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Applications Of G-expectation And BSDE In Stochastic Control, Finance And Insurance

Posted on:2017-06-02Degree:DoctorType:Dissertation
Country:ChinaCandidate:Z Y SunFull Text:PDF
GTID:1319330566453653Subject:Probability theory and mathematical statistics
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Expected utility theory is the foundation of modern mathematical economics,of which the most basic assumptions is that probabilities or expectations are additive or linear.However,there are many uncertainties in financial markets that can not be well described by using additive probabilities or linear expectations.For example,as the famous Allais paradox and Ellsberg paradox were raised,the expected utility theory faced a great challenge.The additivity of probabilities or the linearity of expectations is one of the most important reasons of these paradoxes.In order to avoid expected utility theory not being enough in explaining economic phenomena,more and more scholars devoted to investigating non-linear mathematical expectations.Motivated by measuring risks and other financial problems with uncertainty,Peng introduced the sublinear expectation theory which does not require a probability space framework and can be used to model ambiguous volatility.Based on the sublinear expectation theory,Peng [76,77,78,79] introduced the notions of G-normal distribution and G-Brownian motion,constructed a special type of sublinear expectation space: Gexpectation space,which is the most important sublinear expectation space and takes the role of Wiener space as in classical probability theory.In the G-expectation space,Peng defined the stochastic integral driven by G-Brownian motion,obtained the G-It?o's formula,derived the G-martingale representation theorem,etc.The proof of uniqueness and existence of solutions to G-SDEs or G-BSDEs is similar to the classical case.All these theories constitute a set of complete theoretical system of nonlinear expectations.One important property of the G-expectation is that it can be represented as the supremum of a subset of linear expectations.In most cases,this subset reflects the degree of uncertainty of the decision makers.On the other hand,in order to handle the jump models in financial markets,Hu and Peng [52] first introduced G-L?evy processes under sublinear expectations,which are independent and stationary increment processes but not necessarily continuous.They further obtained the corresponding L?evy-Khintchine formula and proved that the distribution of G-L?evy process satisfies a nonlinear Parabolic integro-partial differential equation.Conversely,the existence of G-L?evy processes can be constructed by the solution of this integro-partial differential equation.Thus,G-L?evy process is a meaningful generalization of G-Brownian motion and we can employ it to solve a large amount of practical problems in financial markets.So the theory of G-expectation,G-Brownian motion,and G-L?evy process have become a powerful tool in the study of asset pricing(Epstein and Ji [35]),mathematical finance(Vorbrink [95]),stochastic recursive utility(see e.g.,Hu et al.[51],Zhang [102],Zheng et al.[107])as well as in the stochastic optimal control(see e.g.,Fei and Fei [38],Hu and Ji [49]).In Chapter 1,we give a brief introduction of G-normal distribution,G-Brownian motion,G-expectation,G-martingale,G-SDE,G-BSDE and G-L?evy process,this will be used many times in the following chapters.Most of the contents are borrowed from Peng [79]and Hu and Peng [52].In Chapter 2 of this thesis,we investigate some stochastic optimal control problems under G-expectation framework.We assume the control system follows a stochastic differential equation driven by a G-Brownian motion,by using the classical Spike variational approach,we obtain a general stochastic maximum principle and the corresponding necessary and sufficient conditions of optimality.In addition,we apply our main results to solve a mean-variance asset management problem in the financial market with ambiguous volatility.All of which give an effective method in solving stochastic control problems under model uncertainty;In Chapter 3,We use the G-expectation framework to investigate some ruin problems for risk models that contain uncertainties on both claim frequency and claim size distribution.We suppose that the reserve process is described as a class of “G-Compound Poisson process”,a special case of the G-L?evy process.By using the exponential martingale approach,we obtain the upper bounds for the two-sided ruin probability as well as the ruin probability involving investment.Furthermore,we derive the optimal investment strategy under the criterion of minimizing this upper bound.It is well known that the investigations of linear backward stochastic differential equation were started from Bismut [8],and then