Dingqian SUN

Abstract The author studies the optimal investment stopping problem in both continuous and discrete cases,where the investor needs to choose the optimal trading strategy and optimal stopping time concurrently to maximize the expected utility of terminal wealth.Based on the work of Hu et al.(2018)with an additional stochastic payofffunction,the author characterizes the value function for the continuous problem via the theory of quadratic reflected backward stochastic differential equations(BSDEs for short)with unbounded terminal condition.In regard to the discrete problem,she gets the discretization form composed of piecewise quadratic BSDEs recursively under Markovian framework and the assumption of bounded obstacle,and provides some useful a priori estimates about the solutions with the help of an auxiliary forward-backward SDE system and Malliavin calculus.Finally,she obtains the uniform convergence and relevant rate from discretely to continuously quadratic reflected BSDE,which arise from corresponding optimal investment stopping problem through above characterization.

Keywords Optimal investment stopping problem,Utility maximization,Quadratic reflected BSDE,Discretely reflected BSDE,Convergence rate

1 Introduction

In this paper,we consider a small trader in an incomplete financial market who can invest in risky stocks and a riskless asset and is also granted the right to stop the whole investment during the finite trading time interval[0,T]to obtain corresponding payoff.The objective of the investor is to maximize her/his exponential utility of terminal wealth,which includes both the profit or loss on investment and the final payoff,by choosing the optimal trading strategy and optimal stopping time simultaneously.For the continuous case,the investor is allowed to stop the investment,which is like exercising an American option,at any time beforeT.While for the discrete case,the investor will be restricted to given discrete exercise time,where the payoffcan be regarded as a kind of Bermudan option.

Such utility maximization problem of mixed optimal stopping/control type was initially studied in[11],which involved both consumption and final wealth under continuous framework and was reduced to a family of related pure optimal stopping problems via duality theory.Similar problems also arise in situations like pricing constrained American contingent claims,see[12]for example,where the closed-form of hedging price of an American-type barrier option under the short-selling constraint was obtained through the solution to a variational inequality.While different from the methods applied in these results,we will proceed by means of the connection between the original utility maximization problem(with a prespecified terminal time)and the theory of quadratic BSDEs,which will be introduced in more detail hereinafter,and pay more attention to the convergence from discrete to continuous problem.

With respect to the continuous problem,if we only consider the optimal strategy on time interval[0,τ]with fixedτ∈[0,T],it will then become the usual exponential utility maximization problem which was widely discussed before,see[9-10,17-18].To be specific,when the terminal payoffatτis bounded,the problem was completely solved in[9]with the help of quadratic BSDE with bounded terminal data.It turns out that the value function of such problem can be characterized by the solution to a particular BSDE,whose generator is of quadratic growth inz-variable.Related theory to quadratic BSDEs can be traced back to[13]with bounded terminal value,where the existence and uniqueness of solutions were established.Then it was extended to unbounded case to obtain the existence in[3],and subsequently the uniqueness with convex generators in[4,6-7].Recently,Hu et al.[10]generalized the previous work,the exponential utility maximization problem with bounded payoff,to the unbounded framework on the basis of above development and studied utility indifference valuation of derivatives with unbounded payoffs as application.

Inspired by the above connection,we adjust the order of optimization and decompose our problem with extra payofffunction into original utility maximization framework,which then reduces to a pure optimal stopping problem,and further obtain the value function in terms of the solution to a quadratic reflected BSDE,where the generator has almost the same form as in utility maximization problem in[10].While the existence and uniqueness of solution to such quadratic reflected BSDEs have been developed,see[14]for bounded terminal value and obstacle and[1,15]for unbounded cases,the main difficulty left is to represent the solution to reflected BSDE via the supremum of solutions to a collection of BSDEs,which have the same quadratic generator as the former,i.e.,in Subsection 2.3.Since here the group of BSDEs has different time horizon[0,τ]and terminal valuegτand thus corresponding different pairs of solutions(Y(τ),Z(τ)),we can not directly apply the optimal stopping representation of reflected BSDEs(see[8,Proposition 2.3]for reference),but need to further use the comparison theorem and uniqueness of quadratic BSDEs to prove such characterization,see Theorem 2.2 for more details.

