Every student of elementary mathematics knows that if a quantity is multiplied each period by a positive number x, the result will evolve geometrically with the number of periods n as xn. It is not much harder to show that if instead, the period multiplying factors Xk are identically distributed positive random variables, the result will tend to evolve according to ean where a = E(lnx1), although it will fluctuate from this value. This is the basis of the famous Kelly result used in investment theory. A major (and deep) generalization of this result due to Bellman is that if a vector is repeatedly multiplied by positive random matrices Xk, each component of the results evolves as ena, although a is hard to compute. This major result of Bellman's has been fruitfully applied to many random systems, including those in biology, inventory control, and population dynmics. It can also be applied in investment problems, including problems of industrial investment, which is our primary interest. In many of the most important situations a vector is transformed by random nonlinear mappings Hk. It might seem impossible to characterize the long term result in this case. However, we have extended Bellman's result to the nonlinear case-obtaining virtually the same conclusion concerning growth. This deep result opens the door to many interesting applications. In this project we propose to further the theory of these processes, apply the results to interesting and important investment problems, and develop effective computational methods for computing the relevant quantities.
