David Assaf

A Diffusion Model for Optimal Portfolio Selection in the Presence of Brokerage Fees

We consider a financial market model with two assets. One has deterministic rate of growth, while the rate of growth of the second asset is governed by a Brownian motion with drift. We can shift money from one asset to another; however, there are losses of money brokerage fees involved in shifting money from the risky to the nonrisky asset. We want to maximize the expected rate of growth of funds. It is proved that an optimal policy keeps the ratio of funds in risky and nonrisky assets within a certain interval with minimal effort.

Multivariate Phase-Type Distributions

A univariate random variable is said to be of phase type if it can be represented as the time until absorption in a finite state absorbing Markov chain. Univariate phase type random variables are useful because they arise from processes that are often encountered in applications, they have densities that can be written in a closed form, they possess some useful closure properties, and they can approximate any nonnegative random variable. This paper introduces and discusses several extensions to the multivariate case. It shows that the multivariate random variables possess many of the properties of univariate phase type distributions and derives explicit formulas for various probabilistic quantities of interest. Some examples are included.

Renewal Decisions when Category Life Distributions are of Phase-Type

A system must operate for t units of time. A certain component is essential for the operation of the system and must be replaced by a new component whenever it fails. There are n types of replacement categories available (with an infinite supply of each) differing only in price and life distribution. The main problem is to select the proper category for replacement at any time a failure occurs, so as to minimize the total expected cost of running the system.In this paper the problem is studied when category life distributions have a common matrix phase type representation. The generalized Erlang and the hyperexponential distributions, as well as coherent structures of the latter distributions, are some special cases of this representation. Our main result is that in many of these cases, the problem of finding an optimal replacement policy is reduced to that of determining a specified number (at most n - 1) of points on the real time axis. Also provided is a condition for identifying categories which should never be used, thus reducing the problem by eliminating them. This work generalizes previous results obtained when all category life distributions were assumed exponential. Several illustrative examples are provided.

The secretary problem : Minimizing the expected rank with I.I.D. random variables

n candidates, represented by n i.i.d. continuous random variables X 1, ...,X n with known distribution arrive sequentially, and one of them must be chosen, using a non-anticipating stopping rule. The objective is to minimize the expected rank (among the ranks of X 1 ,..., X n ) of the candidate chosen, where the best candidate, i.e. the one with smallest X-value, has rank one, etc. Let the value of the optimal rule be V n , and lim V n = V. We prove that V > 1.85. Limiting consideration to the class of threshold rules of the form t n = min {k: X k ≤ a k } for some constants a k , let W n be the value of the expected rank for the optimal threshold rule, and lim W n = W. We show 2.295 < W < 2.327.

Continuous and discrete search for one of many objects

By an easy observation we show that the basic result of Blackwell [2], according to which the most inviting strategy is optimal in a discrete search for one object, is also true when the number of objects is random provided the search is made in continuous time. This result does not hold in the discrete search model even when only tow boxes are present (contrary to a conjecture of Smith and Kimeldorf [7]). For the case of two boxes, a convenient sufficient condition on the distribution of the number of objects is provided which ensures optimality of the most inviting strategy. As a result, this strategy is shown to be optimal for several important distributions.

Two-choice optimal stopping

Let X n ,…,X 1 be independent, identically distributed (i.i.d.) random variables with distribution function F. A statistician, knowing F, observes the X values sequentially and is given two chances to choose Xs using stopping rules. The statisticians goal is to stop at a value of X as small as possible. Let equal the expectation of the smaller of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behaviour of the sequence for a large class of Fs belonging to the domain of attraction (for the minimum) 𝒟(G α), where G α(x) = [1 - exp(-x α)]1(x ≥ 0) (with 1(·) the indicator function). The results are compared with those for the asymptotic behaviour of the classical one-choice value sequence , as well as with the ‘prophet value’ sequence

Optimal cooperative stopping rules for maximization of the product of the expected stopped values

The problem of finding stopping rules which maximize (EXt) (EYt) is considered, for independent pairs (Xi, Yi) of nonnegative r.v.s. with known joint distribution. The solution is compared to that of maximizing E(Xt Yt). When (Xi, Yi) are uniform, a detailed analysis is given for the maximization problem, and for the corresponding minimization and discounted infinite horizon problems.

A New Look at Warning and Action Lines of Surveillance Schemes

Abstract Some authors have suggested an efficient dynamic monitoring procedure for detection of a change in the drift of a Brownian motion. Their procedure can be described as a sequence of extremely short sequential probability ratio tests (SPRTs) done in zero time with infinitesimal time between consecutive SPRTs. The drawbacks of this procedure are that it can be described only as a limit of practical procedures and it does not take into account the cost of initiating a test. In this article, we suggest a procedure that takes both sampling and overhead costs into account and that can reasonably be carried out in practice. In this procedure the process is monitored continuously with a constant sampling rate. The accumulated data are analyzed by the standard cumulative sum (CUSUM) statistics. Whenever the CUSUM procedure raises an alarm, data are accumulated as fast as possible until either the alarm is relaxed or the process is stopped.

INVARIANT PROBLEMS IN DISCOUNTED DYNAMIC PROGRAMMING

Discounted dynamic programming problems whose transition mechanism depends only on the action taken and does not depend on the current state are considered. A value determination operation and method of obtaining optimal policies for the case of finite action space (and arbitrary state space) are presented. The solution of other problems is reduced to this special case by a suitable transformation. Results are illustrated by examples.

Optimal Design of Systems Subject to Two Types of Error

We consider systems such as relay circuits or juries that are composed of components, e.g., relays or jury votes. Each component is in one of two states, with probabilities that depend on the qualities of the components and the requests made of the system. The problem is to construct the system i.e., determine the configurations of the components for which the system is in each of the two states in a way that minimizes the probabilities of errors. We find conditions on the probabilities for which the optimal system is monotone or k out of n. We further study these conditions when the components behave independently.