default probabilities of a holding company with complete and partial information abstract this paper studies the valuation of credit risk for firms that own several subsidiaries or business lines we provide simple analytical approximating expressions for probabilities of default and for equity debt market values both in the case when the information is available in continuous time as well as in the case that it is not instantaneously available the total firm s asset value being modeled as a sum of lognormal random variables we use convex upper and lower approximations to infer these analytical approximating expressions we extend the model to firms financed by multiple stochastic liabilities and conclude by numerical illustrations

exit problems for jump processes with applications to dividend problems this paper investigates the first passage times to flat boundaries for hyper exponential jump diffusion processes explicit solutions of the laplace transforms of the distribution of the first passage times the joint distribution of the first passage times and undershoot overshoot the joint distribution of the process and running suprema infima are obtained the processes recover many models appearing in the literature such as the compound poisson risk models the diffusion perturbed compound poisson risk models and their dual models as applications we present explicit expressions of the dividend formulae for barrier strategy and threshold strategy

higher order corrections in the approximation of low dimensional manifolds and the construction of simplified problems with the csp method in systems of stiff ordinary differential equations odes both fast and slow time scales are encountered the fast time scales are responsible for the development of low dimensional manifolds on which the solution moves according to the slow time scales in this paper methodologies for constructing highly accurate i expressions describing the manifold and ii simplified non stiff equations governing the slow evolution of the solution on the manifold are developed according to an iterative procedure proposed in the computational singular perturbation csp method it is shown that the increasing accuracy achieved with each iteration is directly related to the time rates of change of the csp vectors spanning the manifold along the solution trajectory here an algorithm is presented which implements these calculations and is validated on the basis of two simple examples

the quasi steady state assumption a case study in perturbation the quasi steady state assumption qssa of biochemistry is studied as an approximation that is important in itself and also that exemplifies an approach to ode systems with an initial fast transient simple estimates of the two relevant time scales of the underlying problem are made these estimates lead to a dimensionless formulation containing a different small parameter from that which is customary earlier results on the qssa are surveyed in the context of the new nondimensionalization it is shown that more comprehensive and simpler error estimates can now be made some general methodological hints are drawn from this example

on success runs of a fixed length in bernoulli sequences exact and asymptotic results consider a sequence of n bernoulli success failure or 1 0 trials the exact and limiting distribution of the random variable e n k denoting the number of success runs of a fixed length k email protected email protected n is derived along with its mean and variance an associated waiting time is examined as well the exact distribution is given in terms of binomial coefficients and an extension of it covering exchangeable sequences is also discussed limiting distributions of e n k are obtained using poisson and normal approximations the exact mean and variance of e n k which are given in explicit forms are also used to derive bounds and an additional approximation of the distribution of e n k numbers associated with e n k and related random variables counting binary strings and runs of 1 s useful in applications of computer science are provided the overall study is illustrated by an extensive numerical experimentation

a first passage problem for a bivariate diffusion process numerical solution with an application to neuroscience when the process is gauss markov we consider a bivariate gauss markov process and we study the first passage time of one component through a constant boundary we prove that its probability density function is the unique solution of a new integral equation and we propose a numerical algorithm for its solution convergence properties of this algorithm are discussed and the method is applied to the study of the integrated brownian motion and to the integrated ornstein uhlenbeck process finally a model of neuroscience interest is discussed

asymptotic solution of stiff pdes with the csp method the reaction diffusion equation the computational singular perturbation csp method is employed for the solution of stiff pdes and for the acquisition of the most important physical understanding the usefulness of the method is demonstrated by analyzing a transient reaction diffusion problem it is shown that in the regions where the solution exhibits smooth spatial slopes a simple nonstiff system of equations can be used instead of the full governing equations from the simplified system which is numerically provided by csp and whose structure varies with space and time important physical information comes to light the relation of this method to the class of asymptotic expansion methods is explored it is shown that the csp results are identical to the ones obtained by the asymptotic methods the identifications of the nondimensional parameters and the tedious manipulations needed by the asymptotic methods are performed by programmable numerical or analytic computations specified by csp preliminary numerical results are presented validating the theoretical aspects of the proposed algorithm and providing a measure of its usefulness and its accuracy

on barrier option pricing by erlangization in a regime switching model with jumps abstract we consider the risk neutral pricing of vanilla digital and down and out call options when the underlying asset price evolves like the exponential of a markov modulated brownian motion mmbm with two sided phase type jumps the price of such options is intimately related to the first passage properties of the mmbm to analyse these first passages we randomize the time horizon using erlang distributions with suitable parameters and apply matrix analytic methods this provides us with closed form approximations of the options prices with a very high precision as shown by several numerical illustrations in particular we consider an example in which the phase type jump distribution is constructed in such a way that it mimics fat tails

