Conditioned Stochastic Differential Equations: Theory and Applications

We generalize the notion of brownian bridge. More precisely, we study a standard brownian motion for which a certain functional is conditioned to follow a given law. Such processes appear as weak solutions of stochastic differential equations which we call conditioned stochastic differential equations. The link with the theory of initial enlargement of filtration is made and after a general presentation several examples are studied: the conditioning of a standard brownian motion by its value at a given date, the conditioning of a geometric brownian motion with negative drift by its quadratic variation and finally the conditioning of a standard brownian motion by its first hitting time of a given level. The conditioned stochastic differential equation associated with the quadratic variation of the geometric brownian motion allows us to give a new proof of the extension of the Matsumoto-Yor’s 〈X〉 X theorem. Moreover, we show that the set of all the bridges over a given diffusion Z can be parametrized by a generalized Burger’s equation whose solutions are related by the Hopf-Cole transformation to the positive space-time harmonic functions of Z. As a consequence of this, we deduce that the set of diffusions which have the same bridges as Z is parametrized by the positive eigenfunctions of the generator of Z.