| Backward stochastic differential equations (BSDEs for short in the remaining) were introduced, in the linear case, by Bismut in 1973 [6] and considered general form the first time by Pardoux and Peng in 1990 [41]. The theory of BSDEs has been studied with great interest in the last less than twenty years because of its connections with the non-linear partial differential equations (see, for example Barles and Lesigne [3], Briand [7], Pardoux and Peng [42], and Pardoux and Peng [43], etc.) and more generally the theory of non-linear semi-groups, and stochastic control problems (see Quenez [52], El Karoui, Peng and Quenez [19], Hamadene and Lepeltier [21], Peng [46], etc.). At the same time, in mathematical finance, the theory of the hedging and pricing of a contingent claim is typically expressed in terms of a linear BSDE (see El Karoui, Peng and Quenez [19]). In 1997 Peng [47] introduced a kind of nonlinear expectation: g-expectation via a particular BSDE. Using Peng's g-expectation, it is easy to define conditional expectations. Rosazza [53] considered a type of dynamic risk measures via g-expectations. Peng [45] defined filtration consistent evaluation and p-evaluation. He also proved a theorem that a filtration consistent evaluation satisfying some restrictions is a g-evaluation, that is, whatever model or mechanism used to evaluate, once it satisfies the restrictions, there is a BSDE behind of it, the generator g is its mechanism, and the solution of BSDE is the evaluation.BSDEs are equations of the following type:where (Wt)0≤t≤T is a standard d-dimensional Brownian motion on a probability space (Ω,F,(Ft)0≤t≤T), with (Ft)0≤t≤T the standard Brownian filtration.On the other hand, in the past 40 years, the comparison theorems of two Ito's stochastic differential equations have received a lot of attention, for example, Anderson [1], Gal'cuk and Davis [20], Ikeda and Watanable [24], Mao [35], Skorohod [56], Yamada [63] and Yan [64] gave some sufficient conditions for comparison theorem. Recently, Peng and Zhu [51] has presented a sufficient and necessary condition for comparison theorem by using viability theory. However, so far, there is no result for comparison theorem on stochastic differential delay equations, which this thesis shall cope with. Should our theory be more applicable, we shall establish the comparison theorem of 1-dim stochastic differential delay equations with Markovian switching (SDDEwMSs).Stochastic differential equations with Markovian switching (SDEwMSs) is an important class of hybrid systems. In ecology, engineering and other disciplines it is well known that many systems exhibit such discrete dynamics, due for example to component failures or repairs, changing subsystem interconnections, etc. When coupled with continuous dynamics, these discrete phenomena give rise to what are known as hybrid systems. Of particular interest to the present chapter are hybrid systems whose discrete behavior is driven by continuous-time Markov chains. Such systems have been used to model a number of engineering (and other) systems. For example, Kazangey and Sworder [28] developed a macroeconomics model of the national economy in this framework, to study the effect of federal housing removal policies on the stabilization of the housing sector. The term describing the influence of interest rates was modeled by a finite-state Markov chain to provide a quantitative measure of the effect of interest rate uncertainty on the optimal policy. Athans [2] suggested that such hybrid systems would also become a basic framework in posing and solving control-related problems in Battle Management Command, Control and Communications (BM/C3) systems. Hybrid systems were also considered for modeling electric power systems (Willsky and Levy [62]), the control of a solar thermal central receiver (Sworder and Rogers [61]) and the modeling of subtilin production by Bacillus subtilis (Hu, Wu and Sastry [22]). In his book [39], Mariton discussed how such hybrid systems have also emerged as a convenient mathematical framework for the formulation of various design problems in target tracking, fault tolerant control and manufacturing processes. On the other hand, control engineering intuition suggests that time-delays are common in practical systems and are often the cause of instability and/or poor performance. Moreover, it is usually difficult to obtain accurate values for the delay and conservative estimates often have to be used. The importance of time delay has already motivated several studies on the stability of switching diffusions with time delay, see, for example, [10, 38, 44]. At the same time, time-delays are often found in finance and economics and their applications were studied in Ivanov et al. [25, 26, 27], Kazmerchuk et al. [29] and Swishchuk et al. [58, 59, 60].The paper is organized as follows: Chapter 1 introduces a new type of equations: anticipated backward stochastic differential