The Application Of Backward Stochastic Differential Equations In Optimization Problems Of Economics And Finance

In this dissertation, we research the application of backward stochastic differential equations （BSDEs） in optimization problems of economics and finance. We know that the theory of BSDEs was initiated by the paper of Pardoux and Peng [62]. The main purpose to study BSDEs is to solve a stochastic control problem, and it is the stochastic maximizing principle. Then there are many researches in stochastic control theory relat-ed BSDEs, where the stochastic maximum principle are always the main technical tool in studying the control problem. Many of the researches are extended into the field of economics and finance, which leads that the theory of BSDEs are the interdisciplinary result and related optimization problems in fields of economics and finance can be solved by using BSDE theory.This dissertation fixes three hot problems covering the field of microeconomics, macroeconomics and behavioral finance separately. The first microeconomic problem is about a long-term principal-agent （PA） problem （or contracting problem） with sym-metric uncertainty of the project’s quality and agent’s hidden action assuming that the agent faces ambiguity about the innovation Brownian motion’s distribution, and then faces the distribution of output or cash flow. Then our first problem is that how the risk-neutral principal designs a contract in order to maximize his own utility subject to the risky aversion as well as ambiguity aversion agent’s incentive constraint. In histo-ry, Holmstrom and Milgrom [38] is the first to solve the hidden action problem, or we can say moral hazard problem in continuous time framework. Sannikov [70] gave the tractable way to solve the moral hazard problem assuming that the payment to the agent is at continuous rate. This assmption is commonly used later. Prat and Jovanovic [68] and He et al [37] separately studied the contracting problem with learning about the unknown agent’s quality appeared in the project. Prat and Jovanovic [68] focused on the non-stationary learning and He et al [37] studied stationary learning case otherwise. Miao and Rivera [61] studied a robust contracts case, but they focused on the ambiguity faced by principal instead of the agent, and they only considered the hidden action of the agent without adding the unknown quality in their model. To the contrary, we think it is agent that faces the ambiguity for his experience and knowledge about the environment. Moreover, we use Chen-Epstein [9] approach to describe this ambiguity. As a result, our model consider both the learning and ambiguity case, which can be more realistic in the complicated world.The key point and issue in this problem is how to get the necessary condition for in-centive compatibility （IC condition）. The IC condition is essentially derived from agent’s control problem. We can show that it is essentially to solve a more complex stochastic control problem of forward backward stochastic differential equations （FBSDEs） than the control problem of BSDEs. In detail, we need to use the maximum principle to solve this problem, which is introduced by Cvitanic and Zhang [15]. But FBSDEs induced by our contracting problem under ambiguity violates smooth condition:the drift term of backward equation is not continuous differentiable on state variable, where the varia-tions method used in the process of stochastic maximum principle needs the differential condition, since the first step is to differentiate the state variable. So our technique contribution is to tackle this difficulty using the generalized derivatives in non-smooth analysis, and then get the first-order necessary condition of the agent’s problem. The condition will be presented in two equations involving upper bound and lower bound of a set of information rents separately instead of the single equation as the classical result of control problem of FBSDEs. The set is derived from the ambiguity where there is a set of worst-case beliefs. So our necessary condition presents in a robust form. More importantly, with these conditions as constraints in solving the optimizing problem of principal, we can use dynamic programming approach to get the PDE. Moreover, we solve the PDE in analytical form, which means that we get the specific form of the optimal contract. So we can conveniently analyze the contract in detail and get many important economic results.The results of the first problem is from the work of me visiting the department of economics in Boston University as a visiting scholar for one year. The co-author is Professor Jianjun Miao from Boston University and Professor Shaolin Ji from Shandong University:Dynamic Contracts with Learning under Ambiguity, with Shaolin Ji and Jianjun Miao, Boston University working paper.The second macroeconomic problem is about how to implement an optimal mone-tary policy under commitment with zero lower bound in a continuous-time