hermite polynomial based expansion of european option prices abstract we seek a closed form series approximation of european option prices under a variety of diffusion models the proposed convergent series are derived using the hermite polynomial approach departing from the usual option pricing routine in the literature our model assumptions have no requirements for affine dynamics or explicit characteristic functions moreover convergent expansions provide a distinct insight into how and on which order the model parameters affect option prices in contrast with small time asymptotic expansions in the literature with closed form expansions we explicitly translate model features into option prices such as mean reverting drift and self exciting or skewed jumps numerical examples illustrate the accuracy of this approach and its advantage over alternative expansion methods

saddlepoint approximations for continuous time markov processes this paper proposes saddlepoint expansions as a means to generate closed form approximations to the transition densities and cumulative distribution functions of markov processes this method is applicable to a large class of models considered in finance for which a laplace or characteristic functions but not the transition density can be found in closed form but even when such a computation is not possible explicitly we go one step further by showing how useful approximations can be obtained by replacing the laplace or characteristic functions by an expansion in small time

the delta expansion for the transition density of diffusion models abstract this paper is on the issue of finding a closed form likelihood approximation of diffusion processes and rearranging the hermite expansion in the order of the power of the observational time interval we propose an algorithm that calculates the coefficients of the rearranged expansion that ait sahalia 2002 suggested that is a general expression of the coefficients is provided explicitly which as far as we know has not been given in the existing literature we also introduce a reduced form of the rearranged expansion and call it as the delta expansion in the paper moreover we are able to obtain an explicit expansion of the moments in the order of the power of the observational time interval we examine the delta expansion and the hermite expansion without rearrangement numerically to find that the delta expansion has such advantageous features as the order of the error bound can be more effectively attained it is also found that our expansion gives a comparable numerical accuracy of the approximation to the expansion ait sahalia 1999 suggested while making any symbolic computation unnecessary

efficient estimation of general dynamic models with a continuum of moment conditions there are two difficulties with the implementation of the characteristic function based estimators first the optimal instrument yielding the ml efficiency depends on the unknown probability density function second the need to use a large set of moment conditions leads to the singularity of the covariance matrix we resolve the two problems in the framework of gmm with a continuum of moment conditions a new optimal instrument relies on the double indexing and as a result has a simple exponential form the singularity problem is addressed via a penalization term we introduce hac type estimators for non markov models a simulated method of moments is proposed for non analytical cases

incorporate intelligence into an ecological system an adaptive fuzzy control approach in this paper we incorporate intelligence into kolmogorov s ecological system that is we propose a fuzzy system to combine heuristic rules into the design of nonlinear adaptive controller where the heuristic rules are in the form of fuzzy if then rules by using lyapunov synthesis approach we attain an adaptive law adjusting the free parameters in the controller such that the closed loop system follows a desired trajectory although there exist uncertainty and chaos in komogorov s model application shows that an adaptive fuzzy control is a powerful tool to pursuit sustainable development of ecology

estimation of affine asset pricing models using the empirical characteristic function abstract the known functional form of the conditional characteristic function ccf of discretely sampled observations from an affine diffusion is used to develop computationally tractable and asymptotically efficient estimators of the parameters of affine diffusions and of asset pricing models in which the state vectors follow affine diffusions both time domain estimators based on fourier inversion of the ccf and frequency domain estimators based directly on the ccf are constructed a method of moments estimator based on the ccf is shown to approximate the efficiency of maximum likelihood for affine diffusion and asset pricing models

density approximations for multivariate affine jump diffusion processes we introduce closed form transition density expansions for multivariate affine jump diffusion processes the expansions rely on a general approximation theory which we develop in weighted hilbert spaces for random variables which possess all polynomial moments we establish parametric conditions which guarantee existence and differentiability of transition densities of affine models and show how they naturally fit into the approximation framework empirical applications in option pricing credit risk and likelihood inference highlight the usefulness of our expansions the approximations are extremely fast to evaluate and they perform very accurately and numerically stable

optimal fuzzy control of a poisoning pest model in this paper optimal fuzzy control of a poisoning pest model is considered optimal control has been successfully applied to many biological models we give a rule based fuzzy control for the poisoning pest model by using the pontryagin minimum principle the optimal fuzzy control function can be worked out by a computer with the membership function of the fuzzy sets of the universe of discourse the control parameters of the poisoning pest model will be suitable for practical applications

