The Application Of Backward Doubly Stochastic Differential Equations To SDU

The theory of stochastic differential utility (SDU for short) has been developed only recently, beginning in the early 1990s. In a general form, BSDEs were introduced by Duffie and Epstein.A class of backward doubly stochastic differential equations (BDSDEs for short) were introuced by Pardoux and Peng in 1994, with two different directions of stochastic integrals, i.e.. The equations involve both a standard (forward) stochastic integral dW_t and a backward stochastic integral dB_t. They proved the existence and uniqueness of solution for BDSDEs under uniformly Lipschitz conditions. For one-dimensional BDSDEs, Shi yufeng, Gu yanling and Liu kai proved the comparison theorem of BDSDEs under the production of Pardoux and Peng in 2005.The present thesis mainly consists of third parts.In the first section, I shall introduce backward doubly stochastic differential equation. Firstly, I will give knowledge about BSDEs. Secondly, I will give elementary symbols, the form of BDSBE. Lastly, I will introduce the existence and uniqueness of solution for BDSDEs and the comparison theorem of one-dimensional BDSDE.In the second section, under new conditions, I will introduce the equation, which is satisfied by stochastic differential utility. Then I will give the define of SDU and prove the existence and uniqueness of solution for it.In the last section, before introducing and proving the property of utility process and utility function, I will show the comparison theorem of one-dimensional SDU. The property consist continuity, monotonicity for terminal value, monotonicity in consumption, concavity, risk aversion.