Regarding the discrete problem,we need to restructure the framework and proceed under Markovian system for the sake of following convergence analysis between the two forms.We first give a practical example to illustrate how we get the Markovian structure arising from previous continuous problem.While due to the addition of stochastic factor,the generator we consider herein will be more complicated than that in previous section,i.e.,f(t,x,z)of quadratic growth inzand satisfying locally Lipschitz condition with respect to bothxandz,which is generalized in Assumption 3.1.Then when restricted the exercise time to some given discrete time points,we can deduce recursively from the comparison result of BSDEs to get the backward discretization form,which is composed of piecewise BSDEs and actually a so-called discretely reflected BSDE,see Subsection 3.2 for the form and related properties.

The main result of this paper is the convergence analysis and relevant rate from discrete to continuous optimal investment stopping problem.Thanks to the previous discussion and characterization,we can now transform the problem into the convergence from discretely to continuously reflected BSDE,which has been studied when the generator is uniformly Lipschitz in all the variables,see[16]based on the Euler scheme of forward SDE and[5,Section 3].Whereas originating from the utility maximization problem,we are facing reflected BSDEs with generator of quadratic growth,which brings us new difficulties during obtaining necessary estimates and thus we have to restrict ourselves to the case of bounded and Lipschitz obstacle,and also the deterministic diffusion term in forward SDE at this stage.

Firstly,with the help of the properties of quadratic BSDEs and reflected BSDEs with bounded terminal value,we can inductively prove the boundness ofin discretization form and the relation,which makes it possible to implement the usual techniques using to deal with BSDEs of quadratic growth.

Moreover,in order to handle the additional term coming from reflection,we need further properties ofappearing in piecewise BSDEs of the discretization form.We establish the connection between our discretization form and an auxiliary forward-backward SDE system defined on each time interval[ti−1,ti]with different terminal functions.Recall the existing results in Markovian FBSDE system that the solutionZto quadratic BSDE with bounded and Lipschitz terminalg(XT)is controlled byC(Kg+ 1)(see[19]),whereKgis the Lipschitz constant ofg.And then in[20],the a priori estimate ofZis generalized to the superquadratic case with unbounded terminal condition and also the case with random diffusion term in forward SDE and bounded terminal condition.While unfortunately,neither of them can cover the situation in our assumptions since here the locally Lipschitz coefficient ofxinvolvesz.However,motivated by the proof of these results,we can make use of the BMO property ofZand the representation derived from Malliavin calculus to fill this gap and get the explicit bound ofZ.Together with the uniform Lipschitz continuity of terminal functionsin auxiliary forward-backward SDE system,we can obtain the boundness ofin discretization form at last.

Finally,we give the complete proof of the uniform convergence from discretely to continuously quadratic reflected BSDE and obtain the convergence rate as follows when the obstaclegis Lipschitz:

and

Furthermore,we can achieve double rate whengis in,which is actually the same rate as that with Lipschitz generator in[16].

The paper is organized as follows.We discuss the continuous optimal investment stopping problem in Section 2 and give the characterization of value function in terms of the solution to quadratic reflected BSDE.In Section 3,we focus on Markovian framework and put forward the assumptions based on a practical example,and further obtain the discretization form for corresponding discrete problem.Then in Section 4,after providing some auxiliary results regarding the discretization form with the aid of a forward-backward SDE system,we finally obtain the convergence result of the two forms,and then we conclude the paper in Section 5.

2 Continuous Optimal Investment Stopping Problem

We fix a finite time horizon[0,T]withT >0.LetBbe anm-dimensional standard Brownian motion defined on a complete probability spacebe the augmented natural filtration ofBwhich satisfies the usual conditions.

2.1 Formulation

Consider a financial market consisting of one risk-free bond with interest rate zero andd≤mstocks.In the cased < m,we face an incomplete market.The price process of theith stock is described as

2.2 Results on quadratic reflected BSDEs with unbounded obstacle

and is concave inz,i.e.,it satisfies[1,Assumptions(H1)and(H3)].Consequently,we can get the existence and uniqueness directly from Theorems 3.2 and 4.1 therein.

2.3 Characterization of value function

Now we can characterize the value function of the optimal problem via the solution to the above reflected BSDE.

