s theorem and time dependent quantum invariants the time dependent integrals of motion linear in position and momentum operators of a quantum system are extracted from noether s theorem prescription by means of special time dependent variations of coordinates for the stationary case of the generalized two dimensional harmonic oscillator the time independent integrals of motion are shown to correspond to special bragg type symmetry properties a detailed study for the nonstationary case of this quantum system is presented the linear integrals of motion are constructed explicitly for the case of varying mass and coupling strength they are also obtained from noether s theorem the general treatment for a multi dimensional quadratic system is indicated and it is shown that the time dependent variations that give rise to the linear invariants as conserved quantities satisfy the corresponding classical homogeneous equations of motion for the coordinates

path regularity for solutions of backward stochastic differential equations in this paper we study the path regularity of the adpated solutions to a class of backward stochastic differential equations bsde for short whose terminal values are allowed to be functionals of a forward diffusion using the new representation formula for the adapted solutions established in our previous work 7 we are able to show under the mimimum lipschitz conditions on the coefficients that for a fairly large class of bsdes whose terminal values are functionals that are either lipschitz under the l norm or under the l 1 norm then there exists a version of the adapted solution pair that has at least cadlag paths in particular in the latter case the version can be chosen so that the paths are in fact continuous

representation of integral dispersion relations by local forms the representation of the usual integral dispersion relations idrs of scattering theory through series of derivatives of the amplitudes is discussed extended simplified and confirmed as mathematical identities forms of derivative dispersion relations ddrs valid for the whole energy interval recently obtained and presented as double infinite series are simplified through the use of new sum rules of the incomplete functions being reduced to single summations where the usual convergence criteria are easily applied for the forms of the imaginary amplitude used in phenomenology of hadronic scattering at high energies we show that expressions for the ddr can represent with absolute accuracy the idr of scattering theory as true mathematical identities besides the fact that the algebraic manipulation can be easily understood numerical examples show the accuracy of these representations up to the maximum available machine precision as consequence of our work it is concluded that the standard

numerical algorithms for forward backward stochastic differential equations efficient numerical algorithms are proposed for a class of forward backward stochastic differential equations fbsdes connected with semilinear parabolic partial differential equations as in j douglas jr j ma and p protter ann appl probab 6 1996 pp 940 968 the algorithms are based on the known four step scheme for solving fbsdes the corresponding semilinear parabolic equation is solved by layer methods which are constructed by means of a probabilistic approach the derivatives of the solution u of the semilinear equation are found by finite differences the forward equation is simulated by mean square methods of order 1 2 and 1 corresponding convergence theorems are proved along with the algorithms for fbsdes on a fixed finite time interval we also construct algorithms for fbsdes with random terminal time the results obtained are supported by numerical experiments

correspondence between some wave patterns and lissajous figures we study the connections between some specially entangled stationary states and the classical periodic trajectories of two non interacting oscillators the latter are well known lissajous figures which are shown to run precisely over the apogees of their corresponding probability distributions in the above states we propose in this work a very simple criterion enabling us to obtain the best agreement between the quantum and the classical images finally our results are successfully applied to the interpretation of some experimentally observed wave patterns

quasi linear pdes and forward backward stochastic differential equations weak solutions abstract in this paper we study the existence uniqueness and the probabilistic representation of the weak solutions of quasi linear parabolic and elliptic partial differential equations pdes in the sobolev space h 1 r d for this we study first the solutions of forward backward stochastic differential equations fbsdes with smooth coefficients regularity of solutions and their connection with classical solutions of quasi linear parabolic pdes then using the approximation procedure we establish their convergence in the sobolev space to the solutions of the fbsdes in the space l 2 r d r d l 2 r d r k l 2 r d r k d this gives a connection with the weak solutions of quasi linear parabolic pdes finally we study the unique weak solutions of quasi linear elliptic pdes using the solutions of the fbsdes on infinite horizon

on numerical approximations of forward backward stochastic differential equations a numerical method for a class of forward backward stochastic differential equations fbsdes is proposed and analyzed the method is designed around the four step scheme j douglas jr j ma and p protter ann appl probab 6 1996 pp 940 968 but with a hermite spectral method to approximate the solution to the decoupling quasi linear pde on the whole space a rigorous synthetic error analysis is carried out for a fully discretized scheme namely a first order scheme in time and a hermite spectral scheme in space of the fbsdes equally important a systematical numerical comparison is made between several schemes for the resulting decoupled forward sde including a stochastic version of the adams bashforth scheme it is shown that the stochastic version of the adams bashforth scheme coupled with the hermite spectral method leads to a convergence rate of frac 32 in time which is better than those in previously published work for the fbsdes

