Portfolio Optimization With Nonsmooth Coefficients

Mathematical finance mainly consists of three parts:portfolio selection theory,asset pricing theory and risk measure theory.This thesis is devoted to the portfolio selection problem in continuous time financial markets.Expected utility maximization and mean-variance are by far the two predominant financial portfolio selection theories.Besides,it's interesting to consider the problem of maximizing the probability of reaching a wealth level by a fixed terminal time.This thesis will generalize these portfolio selection theories to some extent.Expected utility theory has been replaced by more general utility theory(recursive utility,multi prior utility)as it has been challenged by many economic phenomenon such as Allais paradox and Ellesberg paradox.First of all,this thesis studies recursive utility optimization from two aspects.On the one hand,we study recursive utility maximization under partial information in which the investor cannot observe directly the stock appreciation rates or the driving Brownian motion and he has only access to the past and present stock-prices.At the same time,the recursive utility equation need not be smooth.We adopt martingale method and turn the primal problem to a game problem for which we give a characterization of the saddle points via convex duality method.Then we obtain the optimal terminal wealth of the primal problem.On the other hand,we study recursive utility optimization with concave coefficients in which the wealth equation and the recursive utility equation need not be smooth.Thus our model can cover different interest rates for borrowing and.lending and the famous K-ignorance model.By martingale method and convex duality method,we give a characterization of its dual game problem and we obtain the optimal terminal wealth for the primal problem.Moreover,we give several examples in which the explicit solutions are available.In the second place,we generalize the mean-variance portfolio selection model from the classical linear wealth equation to a kind of wealth equation which is neither linear nor smooth.When the coefficients of the wealth equation are deterministic,we invoke the dynamic programming principle and give an explicit viscosity solution of the HJB equation.Via this explicit viscosity solution,we obtain explicitly the efficient portfolio strategy and efficient frontier for this problem.When the coefficieints of the wealth equation are stochastic processes,we adopt the completion of squares technique and two stochastic Riccati equations which are highly nonlinear BSDEs to give the optimal feedback portfolios.Furthermore,in order to compare with the sufficient condition of[53],we find the proper subdifferential of the wealth equation via convex duality method.Thirdly,we study the problem of maximizing the probability of reaching a wealth level by a fixeed terminal time under the predicament that the investor has ambiguity about the stock appreciation rates.The investor seeks a robust portfolio strategy to maximize the probability of the specified goal under the worst scenario,thus this is a game problem.But the cost functional is neither concave nor convex,even not continu-ous,and the "min-max" theorem can not applied directly.We will prove the "min-max"can be interchanged and give the saddle point explicitly.Limit theory is a very important component among the classical probability ax-iomatic system.As the emergence of Allais paradox and Ellcsbcrg paradox,expected utility theory is replaced by more general utility theory while the classical probability axiomatic system initiated by Kolmogorov suffers lots of challenges.This inspired the study of law of large numbers and central limit theorem with respect to non-additive probability.But scholars make little progress in the central limit theorem under non-additive probability.Recently,[30]gives a central limit theorem for Bernoulli random variables under belief measures.Although belief measure is the most special nonlinear probability in a scense,the results in[30]is a great progress.Thus,in the last chapter,we generalize the central limit theorem in[30]from Bernoulli random variables to gen-eral bounded random variables.It's worth pointing out that the belief measure on a two-sided interval is characterized by a bivariate normal distribution whose correlation coefficient is difficult to calculate.In the following,we will briefly introduce the structure and important results of each chapter.1.Recursive utility optimization under partial informationSuppose that the stock prices satisfy the follow stochastic differential equation:In this chapter,the investor cannot observe the appreciate rates ?'