ბესიკ ჩიქვინიძე

This book consists of four chapters. In first chapter there is a short review of theory of Backward Stochastic Differential Equations (BSDEs) and Bounded Mean Oscillation (BMO) martingales. In second chapter an interesting connections between theory of BSDEs and BMO martingales is studied. Using the BSDE tool a new proofs of some classical results on BMO martingales are provided. In Third chapter we have studied Backward Stochastic Differential Equations with a convex generator of quadratic growth. Existence and uniqueness of a solution is proved for such equations driven by continuous martingale with unbounded characteristic. Results on the existence and uniqueness for BSDEs with quadratic growth we have used in fourth chapter, to solve the linear-quadratic regulator (LQR) problem in general martingale setting. We derived the corresponding BSDE for LQR problem and expressed the optimal strategy of LQR problem in terms of the unique solution of corresponding BSDE.

Backward stochastic differential equations (BSDEs) with convex generators of quadratic growth are considered. The existence of a solution is proved for such equations driven by a continuous martingale with unbounded characteristic. A suitable optimization problem is formulated and it is shown that the corresponding value process satisfies BSDEs.

Keywords.: Backward stochastic differential equation; value process; semimartingale

Received

Given a continuous local martingale M, the associated stochastic exponential is a local martingale, but not necessarily a true martingale. To know whether is a true martingale is important for many applications, e.g. if Girsanov’s theorem is applied to perform a change of measure. We give a generalization of the well-known Novikov’s and Kazamaki’s criteria which provides a new proof based on the properties of a certain backward stochastic differential equation.

Keywords: Stochastic exponentialGirsanov’s transformationbackward stochastic differential equation

Given a continuous local martingale (Formula presented.), the associated stochastic exponential (Formula presented.) is a local martingale, but not necessarily a true martingale. To know whether (Formula presented.) is a true martingale is important for many applications, e.g., if Girsanov’s theorem is applied to perform a change of measure. We give several generalizations of Kazamaki’s results and finally construct a counterexample which does not satisfy the mixed Novikov–Kazamaki condition, but satisfies our conditions.

Using backward stochastic differential equations we give a new proof of well known Novikov’s criterion.

We establish necessary and sufficient conditions for uniform integrability of thestochastic exponential E(M), where M is a continuous local martingale.

Key words: Stochastic exponential ·Girsanov’s transformation ·Local martingale Mathematics Subject 

We generalize the well known mixed Novikov–Kazamaki condition and introduce a new type sufficient condition for the uniform integrability of the stochastic exponential with jumps.

Keywords: Right continuous filtrationstochastic exponentiallocal martingaleGirsanov's transformation