Optimal investment with stopping in finite horizon

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1 Jian et al. Journal of Inequalities and Applications 04, 04:43 R E S E A R C H Open Access Optimal investment with stopping in finite horizon Xiongfei Jian,XunLi and Fahuai Yi * * Correspondence: fhyi@scnu.edu.cn School of Mathematical Sciences, South China Normal University, Guangzhou, China Full list of author information is available at the end of the article Abstract In this paper, we investigate dynamic optimization problems featuring both stochastic control and optimal stopping in a finite time horizon. The paper aims to develop new methodologies, which are significantly different from those of mixed dynamic optimal control and stopping problems in the existing literature. We formulate our model to a free boundary problem of a fully nonlinear equation. Furthermore, by means of a dual transformation for the above problem, we convert the above problem to a new free boundary problem of a linear equation. Finally, we apply the theoretical results to some challenging, yet practically relevant and important, risk-sensitive problems in wealth management to obtain the properties of the optimal strategy and the right time to achieve a certain level over a finite time investment horizon. MSC: 35R35; 9B8; 93E0 Keywords: optimal investment; optimal stopping; dual transformation; free boundary Introduction Optimal stopping problems, a variant of optimization problems allowing investors freely to stop before or at the maturity in order to maximize their profits, have been implemented in practice and given rise to investigation in academic areas such as science, engineering, economics and, particularly, finance. For instance, pricing American-style derivatives is a conventional optimal stopping time problem where the stopping time is adapted to the information generated over time. The underlying dynamic system is usually described by stochastic differential equations SDEs. The research on optimal stopping, consequently, has mainly focused on the underlying dynamic system itself. In the field of financial investment, however, an investor frequently runs into investment decisions where investors stop investing in risky assets so as to maximize their expected utilities with respect to their wealth over a finite time investment horizon. These optimal stopping problems depend on the underlying dynamic systems as well as investors optimization decisions controls. This naturally results in a mixed optimal control and stopping problem, and Ceci and Bassan [] is one of the typical works along this line of research. In the general formulation of such models, the control is mixed, composed of a control and a stopping time. The theory hasalsobeenstudiedinbensoussanandlions[], Elliott and Kopp [3], Yong and Zhou [4] and Fleming and Soner [5], and applied in finance in Dayanik and Karatzas [6], Henderson and Hobson [7],Li and Zhou [8],Li and Wu [9, 0] and Shiryaev, Xu and Zhou []. 04 Jian et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Jian et al. Journal of Inequalities and Applications 04, 04:43 Page of 4 In the finance field, finding an optimal stopping time point has been extensively studied for pricing American-style options, which allow option holders to exercise the options before or at the maturity. Typical examples that are applicable include, but are not limited to,thosepresented in Chang,Pangand Yong [], Dayanik andkaratzas [6] and Rüschendorf and Urusov [3]. In the mathematical finance literature, choosing an optimal stopping time point is often related to a free boundary problem for a class of diffusions see Fleming and Soner [5] and Peskir and Shiryaev [4]. In many applied areas, especially in more extensive investment problems, however, one often encounters more general controlled diffusion processes. In real financial markets, the situation is even more complicated when investors expect to choose as little time as possible to stop portfolio selection over a given investment horizon so as to maximize their profits see Samuelson [5], Karatzas and Kou [6], Karatzas and Sudderth [7], Karatzas and Wang [8], Karatzas and Ocone [9], Ceci and Bassan [],Henderson [0],Li andzhou [8]andLiandWu[9, 0]. The initial motivation of this paper comes from our recent studies on choosing an optimal point at which an investor stops investing and/or sells all his risky assets see Choi, Koo and Kwak [] and Henderson and Hobson [7]. The objective is to find an optimization process and stopping time so as to meet certain investment criteria, such as, the maximum of an expected utility value before or at the maturity. This is a typical problem in the area of financial investment. However, there are fundamental difficulties in handling such optimization problems. Firstly, our investment problems, which are different from the classical American-style options, involve optimization process over the entire time horizon. Secondly, they involve the portfolio in the drift and volatility terms so that the problem of multi-dimensional financial assets are more realistic than those addressed in finance literature see Capenter []. Therefore, it is difficult to solve these problems either analytically or numerically using current methods developed in the framework of studying American-style options. In our model, the corresponding HJB equation of the problem is formulated into a variational inequality of a fully nonlinear equation. We make a dual transformation for the problem to obtain a new free boundary problem with a linear equation. Tackling this new free boundary problem, we establish the properties of the free boundary and optimal strategy for the original problem. The remainder of the paper is organized as follows. In Section,themathematicalformulation of the model is presented, and the corresponding HJB equation is posed. In Section 3, a dualtransformation convertsthefreeboundaryproblemofa fullynonlinear PDE to a new free boundary problem of a linear equation but with the complicated constraint 3.6. In Section 4 we simplify the constraint condition in 3.6 toobtainanewfree boundary problem with a simple condition 4.4. Moreover, we show that the solution of problem 4.5 must be the solution of problem 3.5. Section 5 is devoted to the study of the free boundary of problem 4.5. In Section 6, we go back to the original problem.6 to show that its free boundary is decreasing and differentiable. Moreover, we present its financial meanings. Section 7 concludes the paper. Model formulation. The manager s problem The manager operates in a complete, arbitrage-free, continuous-time financial market consisting of a riskless asset with instantaneous interest rate r and n risky assets. The

