René Ferland

Integer-Valued GARCH Process

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.

Compactness of the fluctuations associated with some generalized nonlinear Boltzmann equations

In this paper, we develop a new approach to obtain the compactness of the fluctuation processes for Boltzmann dynamics. Our method is applicable to Kacs 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

LAWS OF LARGE NUMBERS FOR PAIRWISE INTERACTING PARTICLE SYSTEMS

We consider a system of Markov processes of finitely-many particles which exchange their energies in pairs at random times. A law of large numbers for this system means that the empirical measures of the processes may be approximated (as the number of particles increases) by the solution of a nonlinear evolution equation (the so-called McKean-Vlasov limit). This work presents two results of this type. The first one concerns the empirical processes and gives a probabilistic method for solving the nonlinear equation. The second is stated in the path scheme and extends classical results of chaos propagation by Kac (1956) and McKean (1967).

Cutoff-type Boltzmann equations: Convergence of the solution

Using an approach similar to Tanakas we prove the convergence toward equilibrium for general classes of models which correspond to Boltzmann equations of the cutoff type. A major step consists in showing a convex inequality involving the Kantorovich-Vasherstein metric. This requires assumptions on the interacting kernels. These assumptions are very natural from a physical point of view. In particular, our classes include models recently developed by physicists to study relaxation of closed oscillator systems.

Absolute continuity, singular measures and asymptotics for estimators

Abstract Consider the simple problem of providing an estimator for θ, θ > 0, from observations (Xn), where (X1,…,Xn,…) is a sequence of independent r.v., Xn ∼ Bernoulli (pn), with p n = 1 2 + θa n , 0

Binomial Boltzmann processes: Convergence of the fluctuations

Abstract Under mild initial assumptions, a functional central limit theorem is obtained for a system of randomly interacting particles regulated by a binomial kernel of interaction. This system is related to a nonlinear Boltzmann-type equation.

Monetary Policy and Interest Rate Caps: A Regime-Shift Approach

In this paper, we derive and empirically test a regime-shifting dynamic term structure model for pricing interest rate caps. The central state variables are the target rate of the Federal Reserve and its latent regime in which it fluctuates. These state variables are driven by a discrete time inhomogeneous Markov chain that captures the timing of FOMC meetings. The Fed Funds rate and the slope of the Libor-swap term structure complete the set of state variables. Their dynamics exhibit regime-shifts based on the latent regime of the target rate in both the physical and risk-neutral measures. The modelling approach for pricing at-the-money cap prices leads to an exponential affine analytical form where the regime-shift risk is priced. Allowing for this feature significantly helps the model to empirically match cap prices. The empirical analysis also indicates that the stable regime where the target rate is frequently at inertia is a state where market operators exhibit a higher level of risk aversion. It is also in this regime that the term structure of the mean Black implied volatilities is the highest and where cap price changes are the most sensitive to jumps in the target rate.

A Propagation of Chaos Related to the Continuity Laws

ABSTRACT We build a system composed of a large number of randomly interacting particles in such a way that the empirical laws converge (when the number of particles tends to infinity) to a weak solution of a nonlinear evolution equation, which is a relaxed kinetic equation related to scalar conservation laws.

The Convergence of the Solution of a Boltzmann Type Equation Related to Quantum Mechanics

McKean (1966), Tanaka (1978), and Sznitman (1984) have obtained existence, uniqueness and asymptotic results for the solution of a Boltzmann type equation, for the cases of Kac’s caricature, Maxwell’s gas and Boltzmann’s gas, respectively. Their methods use Wild’s sums. Here we adapt Tanaka’s method for his asymptotic result to show, with the help of Wild’s sums, the convergence toward the geometric equilibrium of the solution of a Boltzmann type equation related to the Bose-Einstein statistic (r = 1) of quantum mechanics.