In this paper, we analyze the pricing of European option when the riskfree interest rate follows a jump process. An expression for European call price is first obtained in the case of constant volatility. Then we present a general formulation which takes care of the variation of volatility. In this general formulation we utilize a point process filtering technique to estimate the state process. Finally we carry out some numerical simulation of our results

In this paper, we consider the class of general bilinear models given by , where is an i.i.d. sequence of heavy tailed noise variables, and where . By means of a point process analysis, we show that the sample correlation function converges in distribution to a nondegenerate random variable. Thus standard model selection and fitting tools when applied to nonlinear heavy tailed models will be misleading. Also, consistency of the Hill estimator as an estimate of the tail index for this class of bilinear models is proved

In this paper, an entrywise perturbation theory for rate matrices of GI/M/1 type Markov chains is presented. We consider perturbation bounds for the rate matrices if small relative perturbations are introduced into the entries of blocks that define the queues. For the class of GI/M/1 type Markov chains whose A 0 is of rank one, we give the relative error bound for each entry of the rate matrices

An exact matrix representation of the operator associated with a correlated arrival process is obtained from which an accurate numerical procedure is derived for the computation of the dominant eigenvalue . The peakedness function is studied in order to obtain a useful approximation to and to permit the numerical computation of peakedness. This is important in practice because direct observation of data allows peakedness determination but does not lead directly to the eigenvalue; however, performance measures of queueing models depend directly on Large deviation limit formulae are derived for this correlated arrival process, namely, for the time to the l'th arrival and for the number of arrivals in , which are then used to obtain the decrements (critical exponent coefficients) for the waiting time in a single server queue and for the number in system. A consequence of these formulae is a simple relation between the two decrements. Also, approximations for the distributions of complementary waiting time and number in system are constructed

We consider the ergodic Markov chain satisfying where is a sequence of i.i.d. random variables, considered by Bhattacharya and Lee. After modifications of their sufficient conditions for ergodicity, we give slightly different sufficient conditions for geometric ergodicity on finiteness of an exponential moment of |εn|. Our conditions are weaker than theirs concerning the main part h(Xn) of the Markov chain.

The aim of the present paper is to obtain Bayes estimators for the offspring mean of a simple branching process with a power series offspring probability distribution. We study the sensitivity behavior of the obtained estimators with respect to the choice of the loss function. We propose a minimax criterion using the Bayes risk for ranking the effectiveness (in the sense of robustness) of the loss functions that are being used. In particular, we show analytically that the Bayes estimator under the relative squared-error loss is preferable to the posterior mean. The usefulness of the analytical results is illustrated on some actual epidemic data from smallpox outbreaks.

In this paper, generalized bounds are developed for the ruin probability under a Markovian modulated risk model. First, lower and upper bounds in terms of NBU and NWU type distributions or functions are obtained which includes the exponential bound as a special case when the Cramer Condition is satisfied. The technique used here is a combination of the conjugation idea, the Markov renewal theorem and the induction method. Then, a numerical iteration method is proposed for obtaining tightened monotone bounds by using the developed lower and upper bounds. Finally, improved bounds are obtained in terms of the ladder height distribution and an example is used for illustration

Let be a compound Poisson process, i.e. positive random variables and a time homogeneous Poisson jump process. We consider two linear boundaries and and the stopping times and . Laplace Stieltjes transforms are developed for the distributions of TU and T for general distribution of . These transforms are obtained by analyzing the sample path behavior of the process {Yt }. The results are applicable to reliability theory, sequential analysis, queuing theory, dam theory, risk analysis, and more