STOCHASTIC

Parts IV and V of the volume discuss dynamic models with uncertainty
taken into account in the decision-making process. Some models can be reduced
to or analyzed as static models in some way and the papers by Wilson, Fama
and Hakansson discuss three such ways to do this. Wilson's model has a single
decision point. Fama shows the result that many dynamic stochastic models are
equivalent to a static model where the future period's random variables have
been eliminated through the expectation operator and the future period's
decision variables have been optimized out. This notion has been used in the
stochastic programming literature since the early paper of Dantzig (1955) and is
standard in the current theoretical literature, see e.g. Birge and Louveaux (1997)
or Wets and Ziemba (1999). This analysis assumes that the distributions of the
random variables are independent of the decisions made. The results show that
concavity is preserved over maximization and expectation but strict concavity is
not preserved without additional assumptions; see Ziemba (1974, 1977) on this
latter point.
Mossin (1968) showed that for essentially unconstrained dynamic investment
problems with power utility functions, the optimal policy was myopic if
the asset returns were intertemporally independent. Hakansson's paper showed
that if the objective function was logarithmic, then an optimal myopic policy
existed for general dependent assets. This paper and those by Breiman and
Thorp in Section 4 of Part V concern the extremely important but little used
(except by some rather talented people discussed below) Kelly (1956) capital
growth theory.
Section V concerns dynamic models and begins with Vickson's intuitive
discussion of the ltd calculus for stochastic differential equations and stochastic
optimal control used in continuous time finance. The only real continuous time
paper in the volume is Merton's in Section IV. Two major areas have evolved
out of continuous time finance whose father is Robert C. Merton, the Nobel
prize winner in economics in 1997 for his work related to the Black-Scholes
(1973) and Merton (1973) option pricing model. The area has exploded with
thousands of articles and hundreds of books on option pricing models and
applications published since these fundamental contributions were made. Our
book just touches the surface of such work so we refer readers to such option
pricing and derivative security books as Rubinstein (1998), Duffie (2001),
Shreve (2004), Hull (2006), and Wilmott (2006). In this area, the continuous
time modeling is crucial to the option pricing. One area of use is in hedge fund
mispricing that is "buy A and sell A*", a close substitute but which is more
expensive. Then wait in this risk arbitrage until the prices converge within a
xvin
PREFACE AND BRIEF NOTES TO THE 2006 EDITION