Kenneth J. Hochberg

Composition of Stochastic Processes Governed by Higher-Order Parabolic and Hyperbolic Equations

We consider compositions of stochastic processes that are governed by higherorder partial differential equations. The processes studied include compositions of Brownian motions, stable-like processes with Brownian time, Brownian motion whose time is an integrated telegraph process, and an iterated integrated telegraph process. The governing higher-order equations that are obtained are shown to be either of the usual parabolic type or, as in the last example, of hyperbolic type.

The arc-sine law and its analogs for processes governed by signed and complex measures

The question whether the classical arc-sine law of Paul Levy for the proportion of time spent by a Brownian particle on the positive half-line can be extended to generalized higher-order processes governed by signed and complex measures is studied. Both even and odd-order processes are considered, corresponding to heat-type equations with both real and imaginary coefficients. Finally, several mixtures of the earlier cases are analyzed as well.

Asymptotic behavior of roots of random polynomial equations

We derive several new results on the asymptotic behavior of the roots of random polynomial equations, including conditions under which the distributions of the zeros of certain random polynomials tend to the uniform distribution on the circumference of a circle centered at the origin. We also derive a probabilistic analog of the Cauchy-Hadamand theorem that enables us to obtain the radius of convergence of a random power series.

Stability of stochastic jump-parameter semi-Markov linear systems of differential equations

The asymptotic stability of stochastic Itô-type jump-parameter semi-Markov systems of linear differential equations is examined. A system of integral matrix equations is derived which has the property that the existence of a positive definite solution of the system implies the asymptotic stability of the stochastic semi-Markov system. Finally, an illustrative example is presented.

High-density limits of hierarchically structured branching-diffusing populations

We develop a general probabilistic approach that enables one to get sharp estimates for the almost-sure short-term behavior of hierarchically structured branching-diffusion processes. This approach involves the thorough investigation of the cluster structure and the derivation of some probability estimates for the sets of rapidly fluctuating realizations. In addition, our approach leads to the derivation of new modulus-of-continuity-type results for measure-valued processes. In turn, the modulus-of-continuity-type results for hierarchical branching-diffusion processes are used to derive upper estimates for the Hausdorff dimension of support.

Stability and Optimal Control for Semi-Markov Jump Parameter Linear Systems

We consider continuous-time and discrete-time jump parameter linear control systems with semi-Markov coefficients and solution jumps that coincide with jumps of a semi-Markov random process. First, we derive stability conditions for semi-Markov systems of differential equations. We then determine necessary optimality conditions for the solutions of continuous-time and discrete-time control systems.