超前BSDE及SDE中的相关结果
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摘要
倒向随机微分方程(简记为BSDE)的线性形式首先由Bismut在1973年引入，而Pardoux和Peng在1990年研究了Lipschitz条件下非线性BSDE解的存在唯一性定理。在过去的近二十年中，BSDE受到广泛的关注，这由于它和非线性偏微分方程之间存在密切的联系(如参见Barles和Lesigne，Briand，Pardoux和Peng，等等)，并可应用于非线性半群理论及随机控制问题(见Quenez，ElKaroui，Peng和Quenez，Haznaxiene和Lepeltier，及Peng，等等)。同时在金融数学中，套期保值及不确定权益定价理论通常可表为一类线性的BSDE(见ElKaroui，Peng和Quenez)。在1997年Peng引入了一种新的非线性期望：由特殊BSDE导出的g-期望，利用Peng的g-期望，很容易定义对应的条件期望，Rosazza通过g-期望得到一类动态风险测度。Peng定义了信息流相容的估计和g-估计，从而证明了满足某些特定限制的信息流相容的估计就是g-估计，即无论模型或机构如何来估值的，一旦它满足这些限制，在它背后就存在一个BSDE，生成元g是规则，而BSDE的解即是该估计。
BSDE是如下形式的方程：
Y_T=ξ+integral from n=t to T g(s，Y_s，Z_s)ds-integral from n=t to T Z_sdW_s，0≤t≤T，其中(W_t)0≤t≤T是定义在概率空间(Ω，(?)，((?)_t)0≤t≤T)上的标准d-维布朗运动，这里((?)_t)0≤t≤T是该布朗运动生成的信息流。
另一方面，过去40年间，对两个It(?)随机微分方程(SDE)比较定理的研究引起了相当多的关注。比如，Anderson，Gal’cuk和Davis，Ikeda和Watanable，Mao，Skorohod，Yamada及Yan都给出了比较定理的充分条件，最近Peng和Zhu借助生存性定理给出了比较定理的充要条件，尽管如此，到目前为止，随机微分延迟方程的比较定理还没有结果，而这也是本文将要研究的内容。同时，为了让理论贴近实际，本文研究带马尔可夫转换的随机微分延迟方程(SDDEwMS)的比较定理。
带马尔可夫转换的随机微分方程(SDEwMS)是混合系统中一个很重要的类型。在生态系统，工程学以及其他领域中，许多系统都显示出离散动力特性；比如，部件失效或维修，更换子系统的相互关联等等。当和连续动力系统相偶合时，这些离散现象就构成了所谓的混合系统。本文所特别关注的是离散行为是由连续时间马尔可夫链驱动的混合系统。这些系统常用来模拟一些工程和其他领域类的系统．例如，Kazangey和Sworder利用该框架建立了国民经济的宏观模型以研究政府住房减免政策对于住房分区稳定性的影响。描述存款利率的影响所产生的相互关系也可以用有限状态的马尔可夫链来量化度量不确定利率对最优策略的作用。Athans提出这种混合系统应该作为一个基本框架来表述和解决控制相关的问题，比如战斗指令管理，控制和通讯系统等。混合系统也被用来处理电力系统(参见Willsky和Levy)，太阳能集热器的控制(参见Sworder和Rogers)及枯草菌素生产模型(参见Hu，Wu和Sastry)．在Mariton书中，他讨论了该混合系统是怎样作为一种便捷的数学框架来建模解决各种目标跟踪，容错控制和制造过程的设计问题。另一方面，控制工程的直觉告诉我们时间延迟在实际系统中非常普遍，并且经常是系统不稳定和性能不佳的原因。进一步说来，延迟的精确值通常难以获得，故而经常采用保守的估计值。时间延迟的重要性吸引了许多人对带转换扩散的时间延迟的稳定性问题进行研究：见[10，38，44]。同时，时间延迟在经济及金融上也十分常见，其应用在这方面的研究可参见Ivanov etal．[25，26，27]，Kazmerchuk et al．[29]及Swishchuk et al．[58，59，60]。
论文组织和具体安排如下：
第一章主要介绍了一种新型的方程-倒向随机微分超前方程(简记为超前BSDE)。这种方程与随机微分延迟方程(简记为SDDE，参见Kolmanovskii，Myshkis和Mao)之间存在完美的对偶。这种新方程具有以下形式：
实际上，超前BSDE是把未来一段时间的不确定的目标转化为今天确定的解以制定今天的决策。本文给出了超前BSDE适应解的存在唯一性定理。而这正是超前BSDE比BSDE晚出现的原因所在。如果沿用f的传统条件，那么超前BSDE的解将不再具有适应性，故而取f为泛函以代替原来的函数来克服这个困难。同时，我们还得到解对参数的连续依赖性，比较定理，单调收敛定理以及带停时的超前BSDE适应解的存在唯一性定理。需要注意的是超前BSDE的比较定理成立的条件与经典BSDE的有所不同，即在原来条件的基础上还要求f对Y的超前项单调增且不含Z的超前项。通过超前BSDE与SDDE的对偶，还可以解决一类控制问题。
以下是第一章的主要结果：
定理1.2.1.(超前BSDE与SDDE的对偶)设θ＞0是给定常数且μ．，(?)．∈L~(?)~2(t_0—θ，T+θ)，l．∈L_(?)~2(t_0，T)，σ．，(?)．∈L_(?)~2(t_0—(?)，T+θ；R~(d×1))，μ．，(?)．，σ．，(?)．一致有界。则(?)Q．∈S_(?)~2(T，T+θ)，P．∈L_(?)~2(T，T+θ；R~d)，超前BSDE的解Y可用下面的公式显式表达：
Y_t=E~((?)T)[X_TQ_T+integral from n=t to T X_sl_sds+integral from n=T to T+θ(Q_s(?)_(s-θ)+P_s(?)_(s-θ)X_(s-θ)ds]，a．e．，a．s．，
其中X_s是如下SDDE的解：
定理1.4.2.(适应解的存在唯一性)设f满足(H1.1)和(H1.2)，δ及ξ满足(i)和(ii)。则对任给终值条件ξ．∈．S_(?)~2(T，T+K；R~m)，η．∈L_(?)~2(T，T+K；R~(m×d))，超前BSDE(*)存在唯一适应解(Y，Z．)∈，S_(?)~2(0，T+K；R~m)×L_(?)~2(0，T+K；R~(m×d))。
定理1.5.1.(比较定理)考虑如下两一维超前BSDE：
其中j=1，2．设对j=1，2，fj满足(H1.1)和(H1.2)，ξj∈S_(?)~2多(T，T+K)，δ满足(i)和(ii)，且(?)t∈[0，T]，y∈R，z∈R~d，有，f2(t，y，z，·)单调增，即，f2(t，y，z，θ_r)≥f2(t，y，z，θ′_r)，若θ_r≥θ′_r，θ，θ′∈L_(?)~2(t，T+K)，r∈[t，T+K]。如果ξ_s~((1))≥ξ_s~((2))，s∈[T，T+K]，且f1(t，y，z，θ_r)≥，f2(t，y，z，θ_r)，t∈[0，T]，y∈R，z∈R~d，θ∈L_(?)~2(t，T+K)，r∈[t，T+K]，那么Y_t~((1))≥Y_T~((2))，a．e．，a．s．
定理1.6.1.(对偶在控制上的应用)设θ＞0为给定常数，对T∈[0，t]，y∈R，z∈R~d，叩∈L_(?)~2(t，t+θ)，r∈[t，T+θ]，令
f(t，y，z，η_r)=esssup{α(t，u_t)y+zσ(t，u_t)+b(t，u_t)E~((?)t)[η_r]+l(t，u_t)，u∈u}，则超前BSDE
有唯一解(Y，Z)，且Y是控制问题Y~*的值函数，即对任给￡∈[0，T]，
Y_T=Y_t~*=esssup{Y_t~u，u∈u}，
其中Y~u是如下线性超前BSDE的解：
