General Martingale Research Articles

Optimal Consumption by a Bond Investor: The Case of Random Interest Rate Adapted to a Point Process

The problem of optimizing the expected total discounted utility of the rate of consumption by an agent with initial wealth x who, at each instant, can choose a rate of consumption and who invests all his unconsumed wealth in a bond (bank) is studied. The bond is assumed to have a randomly varying interest rate with known probabilistic behavior. A general martingale principle is formulated, according to which the optimal consumption rate is expressed in terms of a positive martingale, if one can be found, satisfying an almost sure integral condition. This martingale will be characterized in the case where the stochastic history of the interest rate is adapted to (i.e., expressible in terms of) an underlying counting process. The problem studied can be viewed as a special case of optimal consumption problems in “incomplete markets,” a terminology introduced to the financial literature by Harrison and Pliska [Stochastic Process. Appl., 11 (1981), pp. 215–260].

Approximation for the solutions of stochastic differential equations: ii strong convergence

A general approximation model for square integrable continuous martingales is considered. One studies the strong approximation (i.e. in probability, uniform with respect to the initial value and to the time in compact sets) for the solutions of stochastic differential equations driven by martingales by means of solutions of integral equations driven by the approximation processes. The strong convergence of the derivatives (with respect to the initial value; of any order) is also proved. Various strong convergence theorems have already been proved by Malliavin, Ikeda, Watanabe, Bismut, Shu and finally, for general approximation models, by Kunita. However, all the above-mentioned results refer to the Brownian motion while we deal with a general martingale.

Stock Selection and Timing—A New Look At Market Efficiency

In this paper it is shown that stock returns do not conform to a random walk model, nor to the more general martingale model, but nevertheless the stock market is weak-form efficient. This result is not surprising when we recall that risk-averse investors are typically concerned with more than the first moment of a security return’s distribution, and market efficiency is appropriately determined in terms of (expected) utility - not profits alone. Extant tests of market efficiency have ignored this point. By examining the first two moments of return distributions, and by not assuming any form of equilibrium pricing model, we provide tests of weakform market efficiency which are more powerful than prior studies. A market is considered informationally efficient if market prices ‘fully reflect’ information. The (sub)set of information presumed to be reflected determines the particular form of market efficiency. The market is said to be efficient in the weak-forms if market prices fully reflect the past realizations of market prices. Extant tests of the weak-form of market efficiency (e.g., Fama, 1970; Fama and Blume, 1966; and Mandelbrot, 1966) examine whether information on past price movements can be exploited to enhance profit. These tests assume that investors looking to profit from ‘trend’ data are risk-neutral. However, the normative theory of portfolio selection due to Markowitz (1952), and positive theories of capital asset pricing (e.g., Sharpe, 1964) are developed assuming that investors are risk-averse, This paper fills a gap in the literature by testing the informational efficiency of the stock market by exploring whether or not gains in CXpGGtcd utility are attainable by utilizing the time series of past stock prices. In particular, since Sharpe (1963), a simple algorithm has been available for risk-averse investors to use in selecting optimal portfolios. Normative Portfolio Theory of Markowitz (1952) (N.2“) suggests the use of historical data to obtain estimates of expected return and risk, and then apply a mathematical programming algorithm to build Mean-Variance (E- V) efficient portfolios. Surprisingly, studies that have tackled the timing question have done so in a vacuum vis-B-vis optimal (E- V) portfolio selection.

Martingales in Markov processes applied to risk theory

A general martingale, related to the theory of Markov processes, is introduced and it is shown how it can be used in risk theory.

A stochastic calculus for continuous N-parameter strong martingales

Let M be a 4 N-integrable, real-valued continuous N-parameter strong martingale. Burkholder's inequalities prove to be an adequate tool to control the quadratic oscillations of M and the integral processes associated with it (i.e. multiple 1-stochastic integrals with respect to M and its quadratic variation) such that a 1-stochastic calculus for M can be designed. As the main results of this calculus, several Ito-type formulas are established: one in terms of the integral processes associated with M, another one in terms of the so-called ‘variations’, i.e. stochastic measures which arise as the limits of straightforward and simple approximations by Taylor's formula; finally, a third one which is derived from the first by iterated application of a stochastic version of Green's formula and which may be the strong martingale form of a prototype for general martingales.

A general Martingale approach to discrete-time stochastic control and estimation

A general method of constructing system models for the solution of discrete time stochastic control and estimation problems is presented. The method involves the application of modern martingale theory and entails the judicious choice of certain sigma-algebras and martingales. General estimation equations are derived for observations taking values in a countable space, and previously obtained estimation equations are exhibited as special cases. Finally, an example of the application of these methods to a stochastic control problem is analyzed.

Limits of directed projective systems of probability spaces

The problem of limits of directed projective systems of probability spaces is treated from the categorical point of view, with certain equivalence classes of measurable measurepreserving mappings, or of regular conditional probabilities playing the role of morphisms. A. o. the existence of limits of such systems is established under the condition that the spaces carry compact generating pavings, without invoking conditions like Bochners' sequential maximality. There are some side results on liftings and general martingales.