3 Jian et al. Journal of Inequalities and Applications 04, 04:43 Page 3 of 4 risky asset prices S i are governed by the stochastic differential equations ds i,t S i,t =r + μ i dt + σ i dw j t, fori =,,...,n,. where the interest rate r, the excess appreciation rates μ i, and the volatility vectors σ i are constants, W is a standard n-dimensional Brownian motion. In addition, the covariance matrix = σ σ is strongly nondegenerate. A trading strategy for the manager is an n-dimensional process π t whose ith component, where π i,t is the holding amount of the ith risky asset in the portfolio at time t. An admissible trading strategy π t must be progressively measurable with respect to {F t } such that X t 0. Note that X t = π 0,t + n i= π i,t,whereπ 0,t is the amount invested in the money market. The value of the wealth X t evolves according to dx t = rx t + μ π t dt + π t σ dw t.. In addition, short-selling is allowed. The manager controls assets with initial value x. The manager s dynamic problem is to choose an admissible trading strategy π t and a stopping time τ to maximize his expected utility of the exercise wealth: V x, t=max π,τ E[ e rτ t UX τ + K ],.3 where r >0istheinterestandK is a positive constant e.g.,afixedsalary, Ux= x, 0< <, is the utility function.. HJB equation Applying the dynamic programming principle, we get the following Hamilton-Jacobi- Bellman HJB equation: min{ t V max π [ π π xx V + μ π x V ] rx x V + rv, V x + K } =0, x >0,0<t < T, V 0, t= K.4, 0<t < T, V x, T= x + K, x >0. Suppose that V x is strictly increasing and strictly concave, i.e., x V >0, xx V <0.Note that the gradient of π π with respect to π π π π = π, then π = μ xv x, t xx V x, t..5

4 Jian et al. Journal of Inequalities and Applications 04, 04:43 Page 4 of 4 Thus.4becomes min{ t V + a xv xx V rx xv + rv, V x + K } =0, x >0,0<t < T, V 0, t= K, 0<t < T, V x, T= x + K, x >0,.6 where a = μ μ. Now we find a condition under which the free boundary exists. A simple calculation shows Ux + K= x + K, x Ux + K=x + K, xx Ux + K= x + K. It follows that t Ux + K+ a xux + K xx Ux + K = a 0. rx x Ux + K+rUx + K x + K rxx + K + r x + K Eliminating x + K yields a x + K rx + rx + K 0, i.e., a r + r x a + r K..7 If a r r,.8 then.7holdsforanyx > 0, the solution to problem.6isux + K. If a r 0,.9 then.7 is impossible for any x > 0. Therefore, in this case, the solution to problem.6 satisfies t V + a x V xx V rx xv + rv =0, x >0,0<t < T, V 0, t= K, 0<t < T,.0 x V +, t=0, 0<t < T, V x, T= x + K, x >0.

5 Jian et al. Journal of Inequalities and Applications 04, 04:43 Page 5 of 4 We summarize the above results in the following theorem. Theorem. In the following cases, problem.6 has a trivial solution. If.8 holds, the solution to problem.6 is Ux + K. If.9 holds, the solution to problem.0 is the solution to problem.6 as well. Recalling.8and.9, in the following we always assume that r < a r <0.. In the case of., there exists a free boundary. 3 Dual transformation Define a dual transformation of V x, tseepham[3] vy, t:=max V x, t xy, 0 y y0. 3. x>0 If x V, t is strictly decreasing, which is equivalent to the strict concavity of V, t we will show this fact in the end of Section 5, then the maximum in 3. willbeattainedat just one point x = Iy, t, 3. which is the unique solution of y = x V x, t. 3.3 Using the coordinate transformation 3.yields vy, t= [ V x, t x x V x, t ] x=iy,t = V Iy, t, t yiy, t. 3.4 Differentiating with respect to y and t,we get y vy, t= x V Iy, t, t y Iy, t y y Iy, t Iy, t= Iy, t, 3.5 yy vy, t= y Iy, t= xx V Iy, t, t, 3.6 t vy, t= t V Iy, t, t + x V Iy, t, t t Iy, t y t Iy, t= t V Iy, t, t. 3.7 Substituting 3.5 into3.4, we have V Iy, t, t = vy, t y y vy, t. 3.8 By the transformation 3.and , the HJB equation in.6becomes min { t v a y yy v + rv, v y y v } K yv =0, 0<y < y 0,0<t < T. 3.9