这里，f~u(t，y，z，η_r)=α(t，u_t)y+zσ(t，u_t)+b(t，u_t)E~((?)t)[η_r]+l(t，u_t)，t∈[0，T]，y∈R，z∈R~d，η∈L_(?)~2(t，T+θ)，r∈[t，T+θ]且
定理1.7.1.(单调收敛定理)设m=1．考虑如下一族超前BSDE：对n=1，2，…，设对每一个n=1，2，…，fn满足(H1.1)和(H1.2)，ξn(·)∈S_(?)~2(T，T+K)，δ满足(i)和(ii)，(?) n∈N，t∈[0，T]，y∈R，z∈R~d，有fn(t，y，z，·)单调增，且存在常数μ＞0使得E[integral from n=0 to T|fn(s，0，0，0)|~2ds]≤μ，(?) n∈N．如果(?) t∈[0，T]，y∈R，z∈R~d，θ∈L_(?)~2(t，T+K)，r∈[t，T+K]，有fn(t，y，z，θ_r)↗ f(t，y，z，θ_r)，当n→∞时，且(?)s∈[T，T+K]，有ξn(s)↗ξ(s)，当n→∞时，及f∈S_(?)~2(T，T+K)，那么超前BSDE
有解(Y，Z．)∈S_(?)~2(0，T+K)×L_(?)~2(0，T；R~d)且Y_t=sup_n Y_t~((n))，a．e．，a．s．
定理1.8.3.(带停时的超前BSDE适应解的存在唯一性)设δ满足(i)和(ii)，f循序可测且满足：E[(integral from n=0 to∞|f(s，0，0，0)|ds)~2]＜∞且存在三个正函数μ_1(·)，μ_2(·)及v(·)使得对任给s∈[0，T]，y，y′∈R，z，z′∈R~d，ξ，ξ′∈L_(?)~2(s，T+K)，r∈[s，T+K]，有且(integral from n=0 to∞μ_1(r)dr)~2+(integral from n=0 to∞μ_2(r)dr)~2+integral from n=0 to∞v~2(r)dr＜∞。则对任意给定终值条件ξ．∈S_(?)~2(T，T+K)，超前BSDE
有唯一解(Y，Z．)∈S_(?)~2(0，T+K)×L_(?)~2(0，T；R~d)。
第二章研究推广的超前BSDE，其形式如下：显然这种方程是第一章中研究方程的推广形式。本文给出了推广的超前BSDE适应解的存在唯一性定理，解对参数的连续依赖性，比较定理和单调收敛定理。还研究了推广的超前BSDE的一个例子，它与SDDE之间存在对偶关系并将该对偶应用于随机控制问题。
以下是第二章的主要结果。
定理2.2.2.(适应解的存在唯一性定理)设f满足(H2.1)，(H2.2)和(H2.3)。则对任给终值条件ξ．∈．S_(?)~2(T，T+K；R~m)，η．∈L_(?)~2(T，T+K；R~(m×d))，如上推广的超前BSDE存在唯一解(Y，Z．)∈．S_(?)~2(0，T+K；R~m)×L_(?)~2(0，T+K；R~(m×d))。
定理2.3.1.(比较定理)令(Y~((1))，Z~((1))和(Y~((2))，Z~((2))分别是下面两一维推广的超前BSDE的解：
其中j=1，2．设对j=1，2，f_j满足(H2.1)，(H2.2)和(H2.3)，ξ_j∈S_(?)~2(T，T+K)且(?) t∈[0，T]，z∈R~d，f2(t，·，z)单调增，即f2(t，θ，z)≥，f2(t，θ′，z)，若θ_r≥θ′_r，θ，θ′∈L_(?)~2(t，T+K)，r∈[t，T+K]。如果ξ_s~((1))≥ξ_s~((2))，s∈[T，T+K]，且f1(t，θ，z)≥f2(t，θ，z)，t∈[0，T]，θ∈L_(?)~2(t，T+K)，z∈R~d，那么Y_t~((1))≥Y_t~((2))，a．e．，a．s．
定理2.5.1.(单调收敛定理)设m=1．考虑如下一族推广的超前BSDE：对n=1，2，…，设对任给n=1，2，…，fn满足(H2.1)，(H2.2)及(H2.3)，ξn(·)∈S_(?)~2(T，T+K)，(?) n∈N，t∈[0，T]，z∈R~d，fn(t，·，z)单调增且存在常数μ＞0使得E[integral from n=0 to T|fn(s，0，0)|~2ds]≤μ，(?)n∈N．如果(?)t∈[0，T]，z∈R~d，θ∈L_(?)~2(t，T+K)，有fn(t，θ，z)↗f(t，θ，z)，当n→∞时，及(?)s∈[T，T+K]，有ξn(s)↗ξ(s)，当n→∞时，且ξ∈S_(?)~2(T，T+K)，那么推广的超前BSDE
有解(Y，Z．)∈S_(?)~2(0，T+K)×L_(?)~2(0，T；R~d)且Y_t=sup_nY_t~((n))，a．e．，a．s．
第三章给出了一维SDDEwMS的比较定理及它的一个应用。且证明该比较定理的方法与第1章一维超前BSDE比较定理的证明方法类似。
以下是第三章的主要结果。
定理3.3.4.(比较定理)考虑如下两一维SDDEwMS：其中j=1，2．设δ(t)满足(A.1)，(A.2)且f1，f2，g满足(H3.1′)，(H3.2′)．如果f1(x，y，t，i)≥f2(z，y，t，i)，x，y∈R，t∈[0，T]，i∈S，且x_1(s)≥x_2(s)，s∈[-T，0]，那么X~((1))(T)≥X~((2))(t)，a．e．，a．s．
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 (W_t)0≤t≤T is a standard d-dimensional Brownian motion on a probability space (Ω,F,(F_t)0≤t≤T), with (F_t)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/C~3) 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 X_S 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, f_j 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],Y_t = Y_t~* = esssup{Y_t~u, u∈U}, where Y~u 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,..., f_n 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 (Y~(1),Z~(1)) and (Y~(2), Z~(2)) be respectively the solutions to the following two 1-dimensional generalized anticipated BSDEs:where j = 1,2. Assume that for j = 1,2, f_j 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,..., f_n satisfies (H2.1), (H2.2) and (H2.3), is increasing, and there exists a constant , and , then generalized anticipated BSDEhas a solution and Y_t = sup_ny_t~(n), 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 f_1,f_2,g satisfy (H3.1'), (H3.2'). , and , then , a.e., a.s.
