Discrete and Continuous Ito-Malliavin Type Calculus with Illustration in Finance

As reliable mathematical methods for finance, various concepts of the stochastic calculus are discussed in detail in this thesis such as the Ito integral, the (continuous and discrete) Malliavin calculus and the Stratonovich integral. The derivative of a natural number and the quantum calculus are also illustrated in this thesis. The Stroock lemma and the duality formula are two methods when the Malliavin calculus is applied to calculate the preceding quantities. To extend the range of application of these rules is a crucial purpose of this thesis. Solving certain equations based on the Ito integral and the Malliavin calculus has also been introduced and analysed in this thesis. This equation, which is also a kind of stochastic differential equation, can be treated as an inverse application of the Malliavin derivative. Finally, the product rule for other derivative operators is extensively introduced and analysed throughout the whole thesis, since this rule in the stochastic calculus or the quantum calculus is sometimes different from the traditional infinitesimal calculus. To explore the idea of differential dynamics with the non-standard and new types of differentiation, the differential operators discussed and introduced in this thesis, such as the continuous and the discrete Malliavin derivative operator and the q-derivation operator are applied as transforms on some state spaces, such as measurable space and the space constructed by the finite fields.