| In reality, different market states are interchanged, which affects the asset price and market, stability. This doctoral dissertation intends to employ the hidden Markov model to clarify and express this market phenomenon, and based on this, the stock option pricing formula will be derived.We first consider the binomial tree option pricing under the hidden Markov model. Based on that there are three market situations including market expansion, market contraction and market steady, we assume that the market will exert three different influences on the stock price:conducive to market development, adverse to market development as well as maintaining the market’s previous state. Based on the assump-tion that these three market states are incompatible, the Markov model is employed to describe the conversion rules of these three market states. We further assume in any market state, the stock price rises or falls in the same magnitude. Then, we obtain the pricing formula of two-period of the European call option as follows:Theorem 1 Under the hidden Markov model, considering the initial steady state of the second-period of the European call option, we obtain the option price at time zero as whereFurthermore, we consider the situation of n-period. Different from the classical binomial tree option pricing model, we need to consider that different market conditions have different interest rates due to the maturity of payoff, and their weighted discount is defined as the option price at the period of 0. Recursively, we obtain the pricing-formula of the n-period European call option with the initial steady state as follows:Theorem 2 We consider the market price of the underlying asset described by the hidden Markov model, which is denoted as Then the price of the n-period European call option with the initial steady state is obtained asCompared with the traditional binomial tree option pricing, the model presented in this thesis is more complicated, but is more in accordance with the reality, which has certain reference significance to the study of financial asset pricing and risk man-agement.On the assumption that two regime switching systems can be conversed under the macro-economic environment, we can further study the multi-period operation characteristics of the stock price based on the Black-Scholes model. Under the condition that the interest rate changes along with the change of regime systems, the option pricing formula under the framework of conditional Black-Scholes model is derived and analyzed.Theorem 3 Assume that the the market initial assets meet Xt0= 1. namely. under the assumption that the initial asset is under the market expansion environment. Then we have the pricing formula of the European call option at 0 times as follows.Compared with the general, purely mathematical model derivation or the empirical research by directly applying the corresponding regime switching model, this thesis focuses more on the characteristics and significance of the mathematical model in economic research, and it plays a certain contributing role in promoting the connection of mathematical model theory and empirical research.Another part of the work of this doctoral dissertation is on the study of well-posedness and controllability of the solutions to a class of nonlinear partial differential equations. As is known to all, the theory of stochastic optimal control has important applications in finance, insurance and other fields, such as the decision of optimal port-folio strategy, optimal consumption ratio or the minimum transaction cost calculation under the expected utility maximization, the determination of optimal quota share or retention in optimal reinsurance etc..Currently, there are mainry two ways to study the stochastic optimal control prob-lems. The first one is to use the variational method. Through the introduction of back-ward stochastic differential equation, and the construction of a random Hamiltonian function, it presents the necessary conditions that the optimal control should satisfy. The stochastic optimal control is described by a type of forward-backward stochas-tic differential equations. The results obtained by this approach is called Pontryagin type stochastic maximum principle. The other way on the study of stochastic optimal control problem is to use stochastic optimality principle. The value function then sat-isfies a class of nonlinear partial differential equations. By solving the nonlinear partial differential equation, it finally characterizes the stochastic optimal control and cost functional. This method is called stochastic dynamical programming method, and the corresponding nonlinear partial differential equation is named as the Hamilton-Jacobi-Bellman equation.In the using of dynamic programming method of the stochastic optimal control, it is very significant to study the well-posedness of the nonlinear partial differential equation solution, as the value function is the solution of the Hamilton-Jacobi-Bellman equation. In general, this class of nonlinear equations no longer exists classical solution. So one has to investigate the various kinds of weak solutions in certain sense, such as the viscous solutions and so on. What’s more, when the more general nonlinear partial differential equations being added a control term, whether the nonlinear partial differential equations is well-posedness and controllable? Under such background, the another part of this thesis is on the researching the well-posedness and controllability of a class of higher order weakly nonlinear partial differential equations, Korteweg-de Vries-Burgers equation. We obtain the well-posedness and local exact controllability results for this class of equations.Korteweg-de Vries-Burgers equation has been derived as a model for the propagation of weakly nonlinear dispersive long waves in some physical contexts when dissipative effects occur [1]. The well-posedness of (1) has been studied in [2-4]. In these works, the existence of the solution is obtained by performing a fixed point argument on the corresponding integral equation.As far as we know, the discussion of the Korteweg-de Vries-Burgers equation is mainly about the well-posedness. In this thesis, we will study the Korteweg-de Vries-Burgers equation from a control point of view with a forcing term h= h(x,t) added to the equation as a control input:It is natural to propose the following problem:For any time T> 0, any two states u0 and u1 in a certain space, can one find an appropriate control h that drives the solution of (1) from u0 at t= 0 to u1 at t= T?The problems were first investigated by Russel and Zhang in [5,6] (also in [7]) for the third-order linear dispersion equation: They obtain that on a given open sets ω(?) C T, there exists a control h such that the equation (3) is globally exactly controllable. For the nonlinear three order dispersion equation with infinite distributed delay, the exact controllability result can be seen in [8].However, because of the linear Korteweg-de Vries-Burgers equation with the reg-ularization effect, the exact controllabilitv may not hold with a localized control. So we consider the control acts on the entire region of T.For any s ∈R, letFor the well-posedness of the Korteweg-de Vries-Burgers equations, we obtain the following results:Proposition 1 For any T > 0 and any u0∈H00(T), (1) admits a unique solution U ∈Y0,T which also satisfies where C is independent of T. Moreover, the corresponding solution map is locally Lip-schitz continuous: for any u0, v0 ∈H00(T), the corresponding solutions u and v of (1) satisfy where α0,T :R+→R+ is a nondecreasing continuous function.Proposition 2 For any u0 ∈H30(T), (1) admits a unique solution u∈ Y3.T. Moreover, there exists a nondecreasing continuous function α3,T : R+→R+ such thatThen using the nonlinear interpolation theory and with similar derivation, we haveTheorem 4 For any u0 ∈H0s(T), (1) admits a unique solution u∈Ys,T. In addition, there exists a nondecreasing continuous function αs,T : R+→R+ such thatIn order to study the controllability of the Korteweg-de Vries-Burgers equation (2), we first consider the linear case, that is the following equation,By using the previous well-posedness results: we fist have the following main resultsProposition 3 Let s ≥0, T>0 be given. For any u0, u1∈H0s(T), there exists a control input h∈L2(0, T; H0s-1(T)) such that (4) admits a solution u ∈Ys,T satisfying for any x∈T.Then we can get the local exact, controllability of Korteweg-de Vries-Burgers e-quations:Theorem 5 Lets ≥0,T>0 be given. There exists a δ> 0 such that for any u0, u1∈ H0s(T) satisfying one may find a control h ∈L2(0, T; H0s-1(T)) such that (1) admits a unique solution u ∈C([0, T], H0s(T))∩ L2(0, T, H0s+1 (T))) for which, u(0)=u0 and u(T)=u1. |
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