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

bernoulli polynomials and asymptotic expansions of the quotient of gamma functions the main subject of this paper is the analysis of asymptotic expansions of wallis quotient function c x t c x s and wallis power function c x t c x s 1 t s when x tends to infinity coefficients of these expansions are polynomials derived from bernoulli polynomials the key to our approach is the introduction of two intrinsic variables a 12 t s 1 and b 14 1 t s 1 t s which are naturally connected with bernoulli polynomials and wallis functions asymptotic expansion of wallis functions in terms of variables t and s and also a and b is given application of the new method leads to the improvement of many known approximation formulas of the stirling s type

asymptotic expansions for the gamma function in terms of hyperbolic functions abstract the smith s approximation formula for the gamma function is given by x 1 2 2 x e x 2 x tanh 1 2 x x 2 1 o 1 x 5 as x in this paper by means of a little known power series we develop this formula to an asymptotic expansions x 1 2 2 x e x 2 x tanh 1 2 x x 2 exp n 3 2 n 2 n 1 2 2 n 1 1 2 1 2 n 2 n 2 n 1 2 n b 2 n x 2 n 1 as x and give an estimate of remainder in the above asymptotic series where b 2 n is the bernoulli number moreover as relevant results ramanujan type asymptotic expansions for the gamma function x 1 2 are obtained an asymptotic expansion for wallis ratio and an estimate of the remainder are established

estimations of psi function and harmonic numbers the asymptotic expansion of digamma function is a starting point for the derivation of approximants for harmonic sums or euler mascheroni constant it is usual to derive such approximations as values of logarithmic function which leads to the expansion of the exponentials of digamma function in this paper the asymptotic expansion of the function exp p x t is derived and analyzed in details especially for integer values of parameter p the behavior for integer values of p is proved and as a consequence a new identity for bernoulli polynomials the obtained formulas are used to improve know inequalities for euler s constant and harmonic numbers

exactification of the poincare asymptotic expansion of the hankel integral spectacularly accurate asymptotic expansions and non asymptotic scales we obtain an exactification of the poincare asymptotic expansion pae of the hankel integral as using the distributional approach of mcclure wong we find that for half integer orders of the bessel function the exactified asymptotic series terminates so that it gives an exact finite sum representation of the hankel integral for other orders the asymptotic series does not terminate and is generally divergent but is amenable to superasymptotic summation i e by optimal truncation for specific examples we compare the accuracy of the optimally truncated asymptotic series owing to the mcclure wong distributional method with owing to the mellin barnes integral method we find that the former is spectacularly more accurate than the latter by in some cases more than 70 orders of magnitude for the same moderate value of b moreover the exactification can lead to a resummation of the pae when it is exact with the resummed poincare series exhibiting again the same spectacular accuracy more importantly the distributional method may yield meaningful resummations that involve scales that are not asymptotic sequences

density analysis of bsdes in this paper we study the existence of densities with respect to the lebesgue measure for marginal laws of the solution y z to a quadratic growth bsde using the by now well established connection between these equations and their associated semi linear pdes together with the nourdin viens formula we provide estimates on these densities

on the asymptotic expansions of the gamma function related to the nemes gosper and burnside formulas in this paper we present new asymptotic expansions of the gamma function related to the nemes gosper and burnside formulas

monotonicity properties inequalities and asymptotic expansions associated with the gamma function we define x by the equality x 1 x e x 8 x 3 4 x 2 x 1 30 1 6 e x we call x the remainder of ramanujan s formula in this paper we present some properties for x including monotonicity properties inequalities and asymptotic expansions furthermore we present some full asymptotic expansions for the gamma function related to the nemes ramanujan and burnside formulas

asymptotic analysis of the iterative power means abstract we investigate the asymptotic expansion of the compound mean obtained by the iterative process of two power means we present the stationary and convergence properties of the coefficients in the expansions