Remark 2.1We need to note here that for convenience,what we discussed in this paper is quadratic reflected BSDE with lower obstacle,whose solution we have proved in the above theorem can be characterized by the supremum of the solutions to a collection of BSDEs with the same generator.Therefore,we require consistency of the supremum whether it is taking inside or outside the exponential in the expression of value function(2.8).To this end,when quoting the result in[10],we have to change the sign ofYappearing in the value function as(2.7)and then the corresponding generator of BSDE.Actually,denoting the generator there asF,one can readily check that they satisfyf(t,z)=−F(t,−z)and that is why we are considering the concave generator in this section.

Remark 2.2In order to simplify the notation,we consider the optimal investment stopping problem starting fromt=0 and state that we can also characterize the optimal stopping time and trading strategy via the solution to reflected BSDE.Firstly,we could directly know from above proof that the optimal stopping time is

3 Discrete Optimal Investment Stopping Problem

From this section,we will concentrate on Markovian framework,that is,the following decoupled forward-backward SDE with reflection:

3.1 A special case as connection

We will see from a special case with the subspace portfolio constraint in this subsection that how we can get the above Markovian structure from the previous general problem.Here for simplicity,we consider a market with a single stock whose coefficients depend on a single stochastic factor driven by a 2-dim Brownian motion,that is,m=2,d=1 and

Combined with(3.2),they constitute a Markovian system as(3.1)and one can easily check thatsatisfies Assumption 3.1.

In order to avoid confusion about the notations,we will still usebandσto denote the coefficients of forward SDE and(X,Y,Z,K)the solution to forward-backward SDE with reflection in the following discussion,and consider the discrete problem and subsequent convergence under the generalized Assumption 3.1.

3.2 Discretization form

Thus as with the characterization of value function,we only need to concentrate on the discretely and continuously quadratic reflected BSDEs hereafter.

4 Convergence Analysis

As we stated before,in consideration of the connections we have built respectively for the continuous and discrete optimal investment stopping problems in previous sections,we may now lay emphasis on the convergence from discretely to continuously quadratic reflected BSDE.

4.1 Auxiliary results

and give a crucial estimate ofZin next lemma.

and define

4.2 Main result

where the last inequality comes from H¨older’s inequality.

We now chooseψwith the following form:

Noting(4.11)(a),we further have

Plugging it back to(4.14)and making use of Lemma 4.5,we have

For the second term in the above inequality,applying B-D-G inequality and using(4.11)(a)and Young’s inequality,we obtain

Applying B-D-G inequality and moment inequality,together with the results proved in the former two steps and the estimate(4.9),we then deduce that

5 Conclusions

We have characterized the continuous and discrete optimal investment stopping problems separately and provided the convergence result,which comes down to the convergence from discretely to continuously quadratic reflected BSDE,via the tools of quadratic BSDE with bounded terminals.While at present,we need the bounded assumption due to technical restriction when we try to apply the method in Lipschitz case to Quadratic,and what we discussed here is actually a specific form of quadratic generator withoutyinvolved,since the BSDE essentially originates from the utility maximization problem.

Also notice that we still need to solve a non-discretized BSDE on each time interval[ti−1,ti)in(3.5),which is not ideal enough for practical application.Thus we may further consider the real discrete scheme for the quadratic reflected BSDE as in Lipschitz case(see[2,16]),which will indicate the way to solve the optimal investment stopping problem numerically,as well as generalize the settings about generator and terminal value in our future research.

Appendix Proof of Lemma 4.2

As in the literature,we suppose that the functionsb,gandfin the forward-backward SDE(4.2)are differentiable with respect toxandzfirstly.Thus the solution(X,Y,Z)is differentiable with respect toxand(∇X,∇Y,∇Z)satisfies the following SDE and BSDE:

We conclude the proof by noting that whenb,gandfare not differentiable,one can also prove the result by a standard approximation and the stability result for BSDEs.

AcknowledgementsThe author would like to thank her Ph.D.supervisors,Prof.Shanjian Tang and Dr.Gechun Liang for their guidance and help during the research and revision of this paper,and also the anonymous referees and the editor for their careful reading and helpful comments.