framework. The zero lower bound restricted on nominal interest rate is commonly used to solve the liquidity trap problem, which is the realistic case that the U.S faces since the crisis of 2008. The monetary policy of U.S cut the interest rate rapidly to zero point to mitigate the recession and the policy remained till 2015. As a result, the nominal interest rate can not be lower any more, and the monetary policy implemented by federal reserve becomes invalid.We use the New Keynesian model to study the optimal monetary policy in the s-tochastic and continuous-time framework. In history, the basic New Keynesian model follows the formulation of Eggertsson and Woodford [19], Woodford [78] and Gali [26], which are discrete case models. Clarida, Gali,and Gerlter [11] studied the optimal mon-etary policy from a New Keynesian Perspective, where they studied the policy in the discrete case without zero lower bound. Adam and M. Billi [1] studied the optimal mon-etary policy under commitment with a zero lower bound on the nominal interest rate in a discrete case. Ivan Werning [75] studied the optimal monetary policy in a liquidity trap with zero lower bound in the continuous setting, but he formulated a deterministic model.Then our economic model can be transformed into a control problem of infinite-time horizon BSDEs. Shi and Peng [66] studied infinite time horizon FBSDEs and Haadem and Oksendal [30] studied the maximum principles in infinite time horizon, but their conditions to get the solution are too strict that can not be used in our model. So our technical contribution in solving this problem is that:（1） we research the existence of the solution of an infinite time horizon BSDE model appeared in our optimal monetary policy model. The key point is how to set the transversality condition to make the available control set nonempty, otherwise the control problem will be ill-posed; （2） We get the necessary conditions, i.e. the corresponding adjoint equations and Hamiltonian of the optimal solution in this control problem in infinite time horizon; （3） We also give a limiting condition used to get the sufficient condition in solving the control problem of BSDEs over infinite time horizon. Then we analyze the condition of the optimal solution, and give some economic implications about the optimal monetary policy.This work is mainly from my working paper with Professor Shaolin Ji from Shandong University:Proposal on optimal monetary policy under commitment with zero lower bound in continuous setting, with Shaolin Ji.The third behavioral finance problem is about solving an optimal portfolio selection problem under g-expectation for an investor with his utility function satisfying the Inada condition. Our model is based on Jin and zhou [46], but we use consider g-expectation, which can be used to describe the ambiguity case like Chapter 1, instead of the nonlinear belief distortion. So in the setup of the model, we replace the Choquet expectation involved in Jin and zhou [46] with the g-expectation introduced and initiated by Peng [64], and we use different S-shaped utility functions and g-functions to represent the investor’s different uncertainty attitudes towards gains and losses.From the perspective of control theory, the behavioral finance problem is modeled with a control problem of BSDEs with terminal variable as control variables. To solve this control problem, we use the method called terminal perturbation method introduced by Ji and Peng [42], and Ji and Zhou [43,44]. However, using this method to get the necessary condition of this control problem brings us some tricky technical problems. Because the first-order derivative of utility function at zero point is infinity under the assumption of Inada condition, then the related results of integrability and convergence in the proof process of terminal perturbation method don’t exist. So the technical contribution in solving this problem is that we deal with these tricky points by using the proof by contradiction. We find a counter example by giving a specific variation that can get the proposition of the integrability and convergence. Also, we give the corresponding sufficient condition, which can be used to analyze the results from the perspective of finance.This work is mainly from my published paper:The optimal Portfolio Selection Model under g-Expectation, Abstract and Applied Analysis, Vol.2014, Article ID 426036.and the working paper with Professor Hanqing Jin from Oxford University and Professor Shaolin Ji from Shandong University:The optimal Portfolio Selection Model under g-Expectation and Utility Function with Inada Condition, with Hanqing Jin and Shaolin Ji.Above is the brief introduction of three optimization problems in economics and fi-nance focused on by my dissertation. My dissertation includes three chapters to illustrate these three problems correspondingly. In the following part, we will briefly show some of most important mathematical results for each problem and introduce the