6 Jian et al. Journal of Inequalities and Applications 04, 04:43 Page 6 of 4 Now we derive the terminal condition for vy, T. Note that V x, T= x + K, 3.0 so x V x, T=x + K, i.e.,[ x V x, T] = x + K. It follows that y K = x = Iy, T= y vy, T, 3. and by 3.8, we have vy, T=V Iy, T, T + y y vy, T = y + y K y = y + Ky. 3. Next, we determine the upper bound y 0 for y.infact,v x, t= x + K in the neighborhood of x =0,sotheupperboundis y 0 = x V 0, t=k. 3.3 In addition, we need to determine the value vy 0, t. By 3.8, we also have vy 0, t=v 0, t+y 0 0= K. 3.4 Combining 3.9and3.-3.4, we obtain min{ t v a y yy v + rv, v y y v K yv } =0, 0<y < K,0<t < T, vk, t= K, 0<t < T, vy, T= y + Ky, 0<y < K. 3.5 In 3.5, the equation is a linear parabolic equation, but the constraint condition v y y v + K yv 3.6 is very complicated. In the following section, we simplify this condition. Remark The equation in 3.5 is degenerate on the boundary y = 0. According to Fichera s theorem see Oleĭnik and Radkević [4], we must not put the boundary condition on y =0. 4 Simplifying the complicated constraint condition Note that in the domain {x, t Vx, t= x + K },wehave x V x, t=x + K, ifv x, t= x + K. 4.

7 Jian et al. Journal of Inequalities and Applications 04, 04:43 Page 7 of 4 By the y coordinate, y =K y v, ifv y y v = K yv. 4. Deriving y v from the first equality in 4.yields y v = K y, 4.3 and then substituting 4.3 into3.6, we have v y + Ky. 4.4 This is the simplified constraint condition. We assume that uy, t satisfies min{ t u a y yy u + ru, u y Ky} =0, y, t Q y, uk, t= K, 0<t < T, uy, T= y + Ky, 0<y < K, 4.5 where Q y = 0, K 0, T. Moreover, we split the domain Q y into two parts; denote see Figure { ER y = uy, t= { CR y = uy, t> y + Ky } y + Ky, exerciseregion, 4.6 }, continuation region. 4.7 Theorem 4. The solution ux, t to problem 4.5 is the solution to problem 3.5 as well. In order to prove this theorem, we first show the following two lemmas. Figure CR y and ER y.

8 Jian et al. Journal of Inequalities and Applications 04, 04:43 Page 8 of 4 Lemma 4. For any y, t Q y, we have y u = K y, y, t ER y, 4.8 y u K y, y, t CR y. 4.9 Proof Equation 4.8 follows from the definition 4.6 directly. Also, in CR y t u a y yy u + ru =0, y, t CR y. 4.0 Differentiating 4.0toy yields Note that t y u a y yy y u a y y y u+r y u=0, y, t CR y. 4. y uy, T=K y, 0<y < K, 4. y uy, t=k y, y, t CR y Q y, 4.3 where CR y istheboundaryofcr y. Denote w = K y, we further show that w is a supersolution to problem by and y w = y = y, yy w = y, t w a y yy w a y y w + rw = a = rk + y a y + r K y a r y > 0 by the first inequality in.. So w is a supersolution of This means that 4.9holds. Lemma 4. The function y y u + K yu is increasing with respect to y uif y u K y.

9 Jian et al. Journal of Inequalities and Applications 04, 04:43 Page 9 of 4 Proof Define a function f z=yz + K z, z K y. Then f z=y K z 0 if z K y. Proof of Theorem 4. Note that, from 4.5, t u a y yy u + ru 0, y, t ER y, 4.4 u = Rewrite 4.5as y + Ky, y, t ER y. 4.5 u = y K y + [ ] K K y, y, t ERy. 4.6 Applying 4.8to4.6, we have u = y y u + K yu, y, t ER y. 4.7 On the other hand, from 4.5, in CR y t u a y yy u + ru =0, y, t CR y, 4.8 u We rewrite 4.9as y + Ky, y, t CR y. 4.9 u y K y + [ ] K K y, y, t CRy. 4.0 Applying 4.9 and Lemma 4.,weget u y y u + K yu, y, t CR y. 5 The free boundary of problem 4.5 Denote W, p,loc Q y= { uy, t:u, y u, yy u, t u L p Q, Q Q y }.