commensurate anisotropic oscillator su 2 coherent states and the classical limit we demonstrate a formally exact quantum classical correspondence between the stationary coherent states associated with the commensurate anisotropic two dimensional harmonic oscillator and the classical lissajous orbits our derivation draws upon the earlier work of louck et al 1973 j math phys 14 692 wherein they have provided a non bijective canonical transformation that maps within a degenerate eigenspace the commensurate anisotropic oscillator on to the isotropic oscillator this mapping leads in a natural manner to a schwinger realization of su 2 in terms of the canonically transformed creation and annihilation operators through the corresponding coherent states built over a degenerate eigenspace we directly effect the classical limit via the expectation values of the underlying generators our work completely accounts for the fact that the su 2 coherent state in general corresponds to an ensemble of lissajous orbits

the asymptotic series of the generalized stirling formula the aim of this paper is to construct some asymptotic expansions which produce increasingly accurate approximations of the gamma function

solving forward backward stochastic differential equations explicitly a four step scheme in this paper we investigate the nature of the adapted solutions to a class of forward backward stochastic differential equations sdes for short in which the forward equation is non degenerate we prove that in this case the adapted solution can always be sought in an ordinary sense over an arbitrarily prescribed time duration via a direct four step scheme using this scheme we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic pde system moreover the uniqueness of the adapted solutions over an arbitrary time duration as well as the continuous dependence of the solutions on the parameters can all be proved within this unified framework some special cases are studied separately in particular we derive a new form of the integral representation of the clark haussmann ocone type for functionals or functions of diffusions in which the conditional expectation is no longer needed

new sequence converging towards the euler mascheroni constant we propose new sequences containing a modified logarithmic term which converge to the euler mascheroni constant faster than sequences known from the literature

best estimates of the generalized stirling formula the aim of this paper is to introduce an approximations family of the factorial function that contains stirling s formula burnside s formula and gosper s formula the parameters which provide the best approximations are indicated finally numerical computations are made to show the superiority of our formulas over other known formulas

mean variance portfolio selection under partial information this paper is concerned with a continuous time mean variance portfolio selection problem in a possibly incomplete market with multiple stocks and a bond only the past price movements of the stocks and the bond are the information available to the investors a separation principle is shown to hold in this setting efficient strategies based on the aforementioned partial information are derived which involve the optimal filter of the stock appreciation rate processes the main methodological contribution of the paper is to employ the particle system representation to develop analytical and numerical approaches in obtaining the filter as well as solving the related backward stochastic differential equation

products of ratios of gamma functions an application to the distribution of the test statistic for testing the equality of covariance matrices abstract an expansion of the ratio of two gamma functions is used to obtain an adequate representation for the product of ratios of gamma functions which enables the derivation of approximating expansions for the distribution of the test statistic used for testing the equality of several covariance matrices these asymptotic approximations are based on mixtures of gamma distributions and are obtained using a new expression of the fourier transform of the density function of the logarithm of the test statistic and also on a matching moments technique numerical studies show that these approximations are simple precise and easy to implement a computational module developed for mathematica software is presented together with an example of application

linear quadratic stochastic differential games open loop and closed loop saddle points in this paper we consider a linear quadratic stochastic two person zero sum differential game the controls for both players are allowed to appear in both drift and diffusion of the state equation the weighting matrices in the performance functional are not assumed to be definite non singular the existence of an open loop saddle point is characterized by the existence of an adapted solution to a linear forward backward stochastic differential equation with constraints together with a convexity concavity condition and the existence of a closed loop saddle point is characterized by the existence of a regular solution to a riccati differential equation it turns out that there is a significant difference between open loop and closed loop saddle points also it is found that there is an essential feature that prevents a linear quadratic optimal control problem from being a special case of linear quadratic two person zero sum differential games

solution to discrete time linear fbsdes with application to stochastic control problem in this paper we consider a class of infinite horizon forward and backward stochastic difference equations fbsdes that are fully coupled the adapted solution is given by establishing an explicit relation between the forward and backward components in terms of a generalized algebraic riccati equation the equivalence between the exponential stabilizability of the stochastic control system and the infinite horizon fbsdes is derived as an application the fbsdes are used to characterize the maximum principle of the infinite horizon stochastic optimal control problem

functional ito calculus and stochastic integral representation of martingales we develop a non anticipative calculus for functionals of a continuous semimartingale using an extension of the ito formula to path dependent functionals which possess certain directional derivatives the construction is based on a pathwise derivative introduced by b dupire for functionals on the space of right continuous functions with left limits we show that this functional derivative admits a suitable extension to the space of square integrable martingales this extension denes a weak derivative which is shown to be the inverse of the ito integral and which may be viewed as a non anticipative lifting of the malliavin derivative these results lead to a constructive martingale representation formula for ito processes by contrast with the clark haussmann ocone formula this representation only involves nonanticipative quantities which may be computed pathwise