BSDE是如下形式的方程：
Y_T=ξ+integral from n=t to T g(s，Y_s，Z_s)ds-integral from n=t to T Z_sdW_s，0≤t≤T，其中(W_t)0≤t≤T是定义在概率空间(Ω，(?)，((?)_t)0≤t≤T)上的标准d-维布朗运动，这里((?)_t)0≤t≤T是该布朗运动生成的信息流。
另一方面，过去40年间，对两个It(?)随机微分方程(SDE)比较定理的研究引起了相当多的关注。比如，Anderson，Gal’cuk和Davis，Ikeda和Watanable，Mao，Skorohod，Yamada及Yan都给出了比较定理的充分条件，最近Peng和Zhu借助生存性定理给出了比较定理的充要条件，尽管如此，到目前为止，随机微分延迟方程的比较定理还没有结果，而这也是本文将要研究的内容。同时，为了让理论贴近实际，本文研究带马尔可夫转换的随机微分延迟方程(SDDEwMS)的比较定理。
带马尔可夫转换的随机微分方程(SDEwMS)是混合系统中一个很重要的类型。在生态系统，工程学以及其他领域中，许多系统都显示出离散动力特性；比如，部件失效或维修，更换子系统的相互关联等等。当和连续动力系统相偶合时，这些离散现象就构成了所谓的混合系统。本文所特别关注的是离散行为是由连续时间马尔可夫链驱动的混合系统。这些系统常用来模拟一些工程和其他领域类的系统．例如，Kazangey和Sworder利用该框架建立了国民经济的宏观模型以研究政府住房减免政策对于住房分区稳定性的影响。描述存款利率的影响所产生的相互关系也可以用有限状态的马尔可夫链来量化度量不确定利率对最优策略的作用。Athans提出这种混合系统应该作为一个基本框架来表述和解决控制相关的问题，比如战斗指令管理，控制和通讯系统等。混合系统也被用来处理电力系统(参见Willsky和Levy)，太阳能集热器的控制(参见Sworder和Rogers)及枯草菌素生产模型(参见Hu，Wu和Sastry)．在Mariton书中，他讨论了该混合系统是怎样作为一种便捷的数学框架来建模解决各种目标跟踪，容错控制和制造过程的设计问题。另一方面，控制工程的直觉告诉我们时间延迟在实际系统中非常普遍，并且经常是系统不稳定和性能不佳的原因。进一步说来，延迟的精确值通常难以获得，故而经常采用保守的估计值。时间延迟的重要性吸引了许多人对带转换扩散的时间延迟的稳定性问题进行研究：见[10，38，44]。同时，时间延迟在经济及金融上也十分常见，其应用在这方面的研究可参见Ivanov etal．[25，26，27]，Kazmerchuk et al．[29]及Swishchuk et al．[58，59，60]。
论文组织和具体安排如下：
第一章主要介绍了一种新型的方程-倒向随机微分超前方程(简记为超前BSDE)。这种方程与随机微分延迟方程(简记为SDDE，参见Kolmanovskii，Myshkis和Mao)之间存在完美的对偶。这种新方程具有以下形式：
实际上，超前BSDE是把未来一段时间的不确定的目标转化为今天确定的解以制定今天的决策。本文给出了超前BSDE适应解的存在唯一性定理。而这正是超前BSDE比BSDE晚出现的原因所在。如果沿用f的传统条件，那么超前BSDE的解将不再具有适应性，故而取f为泛函以代替原来的函数来克服这个困难。同时，我们还得到解对参数的连续依赖性，比较定理，单调收敛定理以及带停时的超前BSDE适应解的存在唯一性定理。需要注意的是超前BSDE的比较定理成立的条件与经典BSDE的有所不同，即在原来条件的基础上还要求f对Y的超前项单调增且不含Z的超前项。通过超前BSDE与SDDE的对偶，还可以解决一类控制问题。
以下是第一章的主要结果：
定理1.2.1.(超前BSDE与SDDE的对偶)设θ＞0是给定常数且μ．，(?)．∈L~(?)~2(t_0—θ，T+θ)，l．∈L_(?)~2(t_0，T)，σ．，(?)．∈L_(?)~2(t_0—(?)，T+θ；R~(d×1))，μ．，(?)．，σ．，(?)．一致有界。则(?)Q．∈S_(?)~2(T，T+θ)，P．∈L_(?)~2(T，T+θ；R~d)，超前BSDE的解Y可用下面的公式显式表达：
Y_t=E~((?)T)[X_TQ_T+integral from n=t to T X_sl_sds+integral from n=T to T+θ(Q_s(?)_(s-θ)+P_s(?)_(s-θ)X_(s-θ)ds]，a．e．，a．s．，
其中X_s是如下SDDE的解：
定理1.4.2.(适应解的存在唯一性)设f满足(H1.1)和(H1.2)，δ及ξ满足(i)和(ii)。则对任给终值条件ξ．∈．S_(?)~2(T，T+K；R~m)，η．∈L_(?)~2(T，T+K；R~(m×d))，超前BSDE(*)存在唯一适应解(Y，Z．)∈，S_(?)~2(0，T+K；R~m)×L_(?)~2(0，T+K；R~(m×d))。
定理1.5.1.(比较定理)考虑如下两一维超前BSDE：
其中j=1，2．设对j=1，2，fj满足(H1.1)和(H1.2)，ξj∈S_(?)~2多(T，T+K)，δ满足(i)和(ii)，且(?)t∈[0，T]，y∈R，z∈R~d，有，f2(t，y，z，·)单调增，即，f2(t，y，z，θ_r)≥f2(t，y，z，θ′_r)，若θ_r≥θ′_r，θ，θ′∈L_(?)~2(t，T+K)，r∈[t，T+K]。如果ξ_s~((1))≥ξ_s~((2))，s∈[T，T+K]，且f1(t，y，z，θ_r)≥，f2(t，y，z，θ_r)，t∈[0，T]，y∈R，z∈R~d，θ∈L_(?)~2(t，T+K)，r∈[t，T+K]，那么Y_t~((1))≥Y_T~((2))，a．e．，a．s．
定理1.6.1.(对偶在控制上的应用)设θ＞0为给定常数，对T∈[0，t]，y∈R，z∈R~d，叩∈L_(?)~2(t，t+θ)，r∈[t，T+θ]，令
f(t，y，z，η_r)=esssup{α(t，u_t)y+zσ(t，u_t)+b(t，u_t)E~((?)t)[η_r]+l(t，u_t)，u∈u}，则超前BSDE
有唯一解(Y，Z)，且Y是控制问题Y~*的值函数，即对任给￡∈[0，T]，
Y_T=Y_t~*=esssup{Y_t~u，u∈u}，
其中Y~u是如下线性超前BSDE的解：
这里，f~u(t，y，z，η_r)=α(t，u_t)y+zσ(t，u_t)+b(t，u_t)E~((?)t)[η_r]+l(t，u_t)，t∈[0，T]，y∈R，z∈R~d，η∈L_(?)~2(t，T+θ)，r∈[t，T+θ]且
定理1.7.1.(单调收敛定理)设m=1．考虑如下一族超前BSDE：对n=1，2，…，设对每一个n=1，2，…，fn满足(H1.1)和(H1.2)，ξn(·)∈S_(?)~2(T，T+K)，δ满足(i)和(ii)，(?) n∈N，t∈[0，T]，y∈R，z∈R~d，有fn(t，y，z，·)单调增，且存在常数μ＞0使得E[integral from n=0 to T|fn(s，0，0，0)|~2ds]≤μ，(?) n∈N．如果(?) t∈[0，T]，y∈R，z∈R~d，θ∈L_(?)~2(t，T+K)，r∈[t，T+K]，有fn(t，y，z，θ_r)↗ f(t，y，z，θ_r)，当n→∞时，且(?)s∈[T，T+K]，有ξn(s)↗ξ(s)，当n→∞时，及f∈S_(?)~2(T，T+K)，那么超前BSDE