phase type fitting of scale functions for spectrally negative levy processes we study the scale function of the spectrally negative phase type levy process its scale function admits an analytical expression and so do a number of its fluctuation identities motivated by the fact that the class of phase type distributions is dense in the class of all positive valued distributions we propose a new approach to approximating the scale function and the associated fluctuation identities for a general spectrally negative levy process numerical examples are provided to illustrate the effectiveness of the approximation method

physical understanding of complex multiscale biochemical models via algorithmic simplification glycolysis in saccharomyces cerevisiae abstract large scale models of cellular reaction networks are usually highly complex and characterized by a wide spectrum of time scales making a direct interpretation and understanding of the relevant mechanisms almost impossible we address this issue by demonstrating the benefits provided by model reduction techniques we employ the computational singular perturbation csp algorithm to analyze the glycolytic pathway of intact yeast cells in the oscillatory regime as a primary object of research for many decades glycolytic oscillations represent a paradigmatic candidate for studying biochemical function and mechanisms using a previously published full scale model of glycolysis we show that due to fast dissipative time scales the solution is asymptotically attracted on a low dimensional manifold without any further input from the investigator csp clarifies several long standing questions in the analysis of glycolytic oscillations such as the origin of the oscillations in the upper part of glycolysis the importance of energy and redox status as well as the fact that neither the oscillations nor cell cell synchronization can be understood in terms of glycolysis as a simple linear chain of sequentially coupled reactions

option pricing under a double exponential jump diffusion model this paper aims to extend the analytical tractability of the black scholes model to alternative models with arbitrary jump size distributions more precisely we propose a jump diffusion model for asset prices whose jump sizes have a mixed exponential distribution which is a weighted average of exponential distributions but with possibly negative weights the new model extends existing models such as hyperexponential and double exponential jump diffusion models because the mixed exponential distribution can approximate any distribution as closely as possible including the normal distribution and various heavy tailed distributions the mixed exponential jump diffusion model can lead to analytical solutions for laplace transforms of prices and sensitivity parameters for path dependent options such as lookback and barrier options the laplace transforms can be inverted via the euler inversion algorithm numerical experiments indicate that the formulae are easy to implement and accurate the analytical solutions are made possible mainly because we solve a high order integro differential equation explicitly a calibration example for spy options shows that the model can provide a reasonable fit even for options with very short maturity such as one day this paper was accepted by michael fu stochastic models and simulation

distribution theory of runs a markov chain approach abstract the statistics of the number of success runs in a sequence of bernoulli trials have been used in many statistical areas for almost a century even in the simplest case of independent and identically distributed bernoulli trials the exact distributions of many run statistics still remain unknown departing from the traditional combinatorial approach in this article we present a simple unified approach for the distribution theory of runs based on a finite markov chain imbedding technique our results cover not only the identical bernoulli trials but also the nonidentical bernoulli trials as a byproduct our results also yield the exact distribution of the waiting time for the mth occurrence of a specific run

on the leading term of the outer asymptotic expansion of van der pol s equation it is proved rigorously that the leading term in the outer asymptotic expansion of the time dependent solution describing relaxation oscillations of the van der pol equation is correct this is accomplished by constructing rigorous estimates of the difference between the exact solution and the outer asymptotic solution as constructed by j d cole these estimates are both rigorous and numerically computable

a second order finite difference method for option pricing under jump diffusion models we develop a finite difference method to solve partial integro differential equations which describe the behavior of option prices under jump diffusion models with localization to a bounded domain of the spatial variable these equations are discretized on uniform grid points over a finite domain of time and spatial variables the proposed method is based on three time levels and leads to linear systems with tridiagonal matrices in this paper the stability of the proposed method and the second order convergence rate with respect to a discrete ell 2 norm are proved numerical results obtained with european put options under the merton and kou models show the behaviors of the stability and the second order convergence rate

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

iterative algorithm for the first passage time distribution in a jump diffusion model with regime switching and its applications for a regime switching model with a finite number of regimes and double phase type jumps jiang and pistorius 2008 derived matrix equations with real parameters for the wiener hopf factorization the laplace transform of the first passage time distribution is expressed in terms of the solution of the matrix equations in this paper we provide an iterative algorithm for solving the matrix equations of jiang and pistorius 2008 with complex parameters this makes it possible to obtain numeric values of the laplace transform with complex parameters for the first passage time distribution the laplace transform with complex parameters can be inverted by numerical inversion algorithms such as the euler method as an application we compute the prices of defaultable bonds under a structural model with regime switching and double phase type jumps