(?,...,?d)and the Brownian motion W in the above SDE,but only the stock prices S.Thus,his/her portfolio strategies must be based on the information of stock prices,that is ?(t)should be gt = ?(S(u),u? t)adapted.As he/she could not observe the Brwonian motion W,his/her recursive utility process could not be defined through W.In order to define utility process based on the filtration {gt}t?0,we introduce the innovation processwhere W is a,Brownian motion under P,and ?(W(s),s ?t)(?)gt.Thus we could define the recursive utility process as follows and the wealth equation can be written asThrough some filtering analysis,our problem in Chapter 1 iswhere X(t)>0 means no-bankruptcy is required.Notice that selecting ? is equivalent to selecting the terminal wealth X(T)by the existence and uniqueness result of BSDEs.By the comparison theorem of BSDEs,the nonnegative terminal wealth keeps the wealth process be nonnegative all the time.Thus our problem(0.0.28)turns to bewhere and "control variable" terminal wealth ? is selected fromThe maximal principle developed in[54,55]does not work here because f is a concave function which is not necessarily smooth enough.Then we turn problem(0.0.29)to a game problem,Maximizewhere function F is the convex duality of f,andWe try to find saddle-point(?,?,?)? × A(x)such thatFor any 0<?<?,we introduce the value functions andIn section 1.3,we get the following results.Lemma 0.1.Suppose(H1.1),(H1.3),(H1.4)hold.For any ?>0,there exists(?,?)?(??,??)? B which attains the infimum of(0.0.33).Lemma 0.2.Suppose(H1.1),(H1.3),(H1.4)hold andThen for any x>0,there exists ? = ?x ?(0,00)which attains the infimum of(0.0.34).Our main result in this chapter is the following theorem.Theorem 0.1.Suppose(H1.1),(H1.3),(H1.4)hold.Let(?,?)is the extre,me point of(0.0.33),? is the extreme point of(0.0.34)?then(?,?,?)is the saddle point of(0.0.31),where ? =In Section 1.4,we give several examples under the case d = 1 and f=K|Z|.Example 0.1.When u(x)= 1-e-?x,x?R ?>0,we have and the optimal terminal wealth of Problem(0.0.29)isExample 0.2 When u(x)=1nx,x>0,we havewhere(z(t),z(t))is the unique solution of the following BSDE,ExampleExample2.Recursive utility optimization under concave coefficientsThe wealth equation in Chapter 1 is linear,but in many important financial markets,the wealth equation is no longer linear,such as the case of different interest rates for borrowing and lending.So in this chapter,we focus on recursive utility optimization problem when the coefficients of wealth equation and utility equation are both concave functions.In the beginning of Section 2.2,we give several wealth equations with nonlinear and nonsmooth coefficients.Regarding the terminal wealth as the "control variable",our problem is,whereandThe maximal principle does not work here because b and f are concave functions which are not necessarily smooth enough.Then we turn problem(0.0.35)into a game problem,MaximizeWe try to find saddle-point(?,?,?)E B × A(x)such thatFor any 0<?<?,we introduce value functionsandIn Section 2.3,we get the following results,Lemma 0.3.Suppose(H2.1),(H2.2),(H2.3)and(H2.4)hold.Then for any ?>0,there exists(?,?)=(??,??)? B and(?,?)=(??,??)? B' which attains the infimum of(0.0.40).Lemma 0.4.Suppose(H2.1),(H2.2),(H2.3)and(H2.4)hold.Then for any x>0,there exists ?x?(0,?)which,attains the infirnum of(0.0.41).Our main result of this chapter is the following theorem.Theorem 0.2.Suppose(H2.1),(H2.2),(H2.3),(H2.4)and(H2.5)hold.Let(?,?,?,?)be the extreme point of(0.0.40),and ? the extreme point of(0.0.17).SetThen we have(?)?(x),(?)(?,?)?B,,That is(?,?,?)is a saddle point of(0.0.14).In Section 2.4,we set f= K'|Z|,u(x)=1/?x?,x>0,0<?