Discrete-time Normal Martingale Generalized Functional And Clark-Ocone Formula

The Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discrete-time normal martingales.In this paper,we aim at extending this formula to generalized functionals of discrete-time normal martingales.Let M be a discrete-time normal martingale that has the chaotic representation property.We first prove a result concerning the regularity of generalized functionals of M,then we use the Fock transform to define some funda-mental operators on generalized functionals of discrete-time normal martingales of M and apply the above mentioned result to prove the continuity of these operators.Finally,we establish the Clark-Ocone formula for generalized functionals of M and show its application results.The paper is organized as follows.In Chapter 1,we briefly describe the background and the state-of-art concern-ing our research topics.We also present some general concepts,and fundamental knowledge of quantum Bernoulli noises.In Chapter 2,we construct the generalized functionals of discrete-time normal martingales of M and show its related theories.Finally,in Chapter 3,we use the Fock transform to define some fundamen-tal operators on generalized functionals of discrete-time normal martingales of M and discuss their basic properties,then we establish the Clark-Ocone formula for generalized functionals of M and show its application results.