有解(Y，Z．)∈S_(?)~2(0，T+K)×L_(?)~2(0，T；R~d)且Y_t=sup_n Y_t~((n))，a．e．，a．s．
定理1.8.3.(带停时的超前BSDE适应解的存在唯一性)设δ满足(i)和(ii)，f循序可测且满足：E[(integral from n=0 to∞|f(s，0，0，0)|ds)~2]＜∞且存在三个正函数μ_1(·)，μ_2(·)及v(·)使得对任给s∈[0，T]，y，y′∈R，z，z′∈R~d，ξ，ξ′∈L_(?)~2(s，T+K)，r∈[s，T+K]，有且(integral from n=0 to∞μ_1(r)dr)~2+(integral from n=0 to∞μ_2(r)dr)~2+integral from n=0 to∞v~2(r)dr＜∞。则对任意给定终值条件ξ．∈S_(?)~2(T，T+K)，超前BSDE
有唯一解(Y，Z．)∈S_(?)~2(0，T+K)×L_(?)~2(0，T；R~d)。
第二章研究推广的超前BSDE，其形式如下：显然这种方程是第一章中研究方程的推广形式。本文给出了推广的超前BSDE适应解的存在唯一性定理，解对参数的连续依赖性，比较定理和单调收敛定理。还研究了推广的超前BSDE的一个例子，它与SDDE之间存在对偶关系并将该对偶应用于随机控制问题。
以下是第二章的主要结果。
定理2.2.2.(适应解的存在唯一性定理)设f满足(H2.1)，(H2.2)和(H2.3)。则对任给终值条件ξ．∈．S_(?)~2(T，T+K；R~m)，η．∈L_(?)~2(T，T+K；R~(m×d))，如上推广的超前BSDE存在唯一解(Y，Z．)∈．S_(?)~2(0，T+K；R~m)×L_(?)~2(0，T+K；R~(m×d))。
定理2.3.1.(比较定理)令(Y~((1))，Z~((1))和(Y~((2))，Z~((2))分别是下面两一维推广的超前BSDE的解：
其中j=1，2．设对j=1，2，f_j满足(H2.1)，(H2.2)和(H2.3)，ξ_j∈S_(?)~2(T，T+K)且(?) t∈[0，T]，z∈R~d，f2(t，·，z)单调增，即f2(t，θ，z)≥，f2(t，θ′，z)，若θ_r≥θ′_r，θ，θ′∈L_(?)~2(t，T+K)，r∈[t，T+K]。如果ξ_s~((1))≥ξ_s~((2))，s∈[T，T+K]，且f1(t，θ，z)≥f2(t，θ，z)，t∈[0，T]，θ∈L_(?)~2(t，T+K)，z∈R~d，那么Y_t~((1))≥Y_t~((2))，a．e．，a．s．
定理2.5.1.(单调收敛定理)设m=1．考虑如下一族推广的超前BSDE：对n=1，2，…，设对任给n=1，2，…，fn满足(H2.1)，(H2.2)及(H2.3)，ξn(·)∈S_(?)~2(T，T+K)，(?) n∈N，t∈[0，T]，z∈R~d，fn(t，·，z)单调增且存在常数μ＞0使得E[integral from n=0 to T|fn(s，0，0)|~2ds]≤μ，(?)n∈N．如果(?)t∈[0，T]，z∈R~d，θ∈L_(?)~2(t，T+K)，有fn(t，θ，z)↗f(t，θ，z)，当n→∞时，及(?)s∈[T，T+K]，有ξn(s)↗ξ(s)，当n→∞时，且ξ∈S_(?)~2(T，T+K)，那么推广的超前BSDE
有解(Y，Z．)∈S_(?)~2(0，T+K)×L_(?)~2(0，T；R~d)且Y_t=sup_nY_t~((n))，a．e．，a．s．
第三章给出了一维SDDEwMS的比较定理及它的一个应用。且证明该比较定理的方法与第1章一维超前BSDE比较定理的证明方法类似。
以下是第三章的主要结果。
定理3.3.4.(比较定理)考虑如下两一维SDDEwMS：其中j=1，2．设δ(t)满足(A.1)，(A.2)且f1，f2，g满足(H3.1′)，(H3.2′)．如果f1(x，y，t，i)≥f2(z，y，t，i)，x，y∈R，t∈[0，T]，i∈S，且x_1(s)≥x_2(s)，s∈[-T，0]，那么X~((1))(T)≥X~((2))(t)，a．e．，a．s．
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 (W_t)0≤t≤T is a standard d-dimensional Brownian motion on a probability space (Ω,F,(F_t)0≤t≤T), with (F_t)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/C~3) 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 X_S 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, f_j 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],Y_t = Y_t~* = esssup{Y_t~u, u∈U}, where Y~u 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,..., f_n 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 (Y~(1),Z~(1)) and (Y~(2), Z~(2)) be respectively the solutions to the following two 1-dimensional generalized anticipated BSDEs:where j = 1,2. Assume that for j = 1,2, f_j 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,..., f_n satisfies (H2.1), (H2.2) and (H2.3), is increasing, and there exists a constant , and , then generalized anticipated BSDEhas a solution and Y_t = sup_ny_t~(n), 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 f_1,f_2,g satisfy (H3.1'), (H3.2'). , and , then , a.e., a.s.
引文
[1] W.J.Anderson, Local behaviour of solutions of stochastic integral equations, Trans. Amer. Math. Soc. 164, 309-321, 1972.
[2] M.Athans, Command and control (c2) theory: A challenge to control science, IEEE Trans. Automat. Control 32, 286-293, 1987.
[3] G.Barles and E.Lesigne, SDE, BSDE and PDE, Backward Stochastic Differential Equation, El Karoui and L.Mazliak, editors,Pitman Research Notes in Mat.Series, 364, Longman, 1997.