10 Jian et al. Journal of Inequalities and Applications 04, 04:43 Page 0 of 4 Theorem 5. Problem 4.5 has a unique solution u W, p,loc Q y Q y \{y =0}, and y + Ky uy, t e AT t y u y Ky t u y Ky y + Ky, 5. 0, 5. 0, 5.3 where A = a. Proof According to the existence and uniqueness of W, p,loc Q y Q y \{y =0}, the solution for system 4.5 can be proved by a standard penalty method see Friedman [5]. Here, we omit the details. The first inequality in 5. follows from 4.5 directly, and now we prove the second inequality in 5.. Denote Wy, t:=e AT t y + Ky, where A > 0 to be determined later on. We first show that Wy, t isasupersolutionto problem 4.5. In fact, if t W a y yy W + rw = Ae AT t e AT t A A = a. y + Ky a + e AT t [ a y =0 + r ] y + rky So, Wy, tisasupersolutiontoproblem4.5. Hence, the second inequality in 5.holds. In addition, inequality 5. follows from 4.8and4.9. In order to prove 5.3, we define wy, t=uy, t δ for small δ >0. From 4.5, we know that wx, tsatisfies min{ t w a y yy w + rw, w wk, t= K, δ < t < T, wy, T=uy, T δ y Ky} =0, y >0,δ < t < T, y + Ky, 0<y < K. 5.4

11 Jian et al. Journal of Inequalities and Applications 04, 04:43 Page of 4 Figure y = ht, ϕy= y + Ky. Applying the comparison principle to variational inequalities 4.5and5.4withrespect to terminal values see Friedman [6], we obtain uy, t wy, t=uy, t δ, y >0,δ < t < T. Thus t u 0and5.3holds. Based on 5., we define the free boundary { ht:=min y uy, t= } y + Ky, 0 t < T. Theorem 5. The free boundary function ht ismonotonicdecreasing Figure with ht:= lim ht= t T a rk r Moreover, ht C[0, T] C [0, T Proof First, from 5.3, ht is monotonic decreasing. Denote In ER y, so ϕy:= y + Ky. t ϕ a y yy ϕ + rϕ = a + r y + rky 0, ht a rk r, 0 t < T.

12 Jian et al. Journal of Inequalities and Applications 04, 04:43 Page of 4 Hence, ht a rk r. Inordertoprove5.5, we suppose ht> a rk r then it is not hard to get t uy, T>0, forht<y <, 5.6 a rk r, which is a contradiction to 5.3. Therefore, the desired result 5.5 holds. Finally, the proof of ht C[0, T] C [0, T is similar to the result in Friedman [5]. Here, we omit the details. Theorem 5.3 For any y, t Q y, we have yy uy, t> Proof If y, t ER y,thenu = y + Ky.Thus, yy u = y >0, y, t ER y. If y, t CR y,then t u a y yy u + ru =0, y, t CR y. 5.8 Differentiating 5.8withrespecttoy twice yields Note that t yy u a y yy yy u a y y yy u+ r a yy u=0, y, t CR y. 5.9 yy uy, t>0, t = T or y = ht. Applying the minimum principle, we obtain yy u = y >0, y, t CR y. Remark From 3.6, we have xx V <0,whichmeansV is strict concave to x.

13 Jian et al. Journal of Inequalities and Applications 04, 04:43 Page 3 of 4 Figure 3 x = gt. 6 The free boundary of original problem.6 Recalling on the free boundary y = ht uy, t= y + Ky, y = ht, 6. y uy, t= y + K, y = ht. 6. From the dual transformation 3.and3.5, we know x = y uy, t. 6.3 Denote the free boundary of.6byx = gt. Applying 6.and6.3yields gt= y u ht, t = ht K. 6.4 Moreover, g t= ht h t>0, 6.5 gt=ht K = a rk Thus, we have following theorem. r K by Theorem 6. The free boundary x = gt of problem.6 is monotonic increasing Figure 3 and gt is determined by 6.6. Moreover, gt C[0, T] C [0, T. Financial meanings At time t, the manager should continue to invest according to.5 if x > gt, while the investor should stop investment if x < gt. 7 Concluding remarks We explore a class of optimal investment problems mixed with optimal stopping in the financial investment. The corresponding HJB equation, a free boundary problem of a fully nonlinear equation, is posed. By means of a dual transformation, we obtain a new free boundary problem with a linear equation under a complicated constraint condition. The

