Bayesian Inference And Stochastic Differential Equations Under Nonlinear Expectations

The study for the problems of probability and statistics under the nonlinear expec-tations is inspired by the development trend of the fundamental research of probability theory, as well as the growing uncertainty problem in the financial markets.Andrey Kolmogorov established the axiom system of the theory of probability in Foundations of the Theory of Probability, and gave the definitions of linear probability measure and expectation. But there exist some phenomenon of uncertainty in the risk measures, super hedge pricing that can not be modeled well by the linear probability. Due to the nonlinear risk problems in finance, many scholars introduced the nonlinear tool to measure the uncertainty. Choquet (1954) introduced the conception of capacity and Choquet expectation (or Choquet integral), and it had been used widely in potential the-ory and decision theory, for more information please see Choquet (1954), Doob (1984),Schmeidler (1989),Gilboa and Schmeidler (1989)?But the conditional Choquet expec-tation is difficult to deal with the dynamic problem in the economics, it had not been applied well. Peng (1997) introduced another important nonlinear expectation called g-expectation via backforward stochastic differential equations to measure the randomicity and risk of the probabilistic uncertainty model, for more information please see Chen and Epstein (2002), Frittelli and Gianin (2004), Peng (2004)?But g-expectation certainly had some limitations because it can only used to deal with uncertain probability measures,which is absolute continuous about the probability measure. The volatility uncertainty problem in finance, however, exists uncountable unknown probability measures and they had singularity.Recently, motivated by the risk measures, superhedge pricing and modelling un-certainty in finance, Peng has introduced a framework of time consistent nonlinear expectation called G-expectation (see Peng (2007, 2010)). Together with the notion of G-expectations Peng also introduced the related G-normal distribution and the G-Brownian motion. The G-Brownian motion is a stochastic process with stationary and independent increments and its quadratic variation process is, unlike the classical case of a linear expectation, a nondeterministic process. Moreover, an Ito calculus for the G-Brownian motion has been developed recently in Peng (2007, 2010), Li and Peng (2011).Since these notions were introduced, many properties of G-Brownian motion have been studied by researchers, see for example, Xu and Zhang (2009), Gao and Jiang (2010),Hu and Zhang (2010), Song (2012), Nutz (2013), Nutz and Van Handel (2013), Hu, Ji,Peng and Song (2014), Peng and Song (2015), Zhang and Chen (2015), Hu, Wang and Zheng (2016).In view of above problems, this doctoral dissertation conducts research on these issues and presents following results, we believe some of them are very interesting and innovative:1. We btain a new algorithm called PSI method to simulate the posterior expectations with uncertainty, and innovatively introduce an auxiliary hyperprior distribution on a parameter that indexes the uncertainty. Furthermore, we widely apply the algorithm to a .variety of uncertain situations.2. We study the asymptotic estimates for the solution of stochastic differential equa-tions driven by G-Brownian motion. And for the first time, we obtain the iterated logarithm estimates for the solution of stochastic differential equations under the nonlinear expectations, and study the estimates for the solution of some special multidimensional G-SDEs and generalize it to the general form.3. We study the properties of the solution of stochastic differential equations under the G-framwork, and obtain the boundedness and exponential stability of the solutions of first and second order stochastic differential equations driven by G-Brownian motion.4. Furthermore, we also extend some convergence and stability results in classical s-tochastic analysis to G-expectation's framework.Now we will introduce above results by chapters. These results come from five aca-demic papers I finished during my Ph.D. study.Chapter One A new computational method is proposed in our paper for computing the range of posterior expectations in Bayesian statistics in order to account for uncertainty in the prior or uncertainly in the likelihood func-tion. This method uses an auxiliary prior to generate a Monte Carlo sample by means of, for example, the Metropolis algorithm, and then estimates the range of the posterior expectations. The method is shown to accommodate the posterior expectations for all possible choices of the uncertainty scenar-ios. Our framework is general and can be useful for many different situations,such as uncertainty for the choice of the likelihood function, or uncertainty for the choice of the prior distribution. The main contribution of this paper is to take only one Monte Carlo sample for all possible choices of uncertainty scenarios, avoiding heavy computation in the traditional sensitivity analysis.· 1.1 PSI Algorithm and Theoretical SupportThe proposed computational method involves three steps (with the mnemonic PSI):1. (Prior Step P) Introduce an auxiliary hyperprior distribution ?(?) on a parameter? that indexes the uncertainty.2. (Sampling Step S) For any parameter of interest X, we derive a sample of (X, ?)from the joint posterior distribution given the observed data, using a Monte Carlo method:{Details: It is noted that often the parameter of interest X is not the same as the structure parameter ? used in the model of generating the data D. The structure parameter ? could be a vector, while X ?F(?,?) of interest could be a scalar function of it. In this more general situation, we can decompose the Sampling Step into the following sub-steps:2.1: Use a Monte Carlo method to sample from the joint posterior distribution of (?,?),where ? is the structural parameter of the data generation process after fixing the uncertainty.2.2: For any parameter of interest X=F(?, ?),we derive the corresponding sample of (X,?)from the posterior distribution.