J. Michael Harrison

Bio: J. Michael Harrison is an academic researcher from Stanford University. The author has contributed to research in topics: Queueing theory & Heavy traffic approximation. The author has an hindex of 45, co-authored 86 publications receiving 15644 citations. Previous affiliations of J. Michael Harrison include University of Florida & University of Bristol.

Dynamic Scheduling of a Two-Class Queue: Small Interest Rates

Abstract: We consider a single server queuing system with two classes of customers who arrive according to independent Poisson processes. The two service time distributions are arbitrary, and we assume a linear holding cost and fixed service reward for each class. The problem is to decide, at the completion of each service and given the state of the system, which class (if any) to admit next into service. We seek a policy, called Blackwell optimal, which will maximize for all sufficiently small interest rates the expected net present value of service rewards received minus holding costs incurred over an infinite planning horizon. For a variety of different cases, it is shown that there exists a Blackwell optimal policy which simply enforces a static priority ranking of the classes, choosing idleness only when the system is empty. Criteria for ranking the classes are derived, extending classical results on optimal priority rules. For other cases, involving one zero holding cost and/or an unstable system, it is shown...

Optimal Funding and Asset Allocation Rules for Defined-Benefit Pension Plans

Abstract: This paper considers a world in which pension funds may default, the cost of the associated risk of default is not borne fully by the sponsoring corporation, and there are differential tax effects. The focus is on ways in which the wealth of the shareholders of a corporation sponsoring a pension plan might be increased if the Internal Revenue Service (IRS) and the Pension Benefit Guaranty Corporation (PBGC) follow simple and naive policies. Under the conditions examined, the optimal policy for pension plan funding and asset allocation is shown to be extremal in a certain sense. This suggests that the IRS and the PBGC may wish to use more complex regulatory procedures than those considered in the paper.

A Stochastic Calculus Model of Continuous Trading: Complete Markets

Abstract: A paper by the same authors in the 1981 volume of Stochastic Processes and Their Applications presented a general model, based on martingales and stochastic integrals, for the economic problem of investing in a portfolio of securities. In particular, and using the terminology developed therein, that paper stated that every integrable contingent claim is attainable (i.e., the model is complete) if and only if every martingale can be represented as a stochastic integral with respect to the discounted price process. This paper provides a detailed proof of that result as well as the following: The model is complete if and only if there exists a unique martingale measure.

Some Stochastic Bounds for Dams and Queues

Abstract: Let X = {Xt, t > 0} be a process of the form Xt = Zt-ct, where c is a positive constant and Z is an infinitely divisible, nondecreasing pure jump process. Assuming E[Xt] < 0. let U be the d.f. of M = sup Xt. As is well known, U is the contents distribution of a dam with input Z and release rate c. If Z is compound Poisson, one can alternately view U as the waiting time distribution for an M/G/1 queue or 1-U as the ruin function for a risk process. Letting X and X0 be two processes of the indicated form, it is shown that U ≤ U0 if the two jump measures are ordered in a sense weaker than stochastic dominance. In the case where EM = EM0, a different condition on the jump measures yields E[fM] ≤ E[fM0] for all concave f, this resulting from second-order stochastic dominance of the supremum distributions. By way of application, processes with deterministic jumps are shown to be extremal in certain ways, and those with exponential jumps are shown to be extremal among the class having IFR jump distribution. Finally, an extremal property of Brownian Motion which is not among the class of processes considered is demonstrated, this yielding simple bounds for E[fM] with f concave or convex. It is shown how all the bounds obtained for U or E[fM] can be further sharpened with additional computation.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Controlled Markov processes and viscosity solutions

Abstract: This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance. In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an