We assume that there exist two kinds of investors in the market, the 𝔽-investors and the 𝔾-investors. The 𝔽-investors have the market information 𝔽, which is given by a <I>d</I>-dimensional Brownian motion W = (W₁,...;W_d)'</I> as well as an integer-valued random measure <I>μ(du, dy)</I>. The market...</i></i>

We review the relations between adjoints of stochastic control problems with the derivative of the value function, and the latter with the value function of a stopping problem. These results are applied to the pricing of contingent claims.

We prove an existence and uniqueness theorem for backward stochastic differential equations driven by a Brownian motion, where the uniform Lipschitz continuity is replaced by a stochastic one.

A market is described by two correlated asset prices. But only one of them is traded while the contingent claim is a function of both assets. We solve the mean-variance hedging prob- lem completely and prove that the optimal strategy consists of a modified pure hedge expressible in terms of the...

Multi-dimensional backward stochastic Riccati differential equations (BSRDEs in short) are studied. A closed property for solutions of BSRDEs with respect to their coefficients is stated and is proved for general BSRDEs, which is used to obtain the existence of a global adapted solution to some...

The optimal control problem is considered for linear stochastic systems with a singular cost. A new uniformly convex structure is formulated, and its consequences on the existence and uniqueness of optimal controls and on the uniform convexity of the value function are proved. In particular, the...

The following backward stochastic Riccati differential equation (BSRDE in short) is motivated, and is then studied. Some properties are presented. The existence and uniqueness of a global adapted solution to a BSRDE has been open for the case D i 6= 0 for more than two decades. Our recent...

It is well known that backward stochastic differential equations (BSDEs) stem from the study on the Pontryagin type maximum principle for optional stochastic control. A solution of a BSDE hits a given terminal value (which is a random variable) by virtue of an additional martingale term and an...