equations (anticipated BSDEs for short). There exists perfect duality between them and stochastic differential delay equations (SDDEs for short, see Kolmanovskii, Myshkis [30] and Mao [36], [37]). The new type of equation is as follows:In reality, by using anticipated BSDEs, we can change the goal of the future through today's policy (the present solution). Existence and uniqueness for an adapted solution to anticipated BSDEs, in my opinion, is the reason why anticipated BSDEs appear later than BSDEs. If we adopt the traditional conditions of f, the solution of anticipated BSDEs is no longer adapted, thus we make f a functional instead of a function to conquer this difficulty. At the same time, continuous dependence property with respect to parameters, comparison theorem, monotonic limit theorem and existence and uniqueness of the adapted solution to anticipated BSDEs with stopping time have been obtained. Notice that the conditions of this comparison theorem are different from those of comparison theorem of BSDEs, that is, f is asked to be increasing in anticipated term of Y and contain no anticipated term of Z expect for the original conditions. Using the duality between SDDEs and anticipated BSDEs, we can solve a type of stochastic control problems.The following are the main results of Chapter 1.Theorem 1.2.1. (Duality between SDDEs and anticipated BSDEs) Supposeθ> 0 is a given constant and . are uniformly bounded. Then , the solution Y to the anticipated BSDEcan be given by the closed formula:where XS is the solution to SDDE Theorem 1.4.2. (Existence and Uniqueness of the Adapted Solution to Anticipated BSDEs) Suppose that f satisfies (#1.1) and (#1.2),δand ( satisfy (i) and (ii). Then for an arbitrary given terminal conditions , anticipated BSDE (*) has a unique solution .Theorem 1.5.1. (Comparison Theorem) Consider the following two 1-dimensional anticipated BSDEs:where j = 1,2. Assume that for j = 1,2, fj satisfies (#1.1), (#1.2), K),δsatisfies (i),(ii), and is increasing, i.e.,, and , then , a.e., a.s.Theorem 1.6.1. (Duality in Control Application) Letθ> 0 be a given constant. Set = esssup. Then the anticipated BSDEhas a unique solution (Y, Z). Moreover, Y is the value function Y* of the control problem, that is, for each t∈[0, T],Yt = Yt* = esssup{Ytu, u∈U}, where Yu is the solution to the linear anticipated BSDEwhere and Theorem 1.7.1. (Monotonic Limit Theorem) Let m = 1. Consider the following anticipated BSDEs: for n = 1,2,...,Assume for each n = 1,2,..., fn satisfies (H1.1) and (#1.2), satisfies (i) and (ii), is increasing, and there exists a constantμ> 0, such that , and , moreover, , then anticipated BSDE has a solution and , a.e., a.s. Theorem 1.8.3. (Existence and Uniqueness of the Adapted Solution to Anticipated BSDEs with Stopping Time) Suppose 5 satisfies (i),(ii), progressively measurable process f satisfies: and there exist three positive functions and v(·), such that , we haveand . Then for an arbitrary given terminal condition , the anticipated BSDEhas a unique solution .Chapter 2 studies generalized anticipated BSDEs as follows: It is obvious that the type of equations is the generalization of equations studied in Chapter 1. Existence and uniqueness for an adapted solution to generalized anticipated BSDEs, continuous dependence property with respect to parameters, comparison theorem and monotonic limit theorem have been obtained in Chapter 2. We also get the duality between SDDEs and a special generalized anticipated BSDEs and its application to a type of stochastic control problems.The following are the main results of Chapter 2.Theorem 2.2.2. (Existence and Uniqueness of the Adapted Solution to Generalized Anticipated BSDEs) Suppose that f satisfies (H2.1), (H2.2) and (H2.3). Then for arbitrary given terminal condition , the above generalized anticipated BSDE has a unique solution .Theorem 2.3.1. (Comparison Theorem) Let (Y1,Z1) and (Y2, Z2) be respectively the solutions to the following two 1-dimensional generalized anticipated BSDEs:where j = 1,2. Assume that for j = 1,2, fj satisfies (H2.1), (H2.2) and (H2.3), is increasing, i.e., , and , then , a.e., a.s. Theorem 2.5.1. (Monotonic Limit Theorem) Let m = 1. Consider the following generalized anticipated BSDEs: for n = 1,2,...,Assume for n = 1,2,..., fn satisfies (H2.1), (H2.2) and (H2.3), is increasing, and there exists a constant , and , then generalized anticipated BSDEhas a solution and Yt = supnytn, a.e., a.s.Chapter 3 gives comparison theorem of 1-dimensional SDDEwMSs. The method is similar to prove comparison theorem of 1-dimensional anticipated BSDEs. And an application of the comparison theorem is also given. The following are the main results of Chapter 3.Theorem 3.3.4. (Comparison Theorem) Consider the following two 1-dimensional SDDEwMSs:where j = 1,2. Supposeδ(t) satisfy (A.1),(A.2) and f1,f2,g satisfy (H3.1'), (H3.2'). , and , then , a.e., a.s.
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