## The Martingale Method in Continuous Time

*This post describes the martingale method in continuous time. It should be read after this one. It illustrates the method on a simple example within the Black and Scholes framework where an investor maximizes the utility of terminal wealth. In this context, the martingale method allows to spell out how optimal terminal wealth depends on the unique stochastic discount factor, or alternatively, how it is obtained as a transformation of the stock return. The transformation hinges on the shape of the utility function. The case of a CRRA utility function is fully spelled out. The results obtained through dynamic programming here are recovered although the martingale method does not easily uncover the trading policy that generates the optimal terminal wealth.*