[4] V.E.Benes, Existence of optimal strategies based on specified information for a class of stochastic decision problem, SIAM J. Control Optim., 179-188, 1970.
[5] V.E.Benes, Existence of optimal stochastic control law, SIAM J. Control Optim., 9, 446-472, 1971.
[6] J.M.Bismut, Theorie probabiliste du controle des diffusion, Mem. Amer. Math. Soc. No.176, 1973.
[7] Philippe Briand, BSDE's and viscosity solutions of semilinear PDE's. Stochastics Stochastics Rep. 64, no. 1-2, 1-32, 1998.
[8] Ph.Briand, F.Coquet, Y.Hu, J.Memin and S.Peng, A converse comparison theorem for BSDEs and related properties of g-expectation, Elect. Comm. Probab. 5, 101-117, 2000.
[9] R.Buckdahn, M.Quincampoix and Aurel Rascanu, Viability property for a backward stochastic differential equation and applications to partial differential equations, Probab. Theory. Relat.Fields 116,485-504, 2000.
[10] Y.Y.Cao, Y.X.Sun and J.Lam, Delay-dependent robust Hoo control for uncertain systems with time-varying delays. IEE Proceeding-Control Theory and Applications, 145, 338-344, 1998.
[11] Zhigang Cao and Jia-An Yan, A comparison theorem for solutions of stochastic differential equations. Adv. Math. (Chinese) 28, 304-308, 1999.
[12] Zengjing Chen, Existence and uniqueness for BSDE with stopping time, Chinese Science Bulletin, Vol.43, No.2, 96-99, January 1998.
[13] Zengjing Chen and Shige Peng, A general downcrossing inequality for g-martingale, Statictics and Probability Letters, 46, 169-175, 2000.
[14] Francois Coquet, Ying Hu, Jean Memin and Shige Peng, A general converse comparison theorem for stochastic differential equations, C. R. Acad. Sci. Paris, t. 333, Serie I, p. 577-581, 2001.
[15] J.Cvitanic and I.Karatzas, Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24, no. 4, 2024-2056, 1996.
[16] C.Dellacherie, Capacites et processus stochasiques, Springer, 67, 1972.
[17] N.Dunford and J.T.Schwartz, Linear operators, Part Ⅰ: General Theory, Interscience Publishers, New York, 1958.
[18] El Karoui, C.Kapoudjian, E.Pardoux, S.Peng, M.C.Quenez, Reflected solution of backward SDE and related obstacle problems for PDEs, Ann. Probab., 25(2),702-737, 1997.
[19] El Karoui, S.Peng and M.C.Quenez, Backward stochastic differential equations in finance, Mathematical Finance, Vol.7, No.1, pp.1-71, 1997.
[20] L.I.Gal'cluk, M.H.A.Davis, A note on a comparison theorem for equations with different diffusions, Stochastics 6, 147-149, 1982.
[21] S.Hamadene and J.P.Lepeltier, Zero-sum stochastic differential games and backward equations, Systems and Control Letters, 24, 259-263, 1994.
[22] J.Hu, W.Wu and S.Sastry, Modeling subtilin production in bacillus subtilis using stochastic hybrid systems. In R. Alur, G. Pappas Eds., Hybrid Systems: Computation and Control, 7th International Workshop, HSCC 2004, 417-431. Springer LNCS vol. 2993, 2004.
[23] Ying Hu and Shige Peng, On the comparison theorem for multidimensional BSDEs, C. R. Acad. Sci. Paris, Ser. I343, 135-140, 2006.
[24] N.Ikeda and S.Watanabe, A comparison theorem for solutions of stochasitic differential equations and its applications, Osaka J. Math. 14, 619-633, 1977.