<1 and study the case of linear wealth equation,the case of different interest rates for borrowing and lending and the price pressure model.Especially,for the last model,we give the optimal portfolio process and the utility intensity process when the coefficients are deterministic functions.3.Mean-variance portfolio selection with a kind of nonlinear wealth equationSuppose that the investor's wealth equation isFor a given expectation level,consider the following continuous time mean-variance portfolio selection problem:In Section 3.2,we study Problem(0.0.42)when d ? 1 and all the coefficients r,?,?,? are deterministic functions.After the dynamic formulation of Problem(0.0.42),the value function v(t,x;d)(d is the Lagrange multiple)satisfy the following HJB equa-tion,We construct a viscosity solution of(0.0.44)which coincide value function v(t,x;d).Theorem 0.3.Suppose(H3.1)hold,then is a viscosity solution of(0.0.20),and the optimal feedback control of(3.2.6)isThen we get the main result of this section.Theorem 0.4.The efficient strategy of Problem(0.0.43)can be written as a function of time t and wealth X:In Section 3.2.3,we give several examples where wealth equation(0.0.42)emerges.In Section 3.3,we study Problem(0.0.42)when d>1 and the coefficients are stochastic processes.As for the feasibility of Problem(0.0.42),we have the following result.Theorem 0.5.For any(?),Problem(0.0.43)is feasible if and only ifThe HJB equation method in Section 3.2 does not work because the coefficients in(0.0.42)are stochastic.And the terminal perturbation method in[53]does not work either because(0.0.42)is not smooth enough.We will adopt the completing square technique which needs two generalized stochastic Riccati equations.whereFor BSDEs(0.0.45)and(0.0.46),we have the following results.Theorem 0.6.There exists unique solution(Pl,Al)(correspondingly(P2,A2))of BSDE(0.0.45)(correspondingly(0.0.46)).Then we haveTheorem 0.7.The feedback control is the optimal control of Problem(3.3.6).And the minimum cost of Problem(3.3.6)isThe main result of this section isTheorem 0.8.The efficient strategy of Proble,m(0.0.19)can be written as a function of time t and wealth X:In Section 3.3.3,we find the subdifferential claimed in Corollary 4.4 in[53]using convex duality method when d = 1.where(Y,Z)is the unique solution of the following BSDE,and4.Reaching goals under uncertaintyIn this chapter,we consider the problem of reaching goals for an investor under ambiguity.The investor seeks a robust portfolio strategy to maximize the probability of a specified goal under the worst scenario.To be specific,we study the following game problem:The cost functional in our problem is of type ?{x?1} which is not concave or convex,even not continuous.We prove that the min-max theorem is still applicable for our problem.Furthermore,we also construct the optimal portfolio strategy and obtain the saddle point for our problem explicitly.SetwhereThrough the following two theorems,Theorem 0.9.(?)??(?),we haveTheorem 0.10.We have x*(T)?x*(T)|?=?*,and the equality holds if a,nd only ifwe get the main result of this chapter.Theorem 0.11.(?*,?*)is a saddle point of(0.0.47),that is to say,5.Central limit theorems under belief measuresIn this chapter,we generalize the central limit theorems for belief measures in[30]from Bernoulli random variable to general bounded random variable.In Section 5.2,we give several preliminaries.And we study unidirectional central limit theorem and obtain the following result.Theorem 0.12.For random variables(?)defined in(H5.1),we have andwhere ?,?,?,? is defined in Lemma 5.1.The above result shows that the belief measure for random variable falls into unidi-rectional intervals will converge to a random normal which looks like the classical central limit theorem.Note that belief functions are NOT additive in general,so central limit theorem for unidirectional intervals can not character the whole property(even on the two-sided intervals)of belief measure.As for the case of two sided intervals,we have the following theorem.Theorem 0.13.For random variables(?)defined in(H5.1),there exists a constant K which independent of ?1,?2 or n such thatwhere p is defined in Lemma 5.2.Moreover,the similar result holds if ?1 and ?2 depend on n.The belief measures of the events where the partial sums of(?)is in a two-sided intervals should be approximated by a bivariate normal distribution.This shows the essential difference with the classical central limit theorem.