Differential Games Of Forward-Backward Stochastic System Under Partial Information And Applications In Finance

This thesis is concerned with differential games driven by forward-backward stochas-tic differential equations under the partial information condition and the related appli-cations in finance.It can be summarized into six chapters as follows.In a control system,solving the optimal control should be based on the information available to decision makers.However,in most of cases,controllers cannot get the complete information of the state process.Then,it is reasonable to assume the strategy to be adapted to the available filtration and to get the filtering equation of the state process.Meanwhile,it is common to consider single control and single cost functional in a control system.However,there exists many cases(e.g.Prisoner's dilemma)that have multiple players with different goals,which can be seen as a typical game system.When making decisions,each player will consider other players' strategy and try to optimize his own cost functional.The aim is to find a "Nash equilibrium point",not the "optimal control".Further,the stochastic differential game problem is the game system driven by dynamic stochastic differential equations,and the corresponding Nash equilibrium point should be studied.In Chapter 1,we make an introduction on our research background,and illustrate main results by each chapter.In Chapter 2,we study a partially observed stochastic differential game based on forward-backward stochastic differential equation where the forward diffusion coefficients can contain control variables and control domains are convex.We assume players can-not fully observe the state equation and each player has his own observation equation.Meanwhile,the observation equation can contain control variable and has correlated noise with the state equation.By virtue of the convex variational method,we introduce the adjoint equation and establish the maximum principle as a necessary condition for the Nash equilibrium point,and derive the verification theorem as a sufficient condition.In Chapter 3,we focus on the stochastic linear-quadratic system and make research on the partially observed differential game.The state equation is driven by the forward and backward stochastic differential equation where forward diffusion coefficients do not contain control variables and control domains are not necessarily convex.We assume players cannot fully observe the real state model,so their decisions should be based on the filtration generated by the observation process.By virtue of the backward separation technique,we overcome the circular dependence that the control process is adapted to a controlled filtration.Applying the spike variational method,we derive a necessary condition and a sufficient condition for the Nash equilibrium point.In addition,we obtain filtering equations by using stochastic filtering formula,and present a feedback form of the equilibrium point through Riccati equations.As a practical application,we solve a partial information risk-minimizing investment problem,involving g-expectation as a convex risk measure.We give the numerical simulation and analysis.Chapter 4 is concerned with a differential game problem of an anticipated forward-backward stochastic differential delayed equation under partial information.We assume game players can only make decisions based on partial information.We establish a maximum principle and a verification theorem for the Nash equilibrium point by virtue of the duality and convex variation approach.Furthermore,we study a linear-quadratic system under partial information and present an explicit form of the Nash equilibrium point.Also,we derive the filtering equations and prove the existence and uniqueness of the Nash equilibrium point.As an application,we solve a time-delayed risk-minimizing consumption problem and obtain the explicit Nash equilibrium solution.Chapter 5 is discussing a partially observed time-inconsistent stochastic linear-quadratic control system,in which the state follows a stochastic differential equation driven by a Brownian motion and an independent Poisson random measure.Differen-t from the classical type of cost functional,it contains a state-dependent term and a quadratic term of the conditional expected state process,which will cause time incon-sistency in the dynamic system.Thus,the Bellman's principle of optimalityis no longer valid,and we cannot use the dynamic principle any more.Considering that different point-in-time has different preference,we introduce the equilibrium definition for the time-inconsistent problem.We derive an explicit expression for the equilibrium in fully-observed model with stochastic coefficients.Then,we obtain the explicit feedback form of equilibrium with Riccati equations in deterministic coefficients case.Finally,we get filtering equations of the partially observed system and verify the equilibrium in some special case.In Chapter 6,combining financial models,we investigate a robust optimal con-sumption and portfolio choice problem under model uncertainty.We assume investors are ambiguity averse,which means they do not know the real distribution of the model and they keep an averse skepticism toward the accuracy of the asset model.A reference model is considered by the investor to describe the data generating model,but he knows that it might be misspecified and other alternative models might be better.Thus,he in-tends to find the investment strategy that are robust to the model misspecification even the worst-case.In the model setting,we consider the asset process follows a stochas-tic volatility jump-diffusion process and investors can have different levels of ambiguity aversion about diffusion risk and the jump risk.Investors' preferences satisfy the re-cursive Duffie-Epstein-Zin utility,which separates the risk aversion and the elasticity of intertemporal substitution(EIS)preference.At the same time,we assume investors can access not only the stock and risk-less bond markets,but also the derivatives mar-ket.This can make the financial market complete since asset processes are affected by multiple risk factors.We make research separately in the complete market and incom-plete market,and obtain an exact analytical solution when investors have unit EIS of consumption and approximate otherwise.By numerical analysis,we find that optimal exposures corresponding to diffusion risks and to jump risk are significantly affected by the ambiguity aversion about their own risk factor in the complete market.In the in-complete market,the ambiguity aversion with respect to diffusion risk on the optimal stock investment is more sensitive than the ambiguity aversion with respect to jump risk.More importantly,by calculating the utility loss,we make the conclusion that con-sidering ambiguity aversion with respect to diffusion risks and participating derivatives market trading are essential to reduce the welfare loss.