Font Size: a A A

Nonlinear Expectation, Optimal Stopping Rule Under Ambiguity And Applications In Finance

Posted on:2011-12-30Degree:DoctorType:Dissertation
Country:ChinaCandidate:G Q ZhaoFull Text:PDF
GTID:1119360302499817Subject:Financial mathematics and financial engineering
Abstract/Summary:PDF Full Text Request
Motivated by the theory of expected utility, Peng (1997) introduced the notions of g-expectation and conditional g-expectation via a nonlinear backward stochastic differential equation (BSDE). Recently, the interest in g-expectation has remarkably widened, and it has important significance in both nonlinear mathematical theory and many practical applications.There is a substantive literature on the inequalities of g-expectation or g-martingale, see, for example, downcrossing inequality (see Chen and Peng (2000)), Jensen's inequal-ity (see Chen et al (2003), Jiang (2005)), maximal inequality (see Wang (2009)). It is well known that Lenglart domination inequalities play an important role in the semi-martingale theory and stochastic calculus. In this first chapter, we mainly investigate, under what conditions, Lenglart domination inequalities for g-expectations hold? The original motivation for considering Lenglart domination inequalities for g-expectations comes from the proofs to obtain asymptotic properties in the framework of g-martingale, as well as applications in risk management. In the chapter 1, we shall give an affirmative answer to this question. In the setting of g-martingale (g-expectation), our Lenglart domination inequalities provide a powerful tool to study the asymptotic properties.The optimal stopping time problem is to find the "best" stopping time, or decision, to maximize an expected reward of the gain (or to minimize an expected loss of the loss). As such, the optimal stopping time problem is pervasive in many areas. There is a substantive literature on the theory of classical optimal stopping, see for example Karatzas and Shrcve (1998),Φksendal (1998), Peskir and Shiryaev (2006).However, owing to ambiguity in markets, some kind risk-based models in some lit-erature Chen and Epstein (2002), Gilboa and Schmeidler (1989) have well documented empirical failures in markets. By the Ellsberg Paradox, it is well-known that the im- portance of ambiguity can't be negligible, and it would be at least as prominent as risk in making investment decisions.Inspired by Riedel (2009) in which a multiprior preference is considered in discrete time, which published in one of international top journals:Econometrica. The purpose of chapter 2 is to develop a counterpart theory of the optimal stopping in continuous time with ambiguity. The technical setting for the theory in continuous time is a g-expectation to represent the ambiguity in a general context. Our purpose is to do so in a context sufficiently general for applications and to encompasses as particular several existing continuous time model under ambiguity. In the continuous time framework, we need to overcome more difficult technical problems, and the framework also has a broader applications. The continuous time theory, presented in chapter 2, offers richer insights and provides more powerful while accessible tools to deal with decision maker's optimal stoping rule problem.As is well known we can solve an optimal stopping time problem, such as American type option, by transforming to solve a PDE free boundary problem in a Markovian setting. In this setting, we characterize the value function and the optimal stopping rule under ambiguty by using viscosity solution of a partial differential equations (PDE) in chapter 3. We obtain a new class of free boundary problem, which is related to our optimal stopping problem under ambiguity. This result can be viewed as a generaliza-tion of the recursive procedure in discrete time (See Riedal (2009)). In some particular cases, we derive the analytical solutions in the infinite time horizon framework.Royer (2006) introduced a new g-expectation via a nonlinear BSDE with the under-lying filtration generated by both a Brownian motion and a Poisson random measure. In chapter 4, motivated by Briand et al (2000), Jiang (2005a,2009), we studied the properties of the new style of g-expectation and generator g, we obtain a new kind of representation theorem and converse theorem; Furthermore, we give the characteriza-tions of the probability measures dominated by g-expectation.In the following, we list the main result of this thesis.Chapter 1:We mainly introduce the Lenglart domination inequalities for g-expectations both with the sub-additive condition and without this condition.Theorem 1.2.4 (Lenglart domination inequality under sub-additive condi-tion) Under hypotheses(H2), (H3) and (H4), a cadlag adapted process Y is L-dominated by an increasing process K under g-expectation, for (?)ε,δ>0 and (?)∈S, then we haveTheorem 1.3.1 (Lenglart domination inequality without sub-additive condi-tion) Suppose a cadlag adapted process Y is L-dominated by an increasing process K under g-expectation, where g only satisfies (H2)-(H3), for (?)∈,δ>0, (?)∈S, thenChapter 2: In this chapter we develop a theory of optimal stopping time problem in continuous time in the presence of ambiguity.There are three major contributions in this chapter. First, we propose a fairly gen-eral framework. We also present characterization of the value process and the optimal stopping rule for the optimal stopping time problem.The value process {Vt}0≤t≤T is characterized by the following theorem: Theorem 2.2.17 (Snell envelope) {Vt}0≤t<T is the Snell envelope under g-expectation of the process {Xt}0≤tφ(t,x)} with regular boundary, points such that (φ,(?)) is the unique viscosity solution to the following free boundary, problem: where A:= b(x)(?)x+(σ(x)σT(x))/2(?)x2;(?) is the boundary, of (?). Then and the first exit time (?) from (?) is an optimal stopping time.And PDE free boundary problem which solve the problem of optimal stopping with random time horizon:Theorem 3.2.5 (Elliptic free boundary) Suppose we could find a continuous func-tionφ=φ(x, c)∈Rp×C and an open set (?)∈Rp×Rl such that (φ, (?)) is the bounded continuous viscosity solution to the following free boundary, problem: where g satisfies the same conditions in Theorem 2.2.23. (?) is the first exit time from (?) with P((?)<∞)=1.Then and (?) is an optimal stopping time.As applications of the theorems,we list two cases here:Proposition 3.3.1(Investment timing under ambiguity)Given the value function where ct=(?)St, t>0; (?)<β-b-κσrepresents the proportion of consumption against the asset. Assume b<β+κσandκ2≤2β. Then the value function of the investment strategy is where x0=(σ2I(1-π1)π2)/(σ2(1-π1)(π2-1)-2(?)). Moreover, the optimal stopping time (?)* is the first hitting time of the project value S that hits the threshold x0, whereπ1,π2 are given byProposition 3.3.2(Optimal enter/exit under ambiguity)We know the value function Thus the value function of the optimal enter/exit strategy is where x0 is given by x0=λIπ1/π1-1. The optimal exit time is the first hitting time when {St}t≥0 hits x0. The parametersπ1,π2 areChapter 4: We obtain a new kind of representation theorem for new style of gen-erator g, and characterizations of the probability measures dominated by g-expectation. We claim the following representation theorem:Theorem 4.2.1 For each (t, x, y, q)∈[0, T[;x Rn×R×Rn, let the assumptions (R2)-(R5) hold for the generator g, 1≤p≤2, then we haveTheorem 4.2.4 Suppose the generator g is independent of y and satisfies (R1), (R3)-(R5), thus it follows that S1=S2, where and (?):={(αt,βt)t∈[0,T]:αtz+∫Bβt((?))ut((?))λ(d(?))≤g(t,z,u)}.
Keywords/Search Tags:Backward stochastic differential equation, Lenglart domination inequality, g-expectation, Optimal stopping, Ambiguity, Ambiguity aversion, Viscosity solution, Free boundary problem, Smooth-fit condition, Partial differential equation
PDF Full Text Request
Related items