construction for each chapter:Chapter 1:Dynamic Contracts with Learning under AmbiguitySection 2 of Chapter 1 builds the model. The main technical contribution and breakthrough is that we give a necessary condition for a control problem of a FBSDE in section 3, the FBSDE is:To get the Lipschitz condition for this FBSDE, we use Girsanov transformation to this FBSDE and get:The agent’s problem is equivalent to the following control problem: where A ={a:[0, T]×Ω�'[0,1]} is a set of stochastic processes that are {FtY}-predictable.We can see that there is a term with absolute value so it is not differential on |Zta|. Then the necessary condition for this contract to be incentive compatibility is:Theorem 0.0.1. Under some technical assumptions in the proof, if the contract c=（a, ω, WT） is incentive compatible, then （α,γc） satisfies where （vc,γc）is the solution to BSDE with vT=U（WT） associated with the contract c, pt=maxPt∈Pcpt,pt=minpt∈Pcpt, and Qa,bc* is some worst-case measure for vc with density bc*∈Bc,i.e.The detailed meaning and explanation of the equations as well as the symbols will be given in Chapter 1. Here we can easily see the robust presentation from the Theorem above. We get a set Pc for fixed contract c which contains different information rents pt. This is because we have a set of worst-case beliefs Qa,bc* derived from the set of worst-case densities Bc and each pt is defined under different worst-case beliefs. So when there is no ambiguity as in Prat and Jovanovic [68], there is only one information rent since there is single probability belief. Also, we give the sufficient condition of IC in section 3:Theorem 0.0.2. A contract c=（a,w, WT） is incentive compatible if there exists pt ∈Pc s.t are true for t E [0,T], where ξ is the predictable process defined uniquely by where （Bta,bc*） is the standard Brownian motion under worst-case measure Qa,bc* gener-ated by the density generator bc* ∈Bc corresponding to pt.Then in section 4, the necessary condition becomes a constraint of the control prob-lem of the principal, where the principal intends to give the optimal contract that max-imize his own profit. Then we divide the process of solution into two steps. The first step is to solve the control problem of principal when assuming that the effort is not zero （there is no shirking effort） under the constraints of necessary condition. Such contracts with positive effort are called incentive contracts, meaning that principal provides incen-tive in order to motivate the agent not to relax. Then in the second step, we give the result of optimal contract. Note that solving the control problem in the first step with necessary condition as constraints actually enlarges the set of IC contracts, since we do not consider the sufficient condition. So after solving the optimal incentive contract, we need to confirm this contract is indeed incentive compatibility. We can show that solving the optimal incentive contracts is equivalent to solve the HJB equation: subject to andNext we also solve the PDEs above, and the solution constructs the content of the optimal incentive contract:Theorem 0.0.3. Assume that （ⅰ） quality η is unknown and the agent is ambiguity averse to the local mean of the output process, （ⅱ） u（w, a） and U（W） are specified in Chapter 1, and （ⅲ） any recommended effort level satisfies at>0 for all t≥0. Then the optimal incentive contract in the infinite-horizon limit as T�'∞ recommends the first-best effort level at*=1. The principal’s value function is given by where the function f（t） is given by and kt is the positive solution of the quadratic equation, The principal delivers the agent initial value v0 and the agent’s continuation value sat-isfies The wage is given byAt the end of section 4, we give the most important result in chapter 1:Theorem 0.0.4. Assume that （ⅰ） quality η is unknown and the agent is ambiguity averse to the local mean of the output process, （ⅱ） u（w, a） and U（W） are specified in Chapter 1. Let F=1-λ+lnK/ρ)/α+ 1/2ρα（λσK）2. （a） If F>0, then there exists a unique h>0. Suppose further that Then for ho< h, there exists a time τ>0 such that h（τ）= h and the optimal contract in the infinite-horizon limit recommends effort a* such that at*=0 for t∈[0,τ) and at*=1 for t≥τ. The principal offers the agent initial value v0 and the wage The agent’s continuation value satisfies vt= v0 for t ∈[0,τ) and follow the equation （24） for t≥τ with vτ=v0. The principal’s value function J* is given by For h0≥h, ht≥h for all positive time t and the optimal contract is just the optimal incentive contract as above theorem said.（b） If F≤0, the optimal contract is to set at*=0 for t≥0 and the optimal wage, the agent’s continuation value and the principal’s value function are given by for all t>0.How to get the value of h>0 will be given in the proof of the Theorem above. In section 5, we illustrate the optimal contract, analyze the effect of ambiguity on the contract and the difference between risk and ambiguity. In section 6, we compare the ambiguity approach used in our model with other approaches.Chapter 2:Proposal on optimal monetary policy under commitment with zero lower bound in continuous settingSection 2 builds the model. The control problem in this macroeconomic optimization problem is: subject to BSDEs over infinite-time horizon: where the feasible set of controls A（π,x,i）（?）