Translating Nested and Multiple Stochastic Integrals in the Wiener Functional Space through the Moment Inequalities

Various techniques have been adopted in the past, in translating nested and multiple stochastic integrals to the Wiener functional space. However, this paper considers a new method of moving nested and multiple stochastic integrals into the Wiener functional space. Based on the existence of nested integrals and multiple stochastic integrals in moment inequalities, we derived a means of representing and expressing them in the Wiener functional space which is referred to as the moment inequalities formula. Hence, this paper established that the multiple stochastic nested integrals in Moment inequalities are one translating formula onto the Wiener functional space. This result was achieved by showing that the log-Sobolev inequality implies the exponential integrability of the square of the Wiener functional whose derivatives are essentially bounded.