the complete gaussian kernel in the multi factor heston model option pricing and implied volatility applications abstract in this paper we propose two new representation formulas for the conditional marginal probability density of the multi factor heston model the two formulas express the marginal density as a convolution with suitable gaussian kernels whose variances are related to the conditional moments of price returns via asymptotic expansion of the non gaussian function in the convolutions we derive explicit formulas for european style option prices and implied volatility the european option prices can be expressed as black scholes style terms plus corrections at higher orders in the volatilities of volatilities given by the black scholes greeks the explicit formula for the implied volatility clearly identifies the effect of the higher moments of the price on the implied volatility surface further we derive the relationship between the vix index and the variances of the two gaussian kernels as a byproduct we provide an explanation of the bias between the vix and the volatility of total returns which offers support to recently proposed methods to compute the variance risk premium via a series of numerical exercises we analyse the accuracy of our pricing formula under different parameter settings for the one and two factor models applied to index options on the s p500 and show that our approximation substantially reduces the computational time compared to that of alternative closed form solution methods in addition we propose a simple approach to calibrate the parameters of the multi factor heston model based on the vix index and we apply the approach to the double and triple heston models

a new delta expansion for multivariate diffusions via the ito taylor expansion abstract in this paper we develop a new delta expansion approach to deriving analytical approximation to the transition densities of multivariate diffusions using the ito taylor expansion of the conditional expectation of the dirac delta function our approach yields an explicit recursive formulas for the expansion coefficients and is universally applicable for a wide spectrum of models particularly the time inhomogeneous non affine irreducible multivariate diffusions we show that this new approach can be viewed as an extension of ait sahalia 2002 and lee et al 2014 to the case of multivariate models the derived expansions are proved to converge to the true probability density as the observational time interval shrinks the obtained approximations can thereby be used to carry out the maximum likelihood estimation for the diffusions with discretely observed data extensive numerical experiments demonstrate the accuracy and effectiveness of our approach

global attractivity and oscillations in a periodic delay logistic equation abstract sufficient conditions are obtained for the global attractivity of a positive periodic solution of the delay logistic equation dx t dt r t x t 1 x t n k t where r and k are positive periodic functions of period and n is a positive integer sufficient conditions are also obtained for all solutions to be oscillatory about k

explicit form of approximate transition probability density functions of diffusion processes abstract a continuous time diffusion process is very popular in modeling and provides useful tools to analyze particularly but not restricted to a variety of economic and financial variables the transition probability density function tpdf of a diffusion process plays an important role in understanding and explaining the dynamics of the process a new way to find closed form approximate tpdfs for multivariate diffusions is proposed in this paper this method can be applied to general multivariate time inhomogeneous diffusion processes as long as roughly speaking they have smooth drift and volatility functions a diffusion process is said to be reducible if it can be converted into a unit diffusion process where the volatility is the identity matrix we have established how to obtain the approximate tpdf of a reducible diffusion explicitly when a diffusion process is not reducible an explicit form of approximate tpdf can be obtained by using the results in ait sahalia 2008 and choi 2013 the tpdf expansion suggested here enables us to obtain a recursive formula for the coefficient of the approximate tpdf for a multivariate jump diffusion monte carlo simulation studies of conducting maximum likelihood estimation mle using our approximations provide convincing evidence that our tpdf expansion can be used for the mle when the true tpdf is unavailable we also applied our approximate tpdfs to option pricing the differences between our option prices and those from the extended black scholes formula are shown to be quite small this implies that our methods can be employed to price assets whose underlying state variables follow general diffusion models

the second main theorem of hypersurfaces in the projective space we deal with a holomorphic map from the complex plane mathbb c to the n dimensional complex projective space mathbb p n mathbb c and prove the nevanlinna second main theorem for some families of non linear hypersurfaces in mathbb p n mathbb c this second main theorem implies the defect relation if the degree of the hypersurfaces are sufficiently large the defect of the map is smaller than one this means that holomorphic maps which omit the irreducible hypersurface of large degree is algebraically degenerate to prove the second main theorem we use a meromorphic partial projective connection which is totally geodesic with respect to these hypersurfaces a meromorphic partial projective connection is a family of locally defined meromorphic connections such which work as an entirely defined meromorphic connection under the wronskian operator by resolving the singularity and pulling back a meromorphic partial projective connection we also prove the second main theorem for singular hypersurfaces in mathbb p n mathbb c and prove the second main theorem for smooth hypersurfaces in mathbb p 2 mathbb c which are not normal crossing

closed form likelihood expansions for multivariate time inhomogeneous diffusions the aim of this paper is to find approximate log transition density functions for multivariate timeinhomogeneous diffusions in closed form there are many empirical evidences that the underlying data generating processes for many economic variables might change over time one possible way to explain the time dependent behavior of state variables is to model the drift or volatility terms as functions of time as well as state variables closed form likelihood expansions for multivariate time homogeneous diffusions have been obtained by ait sahalia 2008 this research is built on his work and extends his results to time inhomogeneous cases simulation study reveals that our method yields a very accurate approximate likelihood function that can be a good candidate when the true likelihood function is unavailable