closed form solutions for perpetual american put options with regime switching this paper studies an optimal stopping time problem for pricing perpetual american put options in a regime switching model an explicit optimal stopping rule and the corresponding value function in a closed form are obtained using the modified smooth fit technique the solution is then compared with the numerical results obtained via a dynamic programming approach and also with a two point boundary value differential equation tpbvde method

the role of slow system dynamics in predicting the degeneracy of slow invariant manifolds the case of vdp relaxation oscillations abstract the development of dissipative time scales which are much faster than the rest force the solution of a system of 1st order odes to land and then evolve on a slow invariant manifold the slow evolution on this manifold can be approximated by a reduced system which is constructed with asymptotic analysis and is free of the fast scales in this work the dynamics of the reduced system are compared to the fast and slow dynamics of the original system the comparison is based on the van der pol system which is typical of those exhibiting relaxation oscillation behavior and is characterized by successive dissipative and explosive phases special attention is given to the dissipative phase during which the solution evolves on a slow invariant manifold and the fast slow time scale gap tends to contract it is shown that one cannot predict from the dynamics of the van der pol system the approaching explosive phase and the associated degeneracy of the manifold such a prediction is possible only through the dynamics of the reduced system which involves an explosive component that is absent in the original system this explosive component accelerates as higher order effects are incorporated in the reduced system resulting in a smaller gap between the fast time scale of the original system and the fast time scale of the reduced one this feature sets a limit on the accuracy singular perturbation techniques can deliver even in the presence of a reasonably large fast slow time scale gap in the dynamics of the original system given a fast slow system predicting forthcoming changes in the dimensions of the manifold is very important when it is desired to identify the underlying major physical mechanisms or to control the process

an efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems the relation between the iterative algorithms based on the computational singular perturbation csp and the invariance equation ie methods is examined the success of the two methods is based on the appearance of fast and slow time scales in the dynamics of stiff systems both methods can identify the low dimensional surface in the phase space slow invariant manifold sim where the state vector is attracted under the action of fast dynamics it is shown that this equivalence of the two methods can be expressed by simple algebraic relations csp can also construct the simplified non stiff system that models the slow dynamics of the state vector on the sim an extended version of ie is presented which can also perform this task this new ie version is shown to be exactly similar to a modified version of csp which results in a very efficient algorithm especially in cases where the sim dimension is small so that significant model simplifications are possible

a note on hurwitz stability of matrices this note proves that every hurwitz stable matrix can be expressed as the product of a symmetric positive definite matrix and a generalised negative definite matrix based on this it is further shown that the entire set of all hurwitz stable matrices of order n is the product of two convex open cones and itself forms a simply connected open cone in the parameter space with a vertex at the origin

a full likelihood procedure for analysing exchangeable binary data a full likelihood procedure is proposed for analyzing correlated binary data under the assumption of exchangeability the binomial and beta binomial models are shown to occur as special cases correspondingly respectively to the choice of degenerate and beta mixing distributions for a finite exchangeable binary sequence of random variables expressions for the joint distribution moments and correlations of all orders are derived maximum likelihood estimates of the moments of all orders are computed and used to estimate correlations and the distribution of the number of responses in a cluster in an application to developmental toxicology data analysis the procedure introduced is compared with a beta binomial and a generalized estimating equation procedure in which mean response and intralitter correlation are linked to dose

on the distribution of runs of ones in binary strings in this paper we derive the number of binary strings which contain for a given i k exactly i k runs of 1 s of length k in all possible binary strings of length n email protected email protected n such a knowledge about the distribution pattern of runs of 1 s in binary strings is useful in many engineering applications for example data compression bus encoding techniques to reduce crosstalk in vlsi chip design computer arithmetic using redundant binary number system and design of energy efficient communication schemes in wireless sensor networks by transformation of runs of 1 s into compressed information patterns among others we present here a generating function based approach to derive a solution to this counting problem our experimental results demonstrate that for most commonly used file formats the observed distributions of exactly i k runs of length k email protected email protected n closely follow the theoretically derived distributions for a given n for n 8 we find that the experimentally obtained values for most file formats agree within 5 of the theoretically obtained values for all i k runs of length k email protected email protected n also the root mean square rms values of these deviations across all file types studied in this paper are less than 5 for n 8 in view of these facts the results presented in this paper could be useful in various application domains like the ones mentioned above

on the mean and extreme distances between failures in markovian binary sequences this paper is concerned with the mean minimum and maximum distances between two successive failures in a binary sequence consisting of markov dependent elements these random variables are potentially useful for the analysis of the frequency of critical events occurring in certain stochastic processes exact distributions of these random variables are derived via combinatorial techniques and illustrative numerical results are presented