[25] A.Ivanov, Yu.Kazmerchuk and A.Swishchuk, Theory, stochastic stability and applications of stochastic differential delay equations" A survey of recent results", Differential Equations and Dynamical Systems 11, no. 1-2, 55-115, 2003.
[26] A.Ivanov and A.Swishchuk, Optimal control of stochastic differential delay equations with application in economics, working paper, 12p, 2004.
[27] A.Ivanov and A.Swishchuk, Global stability of a stochastic differential delay equations, University of Ballarat, SITMS, Ballarat, Australia, Research Report 04/06, 18p, March 2004.
[28] T.Kazangey and D.D.Sworder, Effective federal policies for regulating residential housing, in: Proc. Summer Computer Simulation Conference, San diego, 1120-1128, 1971.
[29] Yu.Kazmerchuk, A.Swishchuk and J.H.Wu, A continuous-time Garch model for stochastic volatility with delay. Can. Appl. Math. Q. 13, no. 2, 123-149, 2005.
[30] V.B.Kolmanovskii and A.Myshkis, Applied theory of functional differential equations, Kluwer Academic Publishers, Dordrecht, 1992.
[31] J.P.Lepeltier and J.San Martin, Backward stochastic differential equations with continuous coefficient. Statistics and Probability Letters, 32, 425-430, 1997.
[32] J.P.Lepeltier and J.San Martin, Backward SDEs with two barriers and continuous coefficient: an existence result, J.Appl.Prob., 41,162-175, 2000.
[33] Qingquan Lin, A comparison theorem for backward stochastic differential equations. (Chinese) J. Huazhong Univ. Sci. Tech. 29, suppl. 1, 1-3, 2001.
[34] Jicheng Liu and Jiagang Ren, Comparison theorem for solutions of backward stochastic differential equations with continuous coefficient. Statist. Probab. Lett. 56, no. 1, 93-100, 2002.
[35] X.Mao, A note on comparison theorems for stochastic differential equations with respect to semimartingales, Stochastics Stochastics Rep. 37, 49-59, 1991.
[36] X.Mao, Exponential stability of stochastic differential equations, Marcel Dekker, New York, 1994.
[37] X.Mao, Stochastic differential equations and applications, Howood Publishing, 1997.
[38] X.Mao, A.Matasov and A.B.Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli 6, 73-90, 2000.
[39] Mariton and M., Jump linear systems in automatic control, Marcel Dekker, New York, 1990.
[40] S-E A Mohammed, Stochastic functional differential equations, London: Research Notes in Mathematics, 1984.
[41] E.Pardoux and S.Peng, Adapted solution of a backward stochastic differential equation, Systems and Control Letters, 14, 55-61, 1990.
[42] E.Pardoux and S.Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic partial differential equations and their applications (Charlotte, NC, 1991), 200-217, Lecture Notes in Control and Inform. Sci., 176, Springer, Berlin, 1992.
[43] E.Pardoux and A.Yu.Veretennikov, Averaging of backward stochastic differential equations, with application to semi-linear PDE's. Stochastics Stochastics Rep. 60, no. 3-4, 255-270, 1997.
[44] P.Park, A delay-dependent stability criterion for systems with uncertain linear systems. IEEE Trans. Automat. Control 44, 1999.
[45] Shige Peng, Dynamical evaluations, C.R.Acad. Sci. Paris, Serie I, 339, 585-589.
[46] Shige Peng, Backward stochastic differential equations and it's application in optimal control, Appl. Math. Optim., 27, 125-144, 1993.
[47] Shige Peng, BSDE and related g-expectation. In Pitman Research Notes in Mathematics Series, No.364, Backward Stochastic Differential Equation, ed. by N.El Karoui and L.Mazliak, 141-159, 1997.
[48] Shige Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type. Probab. Theory Related Fields 113, no. 4, 473-499, 1999.
[49] Shige Peng, NonLinear expectations, nonLinear evaluations and risk measures, Springer-Verlag Berlin Heidelberg 2004, 165-253, 2004.
[50] S.Peng and M.Xu, The smallest g-supermartingale and reflected BSDE with single and double L~2 obstacles, Probabilites et Statistiques, 41, 605-630, 2005.
[51] Shige Peng and Xuehong Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Processes and their Applications, 116, No.3, 370-380, 2006.
[52] M.C.Quenez, Methodes de controle stochastique en Finance, These de doctorat de l'Universite Pierre et Marie Curie, 1993.
[53] E.Rosazza Gianin, Some examples of risk measures via g-expectations, working paper, version of 2004 May.
[54] Y.F.Shi, Y.L.Gu and K.Liu, Comparison theorems of backward doubly stochastic differential equations and applications, Stochastic Analysis and Applications, Issue Volume 23, No.1, 97-110, 2005.
[55] Rong Situ, Comparison theorem of solutions to BSDE with jumps, and viscosity solution to a generalized Hamilton-Jacobi-Bellman equation. Control of distributed parameter and stochastic systems (Hangzhou, 1998), 275-282, Kluwer Acad. Publ., Boston, MA, 1999.
[56] A.V.Skorokhod, Studies in the theory of random process, Addison-Wesley, Reading, MA, 1965.
[57] A.V.Skorohod, Asymptotic methods in the theory of stochastic differential equations, American Mathematical Society, Providence, 1989.
[58] A.V.Swishchuk and A.Ivanov, Stochastic differential delay equations and stochastic stability: A survey of some results. Research Report, 2/99, January 1999, University of Ballarat, Ballarat, Australia, 23p, 1999.
[59] A.V.Swishchuk and Yu.Kazmerchuk, Stability of stochastic Ito equations with delay, Poisson jumps and Markov switchings with applications to finance, Theory Probab. and Mathem. Staffs., v.64, 2001 (translated by AMS, N64, 2002).
[60] A.V.Swishchuk and M.Ya.Svishchuk, Stability of stochastic differential Poisson equations with delay, Ukrain. Math. J., v.53, No 12, 2001.
[61] D.D.Sworder and R.O.Rogers, An LQ-solution to a control problem associated with a solar thermal central receiver, IEEE Trans. Automat. Control 28, 971-978, 1983.
[62] A.S.Willsky and B.C.Levy, Stochastic stability research for complex power systems, DOE Contract, LIDS, MIT, Rep. ET-76-C-01-2295, 1979.
