Applied Research Of Stochastic Analysis In Complex Networks And Finance

For the problem of European option pricing with stochastic interest rates and the problem of synchronization question in complex dynamical networks with random disturbances, stochastic analysis is an important tool of research. In particular, for the relative problems in ?nancial markets, it is such an important tool. This is because of the underlying random nature of ?nancial markets. This dissertation presents some ef?cient theories and techniques of stochastic analysis to discuss the problem of synchronization in complex networks, and the problem of European option pricing with stochastic interest rates. The following Research results are obtained.1. The problem of function projective synchronization in the complex dynamical network with stochastic Markov switching and disturbances is investigated. Adaptive control scheme is designed. By constructing suitable Lyapunov-krasovskii functionals, combining It ?o formula and the techniques of analysis of inequality, we show some criteria for mean square synchronization in complex dynamic networks with switching topology and random disturbances. Furthermore, for almost sure function projective synchronization between two different complex networks with correlated random disturbances, adaptive control scheme is designed. Finally, some numerical examples are provided to demonstrate the effectiveness of the proposed approach.2. Under the assumption that the volatility of a discounted zero-coupon bond is a constant, an explicit closed form formula that permits the interest rate to be random is obtained. Although spread options have been extensively studied in the literature, few papers deal with the problem of pricing spread options with stochastic interest rates. This study presents two novel spread option pricing models that permit the interest rates to be random. We assume that the volatility of the discounted bond price is a function of time t rather than a constant. This dissertation not only presents a good approach to formulate spread option pricing models with stochastic interest rates but also offers a new test bed to understand the dynamics of option pricing with interest rates in a variety of asset pricing models. This dissertation discusses the merits of the models and techniques presented by us in some asset pricing models. The technique of measure changes is useful in some asset pricing models. Two novel spread option pricing formulae that permit the interest rates to be random are obtained. Under the assumption that the interest rates are random, we also present a generalized Black-Scholes-Merton option pricing formula, an exchange option pricing formula and a European option pricing formula in a jump model.Finally, for spread options, some sensitivity analyses are presented. We use regular grid method to the calculation of the formula when underlying stock returns are continuous and a mixture of both the regular grid method and a Monte Carlo method to the one when underlying stock returns are discontinuous. Though numerical experiments and simulations, the study demonstrates that stochastic interest rates play a signi?cant role. The numerical results indicate that the volatilities in the underlying asset prices signi?cantly affects the value of options.3. This study present three novel basket option pricing formulae that permit the interest rates to be random. This dissertation presents a powerful calculation technique for the problem in low-dimensions when underlying stock returns are continuous. Two novel vulnerable option pricing formulae that permit the interest rate to be random are obtained.