nonparametric pricing of interest rate derivative securities we propose a nonparametric estimation procedure for continuous time stochastic models because prices of derivative securities depend crucially on the form of the instantaneous volatility of the underlying process we leave the volatility function unrestricted and estimate it nonparametrically only discrete data are used but the estimation procedure still does not rely on replacing the continuous time model by some discrete approximation instead the drift and volatility functions are forced to match the densities of the process we estimate the stochastic differential equation followed by the short term interest rate and compute nonparametric prices for bonds and bond options

closed form likelihood approximation and estimation of jump diffusions with an application to the realignment risk of the chinese yuan this paper provides closed form likelihood approximations for multivariate jump diffusion processes widely used in finance for a fixed order of approximation the maximum likelihood estimator mle computed from this approximate likelihood achieves the asymptotic efficiency of the true yet uncomputable mle as the sampling interval shrinks this method is used to uncover the realignment probability of the chinese yuan since february 2002 the market implied realignment intensity has increased fivefold the term structure of the forward realignment rate which completely characterizes future realignment probabilities is hump shaped and peaks at mid 2004 the realignment probability responds quickly to economic news releases and government interventions r 2007 elsevier b v all rights reserved

maximum likelihood estimation of partially observed diffusion models this paper develops a maximum likelihood ml method to estimate partially observed diffusion models based on data sampled at discrete times the method combines two techniques recently proposed in the literature in two separate steps in the first step the closed form approach of ait sahalia 2008 is used to obtain a highly accurate approximation to the joint transition probability density of the latent and the observed states in the second step the efficient importance sampling technique of richard and zhang 2007 is used to integrate out the latent states thereby yielding the likelihood function using both simulated and real data we show that the proposed ml method works better than alternative methods the new method does not require the underlying diffusion to have an affine structure and does not involve infill simulations therefore the method has a wide range of applicability and its computational cost is moderate

nonparametric transition based tests for jump diffusions we develop a specification test for the transition density of a discretely sampled continuous time jump diffusion process based on a comparison of a nonparametric estimate of the transition density or distribution function with their corresponding parametric counterparts assumed by the null hypothesis as a special case our method applies to pure diffusions we provide a direct comparison of the two densities for an arbitrary specification of the null parametric model using three different discrepancy measures between the null and alternative transition density and distribution functions we establish the asymptotic null distributions of proposed test statistics and compute their power functions we investigate the finite sample properties through simulations and compare them with those of other tests this article has supplementary material online

efficient estimation and filtering for multivariate jump diffusions abstract this paper develops estimators of the transition density filters and parameters of multivariate jump diffusion models the drift volatility jump intensity and jump magnitude are allowed to be state dependent and non affine it is not necessary to diagonalize the volatility matrix our density and filter estimators converge at the canonical rate typically associated with exact monte carlo estimation our parameter estimators have the same asymptotic distribution as maximum likelihood estimators which are often intractable for the class of models we consider the results of this paper enable the empirical analysis of previously intractable models of asset prices and economic time series

estimating jump diffusions using closed form likelihood expansions the indispensable role of likelihood expansions in financial econometrics for continuous time models has been established since the ground breaking work of ait sahalia 1999 2002a 2008 jump diffusions play an important role in modeling a variety of economic and financial variables as a significant generalization of li 2013 we propose a new closed form expansion for transition density of poisson driven jump diffusion models and its application in maximum likelihood estimation based on discretely sampled data technically speaking our expansion is obtained by perturbing paths of the underlying model correction terms can be calculated explicitly using any symbolic software numerical examples and monte carlo evidence for illustrating the performance of density expansion and the resulting approximate mle are provided in order to demonstrate the practical applicability of the method using the theoretical results in hayashi and ishikawa 2012 some convergence properties related to the density expansion and the approximate mle method can be justified under some standard sufficient but not necessary conditions

the stochastic difference between econometric statistics dans un contexte general et dans divers cas particuliers on calcule l ordre en probabilite de la difference entre des racines consistantes d equations d estimation rivales

a physical parameter identification method of levy driven vibratory systems based on multipower variation processes abstract in this paper we put forward a physical parameter identification method of levy driven engineering structures based on the properties of the quadratic variation and multipower variation processes the structural dynamic equation is decomposed into the gaussian continuous time autoregressive car equation and the pure jump driven car equation both equations have the same unknown parameters as those included in the levy driven system the parameters of the levy driven system are identified by the maximum likelihood estimation method of the gaussian car system the numerical results demonstrate that the method works well

fully nonparametric estimation of scalar diffusion models we propose a functional estimation procedure for homogeneous stochastic differential equations based on a discrete sample of observations and with minimal requirements on the data generating process we show how to identify the drift and diffusion function in situations where one or the other function is considered a nuisance parameter the asymptotic behavior of the estimators is examined as the observation frequency increases and as the time span lengthens we prove almost sure consistency and weak convergence to mixtures of normal laws where the mixing variates depend on the chronological local time of the underlying diffusion process that is the random time spent by the process in the vicinity of a generic spatial point the estimation method and asymptotic results apply to both stationary and nonstationary recurrent processes

