An integer-valued analogue of the classical generalized autoregressive conditional heteroskedastic (GARCH) (p,q) model with Poisson deviates is proposed and a condition for the existence of such a process is given. For the case p = 1, q = 1, it is explicitly shown that an integer-valued GARCH process is a standard autoregressive moving average (1, 1) process. The problem of maximum likelihood estimation of parameters is treated. An application of the model to a real time series with a numerical example is given. Copyright 2006 The Authors Journal compilation 2006 Blackwell Publishing Ltd.

In this paper, we develop a new approach to obtain the compactness of the fluctuation processes for Boltzmann dynamics. Our method is applicable to Kac's model, already studied by Uchiyama, but it covers many other cases. A novelty worth mentioning is the use of the weak topology of a Hilbert space.

We describe the random meeting motion of a finite number of investors in markets with friction as a Markov pure-jump process with interactions. Using a sequence of these, we prove a functional law of large numbers relating the large motions with the finite market of the so-called continuum of agents.

We study a mean-variance investment problem in a continuous-time framework where the interest rates follow Cox-Ingersoll-Ross dynamics. We construct a mean-variance efficient portfolio through the solutions of backward stochastic differential equations. We also give sufficient conditions under which an explicit analytic expression is available for the mean-variance optimal wealth of the investor. problem using the Hamilton-Jacobi-Bellman equation of dynamic programming. About 15 years later, the equivalent martingale measure concept developed by Harrison and Pliska [7] paved the way for solving several utility maximization problems in a market model with stochastic uniformly bounded parameters (see, for example, Karatzas [8]). Unfortunately one cannot adequately address the continuous-time mean-variance problem using one of the above techniques. This may explains in part why significant results were obtained only recently. Alternate methods had to be considered. For example, Duffie and Richardson [9] solved a deterministic market model mean-variance problem via orthogonal projection techniques. In the early 1990s Pardoux and Peng [10] introduced the notion of backward stochastic differential equations (BSDE) which provided the ideal tool for Lim and Zhou [11] to adequately solve the continuous-time mean-variance allocation problem in a Black-Scholes model with stochastic uniformly bounded market coefficients. It is worth mentioning that BSDE theory was also proved to be valuable in utility portfolio selection problems as shown in [12].Unfortunately, in these papers, the uniform boundedness hypothesis assumed for the interest rate process precludes the use of interest models such as Vasicek, Hull-White and Cox-Ingersoll-Ross (CIR) models which are highly valued by practitioners. In order to pass this limit a number of researchers drew on a more general market where in addition to the usual bank account and stocks an individual is allowed to invest part of his wealth in bonds or interest derivatives. Bajeux-Besnainou and Portait [13] incorporated a Vasicek interest rate model and zero-coupons bonds in their model and solved a mean-variance-type problem. But the Vasicek as well as the Hull-White model have a major drawback which is that there is a positive probability that one encounters negative values. Obviously, this is a highly unrealistic and undesirable feature in a real-world financial market. A few years later, Deelstra et al. [14] modified Bajeux-Besnainou and Portait's model by rather opting for a CIR interest rate process which is known to be almost surely positive under mild conditions. Applying essentially the martingale approach they obtained an optimal strategy for a power-utility maximization problem. Now the classical CIR process rely on three constant parameters, thus, from a practical point of view statistical fitting to real financial data may be greatly improved by considering a more flexible time-varying parameter model as shown in Maghsoodi's [15] study of 25 years of U.S. Treasury bill data....

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