A secret to create a complete market from an incomplete market

The Martingale method has been given increasing attention since it was conducted by Cox and Huang in 1989. Martingale method allows us to solve the problems of utility maximization in a very elegant manner. However, the Martingale method is not omnipotent. When the market is incomplete, traditional Martingale method will be problematic. To overcome the problem of incompleteness, Karatzas/Shreve/Xu [Martingale and duality for utility maximization in an incomplete market, SIAM J. Control and Optimization 29 (1991) 702–730] have developed a way to complete the market by introducing additional fictitious stocks and then making them uninteresting to the investor. Nevertheless, to find such fictitious stocks is not straightforward. In particular, when the number of such stocks needed in order to complete the market were very large, it would be very computational, and even may not be possible to be expressed explicitly. To make life easier, we provide an alternative method by directly creating a complete market from the incomplete one such that the dimension of the underlying Brownian motion equals the number of available stocks. Our approach is ready to be used.