simulated likelihood estimators for discretely observed jump diffusions abstract this paper develops an unbiased monte carlo approximation to the transition density of a jump diffusion process with state dependent drift volatility jump intensity and jump magnitude the approximation is used to construct a likelihood estimator of the parameters of a jump diffusion observed at fixed time intervals that need not be short the estimator is asymptotically unbiased for any sample size it has the same large sample asymptotic properties as the true but uncomputable likelihood estimator numerical results illustrate its properties

non gaussian ornstein uhlenbeck based models and some of their uses in financial economics non gaussian processes of ornstein uhlenbeck ou type offer the possibility of capturing important distributional deviations from gaussianity and for flexible modelling of dependence structures this paper develops this potential drawing on and extending powerful results from probability theory for applications in statistical analysis their power is illustrated by a sustained application of ou processes within the context of finance and econometrics we construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive ou processes and we study these models in relation to financial data and theory

variation based tests for volatility misspecification abstract we provide a simple and easy to use goodness of fit test for the misspecification of the volatility function in diffusion models the test uses power variations constructed as functionals of discretely observed diffusion processes we introduce an orthogonality condition which stabilizes the limit law in the presence of parameter estimation and avoids the necessity for a bootstrap procedure that reduces performance and leads to complications associated with the structure of the diffusion process the test has good finite sample performance as we demonstrate in numerical simulations

estimation of continuous time markov processes sampled at random time intervals we introduce a family of generalized method of moments estimators of the parameters of a continuous time markov process observed at random time intervals the results include strong consistency asymptotic normality and a characterization of standard errors sampling is at an arrival intensity that is allowed to depend on the underlying markov process and on the parameter vector to be estimated we focus on financial applications including tick based sampling allowing for jump diffusions regime switching diffusions and reflected diffusions

on the functional estimation of jump diffusion models abstract we provide a general asymptotic theory for the fully functional estimates of the infinitesimal moments of continuous time models with discontinuous sample paths of the jump diffusion type minimal requirements are placed on the dynamic properties of the underlying jump diffusion process i e stationarity is not required our theoretical framework justifies consistent in a statistical sense nonparametric extraction of the parameters and functions that drive the dynamic evolution of the process of interest i e the potentially nonaffine and level dependent intensity of the jump arrival being an example from the estimated infinitesimal conditional moments as suggested in johannes 2003 the statistical and economic role of jumps in continuous time interest rate models journal of finance forthcoming

point processes and queues martingale dynamics i martingales 1 histories and stopping times 2 martingales 3 predictability 4 square integrable martingales references solutions to exercises chapter i ii point processes queues and intensities 1 counting processes and queues 2 watanabe s characterization 3 stochastic intensity general case 4 predictable intensities 5 representation of queues 6 random changes of time 7 cryptographic point processes references solutions to exercises chapter ii iii integral representation of point process martingales 1 the structure of internal histories 2 regenerative form of the intensity 3 the representation theorem 4 hilbert space theory of poissonian martingales 5 useful extensions references solutions to exercises chapter iii iv filtering 1 the theory of innovations 2 state estimates for queues and markov chains 3 continuous states and nontrivial prehistory references solutions to exercises chapter iv v flows in markovian networks of queues 1 single station the historical results and the filtering method 2 jackson s networks 3 burke s output theorem for networks 4 cascades and loops in jackson s networks 5 independence and poissonian flows in markov chains references solutions to exercises chapter v vi likelihood ratios 1 radon nikodym derivatives and tests of hypotheses 2 changes of intensities a la girsanov 3 filtering by the method of the probability of reference 4 applications 5 the capacity of a point process channel 6 detection formula187 references189 solutions to exercises chapter vi vii optimal control 1 modeling intensity controls 2 dynamic programming for intensity controls complete observation case 3 input regulation a case study in impulsive control 4 attraction controls 5 existence via likelihood ratio references solutions to exercises chapter vii viii marked point processes 1 counting measure and intensity kernels 2 martingale representation and filtering 3 radon nikodym derivatives 4 towards a general theory of intensity references solutions to exercises chapter viii a1 background in probability and stochastic processes 1 introduction 2 monotone class theorem 3 random variables 4 expectations 5 conditioning and independence 6 convergence 7 stochastic processes 8 markov processes references a2 stopping times and point process histories 1 stopping times 2 changes of time and meyer dellacherie s integration formula 3 point process histories references a3 wiener driven dynamical systems 1 ito s stochastic integral 2 square integrable brownian martingales 3 girsanov s theorem references a4 stieltjes lebesgue calculus 1 the stieltjes lebesgue integral 2 the product and exponential formulas references general bibliography