}3. (Inference Step I) Based on this posterior sample of (X, A), we estimate the range of posterior expectations (and similarly the range for any posterior quantile).Then we show that the corresponding range of posterior expectations, related to the target distribution conditional on ?, can be consistently estimated given a uniformly consistent estimator of the conditional mean, based on Monte Carlo samples from the joint distribution.Proposition 0.1. Let the target conditional posterior distribution be which is well-defined (with a nonzero denominator) for all ???. Assume ?(·)is an auxiliary prior measure satisfying(H) the density ?(?) > 0 for all ???.Let the conditional distribution derived from the joint posterior Then(i) ?(X|?, D)=f(X?D, ?), for all X, and for all ???.(ii) Let the target maximal posterior expectation be Then for any choice of auxiliary prior ?(?) satisfying (H).(iii) Let ST=(Xt,?t)t?T be a Monte Carlo sample (e.g., based on a Markov Chain,or i.i.d. from ?(X,??D)),and ?(??ST) be a corresponding uniformly consistent condi-tional mean estimator of ?(?) = E(X?D,?),such that for any ? > 0,Then is a consistent estimator of the maximal posterior expectation i.e., for any positive ?,· 1.2 Simulation Analysis under UncertaintyWe present the simulation analysis of the algorithm under three uncertain cases.Chapter Two We study the asymptotic estimates for the solution of stochastic differential equations driven by G-Brownian motion. And for the first time,we obtain the iterated logarithm estimates for the solution of stochastic differential equations under the nonlinear expectations, and study the estimates for the solution of some special multidimensional G-SDEs and generalize it to the general form.Now we consider the d-dimensional stochastic differential equation driven by G-Brownian motion for 1 ?i,j < m, with initial value Xt0 = X0?Rd,where Bt is an m-dimensional G-Brownian motion, and G(A) := E((AB1,B1>) for A ? Sm, where Sm is the space of all m x m symmetric matrices. Moreover, we denote 2G(I) = ?2. Here we use the above repeated indices i and j to imply the summation. Assume that the equation has a unique global solution Xt on [t0, ?). Indeed, it is sufficient to assume coefficients f, hi and9ij satisfy the corresponding conditions in the infinite time horizon setting, see Peng (2007),Bai and Lin (2010).Then we would impose the monotone condition: There are two positive constants? and ? such that, for all (x,t) ? Rd x [t0, ?),and we can estimateTheorem 0.1. Under the monotone condition (0.0.46),the solution of equation (0.0.45)has the property· 2.1 The Iterated Logarithm Estimates for the Solution of G-SDEsIn the following, we consider a special case of equation (0.0.45), i.e.for 1 ?i,j?m,, with initial value Xt0=Xo? Rd, where ?i is the ith column of a given A?Rdxm.Theorem 0.2. Assume that there exists a pair of positive constants ?1,?2 such that for and the solution of equation (0.0.49) satisfies for some ? > 0 and for all t ? t0, ? > 0,Remark 0.1. In fact, Theorerm 0.2 gives a Law of the Iterated Logarithm (LIL) of the solution of (0.0.49) under the framework of nonlinear expectation. In the classical linear space,the LIL is one of the most important limit theorems,i.e.,However, Chen and Hu (2014) gives the LIL under the nonlinear expectations, which converges to an interval but not a fixed point as in classical situation:where v is the corresponding Ch.oquet capacities. Therefore Theorem 0.2 can be considered as a LIL of the solution of G-SDEs under nonlinear conditions.· 2.2 Asymptotic Estimates for the Solution of Multidimensional G-SDEsTheorem 0.3. Suppose there exist three positive constants ?, ?1 and ?2 such that for all and the solutiorn of equation (0.0.49) satisfies for some ? > 0 and for all t ?t0. Then it has the propertyHemark 0.2, Let us emphasis that the conclusion is independent of ?1,?2 and A. In,fact, (0.0.53) implies that when t is sufficiently large, for quasi-surely all ???,Therefore we see thatTheorem 0.4. Assume that there exist three positive constants ?,?1 and ?2 such that for all (x, t) ? Rd x [t0, ?),and the solution of equatio,n (0.0.49) satisfies for some ? > 0 and for all t ? t0, ? ? 0,Then it has the propertyRemark 0.3. In fact, the G-Novikov's conditions in Theorems 0.2-0.4 are satisfied when-ever f and gij have linear growth. In this case, the above results can be extended to equation (0.0.45) as long as the coefficient h(x,t) is bounded. More precisely, if there exists a C > 0 such that then Theorems 0.3-0.4 still hold for the solution of equation (0.0.45), and the correspond-ing in (0.0.51) and (0.0.57) should be replace by C.Chapter Three We study the properties of the solution of stochastic differential equations under the G-framwork, and obtain the boundedness and exponential stability of the solutions of first and second order stochastic differential equations driven by G-Brownian motion.· 3.1 Boundedness and Exponential Stability of First Order G-SDEsWe now study the properties of the solutions of G-SDEs. Let C(R+;R+) denote the family of continuous functions with non-negative domain and non-negative range.Let C1?2(Rd xR+;R+) denote the family of non-negative functions V(x,t) defined on in t. Now we consider the following d-dimensional SDE driven by m-dimensional G-Brownian motion with initial value Xt0=x0?Rd and t0?0.