Researches On Forward Backward Stochastic Control System With Lévy Process

Up to now,forward backward stochastic control problems driven by Brownian motion or by Brownian motion and Poisson process have been extensively studied and fruitful results have been derived.However,for forward backward stochastic control problems with Lévy process,they are faced with huge challenges and there still exists many difficult problems which need to be solved,such as the maximum condition of the control system,the control problem with constraints and so on.On the other hand,with the development of the financial market and the demand of controlling risks in financial market,options get further development in Chinese market.How to construct reasonable option pricing model and how to obtain an accurate and efficient numerical algorithm have became a hot topic of widespread concerns.These problems are what this thesis is trying to answer.The content of this thesis is mainly divided into the following parts.(1)Using convex variable,dual technology and some another technologies,chapter three proves the maximum principle of forward backward stochastic control system with Lévy process,and under some necessary convex assumptions,verifies this maximum condition is also a sufficient condition.In particular,for linear quadratic(LQ)optimal control problem,this chapter proves the existence and uniqueness of the optimal control and gives the expression for this optimal control.In the case of non-randomness of the coefficient matrices,the optimal linear state feedback regulator of this system is obtained by introducing a generalized Riccati equation.(2)A forward backward stochastic control system driven by Lévy process with terminal state constraints is considered,this problem requires the status terminal of forward part in a convex set.The maximum principle of this control problem is established by using a backward lemma and Ekeland variational principle,and a class of stochastic LQ control problem and stochastic recursive utility problem with constraints is discussed by the maximum principle established above.(3)On the basis of these theoretical researches,in the fifth chapter,by introducing a feasible control set to describe Knight uncertainty on the financial market,an option pricing model in Lévy market under Knight uncertainty is established and the pricing intervals are obtained.The influence of Knight uncertainty parameter on the pricing range is studied by numerical analysis method.At the same time,this chapter also establishes a bank insurance pricing model under Knight uncertainty,and then employs this model to simulate the deposit insurance premiums intervals of 16 listed banks.(4)From the point view of forward backward stochastic differential equation(FBSDE),three option pricing algorithms and comparative analysis are given by using Theta scheme and predictive correction algorithm of FBSDE.