stochastic target problems dynamic programming and viscosity solutions in this paper we define and study a new class of optimal stochastic control problems which is closely related to the theory of backward sdes and forward backward sdes the controlled process x nu y nu takes values in mathbb r d times mathbb r and a given initial data for x nu 0 then the control problem is to find the minimal initial data for y nu so that it reaches a stochastic target at a specified terminal time t the main application is from financial mathematics in which the process x nu is related to stock price y nu is the wealth process and nu is the portfolio we introduce a new dynamic programming principle and prove that the value function of the stochastic target problem is a discontinuous viscosity solution of the associated dynamic programming equation the boundary conditions are also shown to solve a first order variational inequality in the discontinuous viscosity sense this provides a unique characterization of the value function which is the minimal initial data for y nu

the maximum principle for global solutions of stochastic stackelberg differential games national science foundation dms 1303775 research grant council of the hksar cityu 500113

asymptotic formulas for gamma function with applications in the article we present several asymptotic formulas for the gamma function in terms of the bivariate means as applications some sharp upper and lower bounds for the gamma function and factorial n are given

a new asymptotic series for the gamma function the famous stirling s formula says that s 1 2 s s e se s 2 s e se s 12s in this paper we obtain a novel convergent asymptotic series of s and proved that s is increasing for s 0

asymptotic expansions of integral means and applications to the ratio of gamma functions integral means are important class of bivariate means in this paper we prove the very general algorithm for calculation of coefficients in asymptotic expansion of integral mean it is based on explicit solving the equation of the form b a x c x where asymptotic expansions of b and c are known the results are illustrated by calculation of some important integral means connected with gamma function

backward sdes with constrained jumps and quasi variational inequalities we consider a class of backward stochastic differential equations bsdes driven by brownian motion and poisson random measure and subject to constraints on the jump component we prove the existence and uniqueness of the minimal solution for the bsdes by using a penalization approach moreover we show that under mild conditions the minimal solutions to these constrained bsdes can be characterized as the unique viscosity solution of quasi variational inequalities qvis which leads to a probabilistic representation for solutions to qvis such a representation in particular gives a new stochastic formula for value functions of a class of impulse control problems as a direct consequence this suggests a numerical scheme for the solution of such qvis via the simulation of the penalized bsdes

on an open problem of chen and mortici concerning the euler mascheroni constant abstract chen and mortici ch p chen c mortici new sequence converging towards the euler mascheroni constant comput math appl 2011 http dx doi org 10 1016 j camwa 2011 03 099 proposed an open problem for a given positive integer s find the constants a i i 0 1 2 s such that k 1 n 1 k log n i 0 s a i n i is the fastest sequence which would converge to the euler mascheroni constant using logarithmic type bell polynomials we solve this problem the main result shows that the a i s can be recursively determined

oscillator eigenstates concentrated on classical trajectories the highly degenerate eigenspaces of the two dimensional isotropic harmonic oscillator are proven to contain eigenstates that are optimally localized on the closed trajectories of the classical dynamics as h cross to 0 their phase space probability density converges to the unique probability density on the corresponding trajectory which is invariant under the classical flow

some inequalities and monotonicity properties associated with the gamma and psi functions abstract in this paper we present new completely monotonic functions and bernstein functions relating to the gamma and psi functions as consequences we establish new upper and lower bounds for the gamma function in terms of function also we present some estimates for stirling s formula remainder

bid price controls for network revenue management martingale characterization of optimal bid prices we consider a continuous time rate based model of network revenue management under mild assumptions we construct a simple e optimal bid price control which can be viewed as a perturbation of a bid price control in the classical sense williamson e l 1992 airline network seat control ph d thesis mit cambridge ma we show that the associated bid price process forms a martingale and the corresponding booking controls converge in an appropriate sense to an optimal control as e tends to 0 moreover we show that there exists an optimal generalized bid price control where the bid price process forms a martingale and is used in conjunction with a capacity usage limit process we also discuss its connection to the bid price controls in the classical sense and sufficient conditions for the near optimality of the latter

on some inequalities for the gamma and psi functions we present new inequalities for the gamma and psi functions and we provide new classes of completely monotonic star shaped and superadditive functions which are related to and