[63] T.Yamada, On comparison theorem for solutions of stochastic differential equations and its applications, J. Math. Kyoto Univ. 13,497-512, 1973.
[64] Jiaan Yan, A comparison theorem for semimartingales and its applications, Seminaire de Probabilites, ⅩⅩ, Lecture Notes in Mathematics, vol. 1204, Springer, Berlin, 349-351, 1986.
[65] S.J.Yan, J.X.Wang and X.F.Liu, The definition and proposition of integral, The Foundation of Probability, 1999.
[66] Juliang Yin and Rong SiTu, On solutions of forward-backward stochastic differential equations with poisson jumps, Stochastic Analysis and Applications, Issue Volume 21, No.6, 1419-1448, 2003.
[67] T.S.Zhang, A comparison theorem for solutions of backward stochastic differential equations with two reflecting barriers and its applications. Probabilistic methods in fluids, 324-331, World Sci. Publ., River Edge, NJ, 2003.
[68] Shaofu Zhou and Zhenghong Wei, A comparison theorem for an infinite-horizon backward stochastic differential equation. (Chinese) J. Huazhong Univ. Sci. Technol. Nat. Sci. 30, no.1, 109-110, 113, 2002.
[2] M.Athans, Command and control (c2) theory: A challenge to control science, IEEE Trans. Automat. Control 32, 286-293, 1987.
[3] G.Barles and E.Lesigne, SDE, BSDE and PDE, Backward Stochastic Differential Equation, El Karoui and L.Mazliak, editors,Pitman Research Notes in Mat.Series, 364, Longman, 1997.
[4] V.E.Benes, Existence of optimal strategies based on specified information for a class of stochastic decision problem, SIAM J. Control Optim., 179-188, 1970.
[5] V.E.Benes, Existence of optimal stochastic control law, SIAM J. Control Optim., 9, 446-472, 1971.
[6] J.M.Bismut, Theorie probabiliste du controle des diffusion, Mem. Amer. Math. Soc. No.176, 1973.
[7] Philippe Briand, BSDE's and viscosity solutions of semilinear PDE's. Stochastics Stochastics Rep. 64, no. 1-2, 1-32, 1998.
[8] Ph.Briand, F.Coquet, Y.Hu, J.Memin and S.Peng, A converse comparison theorem for BSDEs and related properties of g-expectation, Elect. Comm. Probab. 5, 101-117, 2000.
[9] R.Buckdahn, M.Quincampoix and Aurel Rascanu, Viability property for a backward stochastic differential equation and applications to partial differential equations, Probab. Theory. Relat.Fields 116,485-504, 2000.
[10] Y.Y.Cao, Y.X.Sun and J.Lam, Delay-dependent robust Hoo control for uncertain systems with time-varying delays. IEE Proceeding-Control Theory and Applications, 145, 338-344, 1998.
[11] Zhigang Cao and Jia-An Yan, A comparison theorem for solutions of stochastic differential equations. Adv. Math. (Chinese) 28, 304-308, 1999.
[12] Zengjing Chen, Existence and uniqueness for BSDE with stopping time, Chinese Science Bulletin, Vol.43, No.2, 96-99, January 1998.
[13] Zengjing Chen and Shige Peng, A general downcrossing inequality for g-martingale, Statictics and Probability Letters, 46, 169-175, 2000.
[14] Francois Coquet, Ying Hu, Jean Memin and Shige Peng, A general converse comparison theorem for stochastic differential equations, C. R. Acad. Sci. Paris, t. 333, Serie I, p. 577-581, 2001.
[15] J.Cvitanic and I.Karatzas, Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24, no. 4, 2024-2056, 1996.
[16] C.Dellacherie, Capacites et processus stochasiques, Springer, 67, 1972.
[17] N.Dunford and J.T.Schwartz, Linear operators, Part Ⅰ: General Theory, Interscience Publishers, New York, 1958.
[18] El Karoui, C.Kapoudjian, E.Pardoux, S.Peng, M.C.Quenez, Reflected solution of backward SDE and related obstacle problems for PDEs, Ann. Probab., 25(2),702-737, 1997.
[19] El Karoui, S.Peng and M.C.Quenez, Backward stochastic differential equations in finance, Mathematical Finance, Vol.7, No.1, pp.1-71, 1997.
[20] L.I.Gal'cluk, M.H.A.Davis, A note on a comparison theorem for equations with different diffusions, Stochastics 6, 147-149, 1982.
[21] S.Hamadene and J.P.Lepeltier, Zero-sum stochastic differential games and backward equations, Systems and Control Letters, 24, 259-263, 1994.
[22] J.Hu, W.Wu and S.Sastry, Modeling subtilin production in bacillus subtilis using stochastic hybrid systems. In R. Alur, G. Pappas Eds., Hybrid Systems: Computation and Control, 7th International Workshop, HSCC 2004, 417-431. Springer LNCS vol. 2993, 2004.
[23] Ying Hu and Shige Peng, On the comparison theorem for multidimensional BSDEs, C. R. Acad. Sci. Paris, Ser. I343, 135-140, 2006.
[24] N.Ikeda and S.Watanabe, A comparison theorem for solutions of stochasitic differential equations and its applications, Osaka J. Math. 14, 619-633, 1977.
[25] A.Ivanov, Yu.Kazmerchuk and A.Swishchuk, Theory, stochastic stability and applications of stochastic differential delay equations" A survey of recent results", Differential Equations and Dynamical Systems 11, no. 1-2, 55-115, 2003.
[26] A.Ivanov and A.Swishchuk, Optimal control of stochastic differential delay equations with application in economics, working paper, 12p, 2004.
[27] A.Ivanov and A.Swishchuk, Global stability of a stochastic differential delay equations, University of Ballarat, SITMS, Ballarat, Australia, Research Report 04/06, 18p, March 2004.
[28] T.Kazangey and D.D.Sworder, Effective federal policies for regulating residential housing, in: Proc. Summer Computer Simulation Conference, San diego, 1120-1128, 1971.
[29] Yu.Kazmerchuk, A.Swishchuk and J.H.Wu, A continuous-time Garch model for stochastic volatility with delay. Can. Appl. Math. Q. 13, no. 2, 123-149, 2005.
[30] V.B.Kolmanovskii and A.Myshkis, Applied theory of functional differential equations, Kluwer Academic Publishers, Dordrecht, 1992.