Pardoux and Peng [71] proved the existence and uniqueness of solutions in the general case.Nonlinear BSDEs in the framework of Brownian motion and Poisson random jumps were first introduced by Tang and Li [92].Independently,Duffie and Epstein [29] developed a formulation of recursive utility by a special type of BSDE.In recent years,the theory of BSDEs has developed quickly and found various applications,namely in stochastic control,financial mathematics,partial differential equations(PDE),stochastic differential utility,etc.Since the pioneering work of Duffie and Epstein [29],recursive utility has been investigated by many researchers.The recursive utility means the instantaneous utility depends not only on the instantaneous consumption rate but also on the future utility,it is defined from BSDE and is really a generalization of the standard additive utility.This triggered the investigations of stochastic control problems when the objective is to maximize the recursive utility.A considerable amount of work has been devoted to studying this kind of stochastic recursive optimal control problems.See e.g.,?ksendal and Sulem [69],Pamen [70],Peng [74],Peng and Wu [80],Shi and Wu [86,87],Shi and Yu [90],Tao and Wu [93],Wu [98],etc.In chapter 4,we consider a stochastic system which consists of a forward-backward stochastic differential equation with both Poisson jumps and regime-switching,a general sufficient stochastic maximum principle and its relation to dynamic programming principle have been established.At last,the result is applied to a recursive utility asset management problem as an explicitly illustrated example;In chapter 5,we consider a mean-variance investment and reinsurance problem for an insurer in a stochastic environment.The risks in the market come from the stochastic volatility of the stock,which is described by the CIR model.By the classical stochastic linear quadratic technique,we first consider a backward stochastic differential equation.Then closed-form expressions of the efficient frontiers and efficient strategies are related to the solution of the BSDE;In Chapter 6,we first develop a general risk-sensitive type maximum principle for Markov regime-switching jump-diffusion processes.Then,we apply the maximum principle to study a risk-sensitive benchmarked asset management problem for Markov regime-switching models.The optimal control is of feedback form and is given in terms of solutions to a Markov regime-switching Riccatti equation and an ordinary Markov regime-switching differential equation.This dissertation is mainly devoted to the applications of G-expectation and BSDE in stochastic control,finance and insurance.The innovation points of this dissertation are as follows:First,we use the notion G-expectation and the corresponding G-Brownian motion to study model risk caused by uncertain volatilities.The stochastic maximum principle under G-expectation framework extends the previous studies,which was only considered on a classical probability space(corresponding to G is linear in our case).Besides,the main result can be applied to study the portfolio selection problems in the financial market with ambiguous volatilities.Second,we use the “G-Compound Poisson process” to investigate some ruin problems for risk models that contain uncertainties on both claim frequency and claim size distribution.By using the exponential martingale approach,we obtain the upper bounds for the two-sided ruin probability as well as the ruin probability involving investment.Furthermore,we derive the optimal investment strategy under the criterion of minimizing this upper bound.This enriches the results of the classical risk theory.Third,we consider a mean-variance investment and reinsurance problem for an insurer in a stochastic environment.By the classical stochastic linear quadratic and backward stochastic differential equation technique,we give closed-form expressions of the efficient frontiers and efficient strategies by the solution of the BSDE.This avoids proving a verification theorem with viscosity solutions in the classical dynamic programming approach,which is difficult to tackle under many stochastic volatility models.Fourth,we use backward stochastic differential equations and the stochastic maximum principle to investigate the recursive utility maximization problems and the risk-sensitive asset management problems under Markov regime-switching models.These two methods can deal with not only the Markovian control problems but also some non-Markovian systems,which scales the application of them greatly.
Keywords/Search Tags:G-normal distribution, G-expectation, G-Brownian motion, G-L(?)vy process, G-Compound Poisson process, Backward stochastic differential equations, Stochastic optimal control, Recursive utility, Adjoint process, Stochastic maximum principle
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