Anticipative Stochastic Calculus with Applications to Financial Markets

In this thesis, we study both local time and Malliavin calculus and their application to stochastic calculus and finance. In the first part, we analyze three aspects of applications of local time. We first focus on the existence of the generalized covariation process and give an approximation when it exists. Thereafter, we study the decomposition of ranked semimartingales. Lastly, we investigate an application of ranked semimartingales to finance and particularly pricing using Bid-Ask. The second part considers three problems of optimal control under asymmetry of information and also the uniqueness of decomposition of “Skorohod-semimartingales”. First we look at the problem of optimal control under partial information, and then we investigate the uniqueness of decomposition of “Skorohod-semimartingales” in order to study both problems of optimal control and stochastic differential games for an insider.