14 Jian et al. Journal of Inequalities and Applications 04, 04:43 Page 4 of 4 key step is to simplify this complicated constraint condition. In this way we study the properties of the free boundary and optimal strategy for investors. Remark on constant K salary If K is a function of time t, K = Kt, the unique difficulty is the proof of 5.3. If Ktisdecreasing,then5.3 is still right and all results hold as well. In general case if Kt is not decreasing, then the free boundary may be not monotonic. We will consider this problem in the future. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final version. Author details School of Mathematical Sciences, South China Normal University, Guangzhou, China. Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China. Acknowledgements Xiongfei Jian and Fahuai Yi are supported by NNSF of China No. 743, No and No. 4776, University Special Research Fund for Ph.D. Program of China and Xun Li is supported by Research Grants Council of Hong Kong under grants 560 and The authors are grateful to the editor and anonymous reviewers for their careful reviews, valuable comments, and remarks to improve this manuscript. Received: August 04 Accepted: 0 October 04 Published: 3 Oct 04 References. Ceci, C, Bassan, B: Mixed optimal stopping and stochastic control problems with semicontinuous final reward for diffusion processes. Stoch. Stoch. Rep. 76, Bensoussan, A, Lions, JL: Impulse Control and Quasi-Variational Inequalities. Gauthier-Villars, Paris Elliott, RJ, Kopp, PE: Mathematics of Financial Markets. Springer, New York Yong, J, Zhou, XY: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York Fleming, W, Soner, H: Controlled Markov Processes and Viscosity Solutions, nd edn. Springer, New York Dayanik, S, Karatzas, I: On the optimal stopping problem for one-dimensional diffusions. Stoch. Process. Appl. 07, Henderson, V, Hobson, D: An explicit solution for an optimal stopping/optimal control problem which models an asset sale. Ann. Appl. Probab. 8, Li, X, Zhou, XY: Continuous-time mean-variance efficiency: the 80% rule. Ann. Appl. Probab. 6, Li, X, Wu, ZY: Reputation entrenchment or risk minimization? Early stop and investor-manager agency conflict in fund management. J. Risk Finance 9, Li, X, Wu, ZY: Corporate risk management and investment decisions. J. Risk Finance 0, Shiryaev, A, Xu, ZQ, Zhou, XY: Thou shalt buy and hold. Quant. Finance 8, Chang, MH, Pang, T, Yong, J: Optimal stopping problem for stochastic differential equations with random coefficients. SIAM J. Control Optim. 48, Rüschendorf, L, Urusov, MA: On a class of optimal stopping problems for diffusions with discontinuous coefficients. Ann. Appl. Probab. 8, Peskir, G, Shiryaev, A: Optimal Stopping and Free-Boundary Problems, nd edn. Birkhäuser, Berlin Samuelson, PA: Rational theory of warrant pricing. With an Appendix by HP McKean: A free boundary problem for the heat equation arising from a problem in mathematical economics. Ind. Manage. Rev. 6, Karatzas, I, Kou, SG: Hedging American contingent claims with constrained portfolios. Finance Stoch., Karatzas, I, Sudderth, WD: Control and stopping of a diffusion process on an interval. Ann. Appl. Probab. 9, Karatzas, I, Wang, H: Utility maximization with discretionary stopping. SIAM J. Control Optim. 39, Karatzas, I, Ocone, D: A leavable bounded-velocity stochastic control problem. Stoch. Process. Appl. 99, Henderson, V: Valuing the option to invest in an incomplete market. Math. Financ. Econ., Choi, KJ, Koo, HK, Kwak, DY: Optimal stopping of active portfolio management. Ann. Econ. Financ. 5, Capenter, JN: Does option compensation increase managerial risk appetite? J. Finance 50, Pham, H: Continuous-Time Stochastic Control and Optimization with Financial Applications. Springer, Berlin Oleĭnik, OA, Radkević, EV: Second Order Equations with Nonnegative Characteristic Form. Plenum, New York Friedman, A: Parabolic variational inequalities in one space dimension and smoothness of the free boundary. J. Funct. Anal. 8, Friedman, A: Variational Principles and Free-Boundary Problems. Wiley, New York /09-4X Cite this article as: Jian et al.: Optimal investment with stopping in finite horizon. Journal of Inequalities and Applications 04, 04:43