Control Theorem Of BSDE's Solution

In 1990, Pardoux and Peng (see [5]) introduced a class of nonlinear backward stochastic differential equations (BSDEs):Under some suitable assumptions on ξ and g, Pardoux and Peng showed that there exist a pair of adapted solution (Y_t,Z_t) solving the above kind of BSDE.According to Pardoux and Peng's theorem, the solution of the above BSDE is a solution pair (Y_t, Z_t), which consist of two parts. Since this theory was put forward, it have been widely applied in mathematical finance, stochastic control, partial differential equation and mathematical economics. Due to its broad application, many researchers begin to have a deep interest in this subject. Many properties and results on the solutions of BSDEs have been obtained since the BSDE theory was proposed in 1990. However, so far, most research was focused on the first part of the solution Y_t. Few people has concentrated on the second part Z_t, which is actually the volatility of BSDEs.In recent years, people begin to explore the property of Z_t. A great progress in this direction is Chen's comonotonic theorem in 2005 (see [1]), in whichChen proved the comonotonicity of Zt. That is, let (YJ1, Z\) and (Vt2, Z\) be the solutions of BSDEs corresponding to terminal values ￡1 = $i(Xf<) and 6 = $%{%%), respectively. X\ and X\ are generated by the following two Markov processesdX\ = b{t, Xi)dt + a% XI) dWu te [0, T]where i = 1,2, then Z\ 0 Z\ > 0, as $i and $2 are comonotonic andIn this paper, we continue to explore the property of Zt. We will consider the following questions: Under which conditions we have Zt > k{ox Zt < k), for any constant k? Under which conditions the volatility of backward SDE could be controlled by the volatility of forward SDE, that is Zt > ka(t, Xt)(or Zt < ka(t,Xt))'? In the following, we will give several results about these questions.This paper is arranged into 5 sections: Section 1: Basic Notation and Results on BSDEsIn this section, we give some basic notation and classical results in BSDE theory, which will be used in the following sections.Section 2: Comonotonic TheoremWe here give Chen's comonotonic theorem, which is the foundation of this paper.Section 3: The Simple Version of Control TheoremIn this section, we discuss the first question that under which conditions, we have Zt > fc(or Zt < k). In the last of this section, we will give an example for the application of this theorem.Section 4: Control Theorem of BSDEsIn this part, we take into account the second question, that is under which conditions, we have Zt > ka(t, Xt){ot Zt