{（πt,xt,it）0≤t<∞:（πt,xt,it）satisfies BSDEs （26）,it≥0,E（∫0∞e-ρt[λπt2+xt2]dt）<∞}.Our technical contribution and breakthroug in this problem is that we give the necessary and sufficient condition in section 3.First the corresponding Hamiltonian H: R6�'R is: where （pπ（·）,px（·））is the first-order adjoint process corresponding to one pair（π（t）,x（t）） and satisfies a SDE: Then the necessary condition is:Theorem 0.0.5.Let（π*（t）,x*（t）,i*（t））∈A（π,x,i）be the optimal triple of this control problem.（pπ（·）,px（·））is the the solution of SDE（27）corresponding to this optimal triple. Then ve have: Hi（t,π*（t）,x*（t）,i*（t）,px（t）,pπ（t））×（i-i*（t））≤0 （?）i≥0,t∈[0,00),P-a.s.（28） From eguation（28）,we can get:The sufficient condition is:Theorem 0.0.6.Let the triple （π*（t）,x*（t）,i*（t））∈0≤t<∞ satisfies the equation（26）and belongs to A（π,x,i）.Also we give the following adjoint equation: Suppose that The Hamiltonian then we have （π*（t）,x*（t）,i*（t））0≤t<∞ is the optimal solution of problem.Then we can see under some conditions, the equation （29） is the sufficient and nec-essary condition. So in the later part of section 3, we interpret the equation by providing the suggestion about the monetary policy. In section 4, we combine the monetary policy and fiscal policy and give a suggestion about the government spending.Chapter 3:The optimal portfolio selection model under g-expectation with utility function satisfying Inada conditionIn section 2 of Chapter 3, we give the detailed behavioral problem and build the corresponding mathematical model. We divide the terminal wealth into two part-s, the positive part and negative part, and then evaluate the terminal wealth sepa-rately. We use symbol u+（X+） and u_X- to represent two utilities for gains and losses, and use V+（X+） and V_X- to represent two value functions for gains and losses, which are defined by two g-expectations:V+（X+）=εg1[u+（X+）]=x1（0）, V)（X-）=εg2[u_X-]=x2（0） corresponding the following two BSDEs:The problem under g-expectation’s decision rule can be introduced as: Here G0,T[X] is the solution of BSDE with respect to function g0 and terminal variable X and actually it is a cost constraint.In section 3, we give three subproblems related to the original problem. For any given （x+,A）, the three subproblems are: Subproblem one: Subproblem two: We write the extreme values of subproblem （34） and subproblem （35） as:V+（x+,A）, V_x+,A. Subproblem three: Then we give the following theorem:Theorem 0.0.7. Let g0 be linear about （x,z）. Given X*, we define A*=（ω:X*≥0） and x+*=G0,T[（X*）+].Then X* is the optimal solution of problem （32） if and only if （x+*,A*） is the optimal solution of problem （36）, also （X*）+ and （X*）- are, respectively, the optimal solutions of problems （34） and （35） with respect to （x+*,A*）.Section 4 deals with each problem separately and give the optimal solutions. The main technical contribution in studying this problem is that we give the necessary con-dition for each subproblem under the assumption of u+’（0+）=u’₀₊=∞:Theorem 0.0.8. For any given （x+,A）, where x+≥x0, if X* is the optimal solution of problem （34） with these parameters, then X* has the following form: where A’is the subset of A. m（t）,n1（t） are respective solutions of the stochastic differ-ential equations: where h01<0 and h11>0,|h01|2+|h11|2= 1. Functions gx0, gz0, gx1, gz1 are respective derivatives of g0, g1 with respect to x or z, （x0*（t）,z0*（t）） and （x1*（t）,z1*（t）） are respective solutions of BSDEs （33） and （31） with terminal random variables X* and u+（X*）.Theorem 0.0.9. For any given （x+,A）, where x+≥x0,if X* is the optimal solution of problem （35） with these parameters, then X* has the following form: where h1<0,h02>0 and|h02|2+|h12|2=1.Ac is a subset of Ac. （x0*（t）,z0*（t）） and {x2*（t）,z2*（t）) are respective solutions of BSDEs （33） and （31） with terminal random variables X* and u_X*. m（t）,n2（t） are respectively the solutions of the stochastic differential equations:Then we further analyze the form of the optimal solution and give the sufficient condition:Theorem 0.0.10. For any given pair （x+, A）, let go be a convex function and g1 be a concave function. If there are two constants h01<0 and h11>0, and a random variable strictly positive on A:X*> 0 on A satisfying the constraint condition of problem（34） such that where n1（T）, m（T） are defined above. Then X* is the solution of the maximizition problem （34）.At the end of this section, we give the method of solving the original problem （32）. Secition 5 compares our model with Jin and Zhou [46] and explains the solution from perspective of finance. Section 6 and section 7 gives a specific example to solve under the results of section 4.