[31] J.P.Lepeltier and J.San Martin, Backward stochastic differential equations with continuous coefficient. Statistics and Probability Letters, 32, 425-430, 1997.
[32] J.P.Lepeltier and J.San Martin, Backward SDEs with two barriers and continuous coefficient: an existence result, J.Appl.Prob., 41,162-175, 2000.
[33] Qingquan Lin, A comparison theorem for backward stochastic differential equations. (Chinese) J. Huazhong Univ. Sci. Tech. 29, suppl. 1, 1-3, 2001.
[34] Jicheng Liu and Jiagang Ren, Comparison theorem for solutions of backward stochastic differential equations with continuous coefficient. Statist. Probab. Lett. 56, no. 1, 93-100, 2002.
[35] X.Mao, A note on comparison theorems for stochastic differential equations with respect to semimartingales, Stochastics Stochastics Rep. 37, 49-59, 1991.
[36] X.Mao, Exponential stability of stochastic differential equations, Marcel Dekker, New York, 1994.
[37] X.Mao, Stochastic differential equations and applications, Howood Publishing, 1997.
[38] X.Mao, A.Matasov and A.B.Piunovskiy, Stochastic differential delay equations with Markovian switching, Bernoulli 6, 73-90, 2000.
[39] Mariton and M., Jump linear systems in automatic control, Marcel Dekker, New York, 1990.
[40] S-E A Mohammed, Stochastic functional differential equations, London: Research Notes in Mathematics, 1984.
[41] E.Pardoux and S.Peng, Adapted solution of a backward stochastic differential equation, Systems and Control Letters, 14, 55-61, 1990.
[42] E.Pardoux and S.Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations. Stochastic partial differential equations and their applications (Charlotte, NC, 1991), 200-217, Lecture Notes in Control and Inform. Sci., 176, Springer, Berlin, 1992.
[43] E.Pardoux and A.Yu.Veretennikov, Averaging of backward stochastic differential equations, with application to semi-linear PDE's. Stochastics Stochastics Rep. 60, no. 3-4, 255-270, 1997.
[44] P.Park, A delay-dependent stability criterion for systems with uncertain linear systems. IEEE Trans. Automat. Control 44, 1999.
[45] Shige Peng, Dynamical evaluations, C.R.Acad. Sci. Paris, Serie I, 339, 585-589.
[46] Shige Peng, Backward stochastic differential equations and it's application in optimal control, Appl. Math. Optim., 27, 125-144, 1993.
[47] Shige Peng, BSDE and related g-expectation. In Pitman Research Notes in Mathematics Series, No.364, Backward Stochastic Differential Equation, ed. by N.El Karoui and L.Mazliak, 141-159, 1997.
[48] Shige Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type. Probab. Theory Related Fields 113, no. 4, 473-499, 1999.
[49] Shige Peng, NonLinear expectations, nonLinear evaluations and risk measures, Springer-Verlag Berlin Heidelberg 2004, 165-253, 2004.
[50] S.Peng and M.Xu, The smallest g-supermartingale and reflected BSDE with single and double L~2 obstacles, Probabilites et Statistiques, 41, 605-630, 2005.
[51] Shige Peng and Xuehong Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Processes and their Applications, 116, No.3, 370-380, 2006.
[52] M.C.Quenez, Methodes de controle stochastique en Finance, These de doctorat de l'Universite Pierre et Marie Curie, 1993.
[53] E.Rosazza Gianin, Some examples of risk measures via g-expectations, working paper, version of 2004 May.
[54] Y.F.Shi, Y.L.Gu and K.Liu, Comparison theorems of backward doubly stochastic differential equations and applications, Stochastic Analysis and Applications, Issue Volume 23, No.1, 97-110, 2005.
[55] Rong Situ, Comparison theorem of solutions to BSDE with jumps, and viscosity solution to a generalized Hamilton-Jacobi-Bellman equation. Control of distributed parameter and stochastic systems (Hangzhou, 1998), 275-282, Kluwer Acad. Publ., Boston, MA, 1999.
[56] A.V.Skorokhod, Studies in the theory of random process, Addison-Wesley, Reading, MA, 1965.
[57] A.V.Skorohod, Asymptotic methods in the theory of stochastic differential equations, American Mathematical Society, Providence, 1989.
[58] A.V.Swishchuk and A.Ivanov, Stochastic differential delay equations and stochastic stability: A survey of some results. Research Report, 2/99, January 1999, University of Ballarat, Ballarat, Australia, 23p, 1999.
[59] A.V.Swishchuk and Yu.Kazmerchuk, Stability of stochastic Ito equations with delay, Poisson jumps and Markov switchings with applications to finance, Theory Probab. and Mathem. Staffs., v.64, 2001 (translated by AMS, N64, 2002).
[60] A.V.Swishchuk and M.Ya.Svishchuk, Stability of stochastic differential Poisson equations with delay, Ukrain. Math. J., v.53, No 12, 2001.
[61] D.D.Sworder and R.O.Rogers, An LQ-solution to a control problem associated with a solar thermal central receiver, IEEE Trans. Automat. Control 28, 971-978, 1983.
[62] A.S.Willsky and B.C.Levy, Stochastic stability research for complex power systems, DOE Contract, LIDS, MIT, Rep. ET-76-C-01-2295, 1979.
[63] T.Yamada, On comparison theorem for solutions of stochastic differential equations and its applications, J. Math. Kyoto Univ. 13,497-512, 1973.
[64] Jiaan Yan, A comparison theorem for semimartingales and its applications, Seminaire de Probabilites, ⅩⅩ, Lecture Notes in Mathematics, vol. 1204, Springer, Berlin, 349-351, 1986.
[65] S.J.Yan, J.X.Wang and X.F.Liu, The definition and proposition of integral, The Foundation of Probability, 1999.
[66] Juliang Yin and Rong SiTu, On solutions of forward-backward stochastic differential equations with poisson jumps, Stochastic Analysis and Applications, Issue Volume 21, No.6, 1419-1448, 2003.
[67] T.S.Zhang, A comparison theorem for solutions of backward stochastic differential equations with two reflecting barriers and its applications. Probabilistic methods in fluids, 324-331, World Sci. Publ., River Edge, NJ, 2003.
[68] Shaofu Zhou and Zhenghong Wei, A comparison theorem for an infinite-horizon backward stochastic differential equation. (Chinese) J. Huazhong Univ. Sci. Technol. Nat. Sci. 30, no.1, 109-110, 113, 2002.