Introduction to stochastic integration

the preferences are partial orders such as interval orders. G. Chichilinsky states an Arrow typetheorem for which independence of irrelevant alternatives is replaced by continuity, with continuous spaces of alternatives and preferences, for instance when preferences are given by linear utility functions in the plane. She gives similar results for the existence of games having Nash equilibria. G. Heal in a similar context studies social choice rules on contractible spaces of preferences (contractibility is a necessary and sufficient condition for existence). K. Suzumura defines collective choice rules by composing the Pareto optimal set with transitive closure of majority voting in different orders satisfying conditions of justice. Y.-K. Ng argues that in Harsanyi’s utilitarian social welfare the sum of utilities and the average utility should be the criteria for varying populations on the basis that an additional person is like oneself having an experience and then forgetting it. P. Hammond studies ex post optimality for social welfare over time. Ex ante optimality is optimality of W( C pi,&, where pin and Vinare subjective probability and utility of state n to person i, and W is the welfare function. Ex post optimality is optimality of C p,W(v,,). Hammond argues that ex post optimality is the better criterion. P. Grout shows that sophisticated planning (in which changes in taste over time are considered) may be Pareto dominated by naive planning. T. Bandyopadhyay and R, Deb obtain results on nonstrategyproofness, extending those of Ma&tire and Pattanaik. B. Dutta constructs a neutral exactly and strongly consistent voting rule and proves only dictatorial social choice functions can be implemented with type 2 equilibrium. B. Peleg proves results about tight representations of simple games by social choice functions. N. Schofield considers voting games with a continuous space of alternatives and proves a conjecture of Tullock that continuous voting cycles are Paretian in two-dimensions, by relating local cycles to the Nakamura number. P. Coughlin and S. Nittan relate local electoral equilibria at a status quo to local maxima of the mean log-likelihood function, which must exist. R. Gardner shows that in disequilibrium (fixed prices) the A-transfer value for the Bohm-Bawerk horse market is generally consistent only with uniform rationing. KM. Kim Mathematical Social Sciences Alabama Store University

Montgomery, Alabama 36195 U.S.A.

K.L. Cbuag and R.3. Williams, Introdw*ion Birklmuser,1983, 191 pages,$19.95.

to Stochastic Integmtion. Boston:

Let Br C&C --- be o-algebras of events. A martingale is a sequence of random variables satisfying E( V,+ I 1B,)t - V,. The 5, andard example is the sum Y, =

Xi + - -- + Xn of variables

Xn. where E(& _ I XI. __. . Xc? -- 0. Esamples include Brownian motion and Poisson processes normalized by ubtracting CU. Thi< bwk is concerned with stochastic integrals

where M is a martingale and x is a stochastic process. The integral is ag-ain a martingale (as a function of f) and integration gives an isometr\-. lt is defined in Chapter 2 extending

where my__, denotes the characteristic fur&on of a set A. The integral w-as defined by Ito for Brotynian motion. Chapter 3 ewnds ;hc In Chapter 4. the quadratic variation proccsszs [.\I).. = class of integrands. (A&)‘- (MO)’ - 2 g MdM are studied. In Chapter 5, ths lto formula 6 pcowd. which is

Chapter 6 applies this to characterize Brownian motion by its quadratic \ariati~rr to prove that the exponential processes of martingales are continuous locaf martingales, and to show solutions of the Schrodinger equation are giwn by Feym~arn-l&c functionals. Chapter 7 derives Tanaka’s formula that B - s for a Brow nian mut ion B is the sum of another Brownian moilon and a process L( -. A-)known as losai time measuring the amount of time spent near A-.Chapter 8 studies reflected Brownian motion, as may be used in hydropower stcsrage. Chapter 9 generaiises the ltc) formula to convex f.

A- Coleman, Game Theory and Experimental Games. Oxford: Pergamon.

WC.

301 pages, $17.!6.

The author states that the object of the book is to give a critical survey of tunic game theory and experimental games. In the reviewer’s opinia>n he does an rt~ellent job of relating the two areas. The theoretical portion is elementary and quite readable for a neophyte in the area, with many examples of games from life. Chapter 1 is an introduction and Chapter 2 is on decision theory . Chapter : is