Optimal Consumption with Reference to Past Spending Maximum
aa r X i v : . [ q -f i n . M F ] J u l Optimal Consumption with Reference to Past Spending Maximum
Shuoqing DENG ∗ Xun LI † Huyˆen PHAM ‡ Xiang YU § Abstract
This paper studies an infinite horizon optimal consumption problem under exponential util-ity, together with non-negativity constraint on consumption rate and a reference point to thepast consumption peak. The performance is measured by the distance between the consump-tion rate and a fraction 0 ≤ λ ≤ x and the reference variable h . The associated Hamilton-Jacobi-Bellman (HJB) equationis expressed in the piecewise manner across different regions to take into account constraints.By employing the dual transform and smooth-fit principle, the classical solution of the HJBequation is obtained in an analytical form, which in turn provides the feedback optimal invest-ment and consumption. For 0 < λ <
1, we are able to find four boundary curves x ( h ), ˘ x ( h ), x ( h ) and x ( h ) for the wealth level x that are nonlinear functions of h such that the feedbackoptimal consumption satisfies: (i) c ∗ ( x, h ) = 0 when x ≤ x ( h ); (ii) 0 < c ∗ ( x, h ) < λh when x ( h ) < x < ˘ x ( h ); (iii) λh ≤ c ∗ ( x, h ) < h when ˘ x ( h ) ≤ x < x ( h ); (iv) c ∗ ( x, h ) = h butthe running maximum process remains flat when x ( h ) ≤ x < x ( h ); (v) c ∗ ( x, h ) = h and therunning maximum process increases when x = x ( h ). Similar conclusions can be made in asimpler fashion for two extreme cases λ = 0 and λ = 1. Numerical examples are also presentedto illustrate some theoretical results and financial insights. Keywords : Exponential utility, non-negative consumption, historical consumption maximum,path dependence, dual transform, free boundary.
The Merton problem, also known as continuous time optimal portfolio and consumption via utilitymaximization firstly studied in [17] and [18], has been one of the milestones in quantitative finance,which bridges the investment decision making and some advanced mathematical tools such as PDEtheories and stochastic analysis. The celebrated dynamic programming principle enables one tosolve the stochastic control problem by looking for the solution of the associated HJB equation.Isoelastic utility and exponential utility have attracted dominant attention in academic researchas they enjoy the merits of homogeneity and scaling property. In abundant work on terminalwealth optimization, the value function can be conjectured in some convenient separation forms ∗ Department of Mathematics, University of Michigan, Ann Arbor, USA. Email: [email protected] † Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.Email: [email protected] ‡ LPSM, Universit´e de Paris and CREST-ENSAE, Paris, France. Email: [email protected] § Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.Email: [email protected]
1r the change of variables can be exercised, the dimension reduction can consequently be appliedto simplify the HJB equation. When intermediate consumption is taken into account, the studyof exponential utility becomes relatively rare in the literature due to its unnatural allowance ofnegative consumption behavior. To be precise, as the exponential utility is defined on the wholereal line, the resulting optimal consumption from the first order condition can be negative in general.For technical convenience, some existing literature such as [17], [22], [16] and many subsequent worksimply ignore the constraint or interpret the negative consumption as the infusion of funds, i.e., thenegative consumption control is assigned by different financial meanings so that the non-negativityconstraint can be avoided in their mathematical problems.The case of exponential utility with non-negative consumption has been examined before by [5]using the martingale method, in which the optimal consumption can be expressed in an integral forminvolving the state price density process. As illustrated in [5], the structure of the value functionand the optimal consumption differ substantially from the case when the constraint is neglected.Some technical endeavors are actually required to fulfill the non-negativity constraint on the controlprocess. In the present paper, we aim to revisit this problem under the exponential utility bindingstrictly with the constraint that the consumption rate must be non-negative. Moreover, unlike thetime separable utility studied in [5], our paper further attempts to go beyond the conventionalpreference and investigate the consumption behavior when a reference point is combined in theutility as well. In particular, our new preference essentially concerns how far the investor is awayfrom the past consumption maximum level, and this intermediate gap is chosen as the metric togenerate the utility of the investor in a dynamic way. Due to the consumption running maximumprocess inside the utility function, the martingale method developed in [5] can no longer handleour path-dependent optimization problem because it turns to be difficult to conjecture the validdual processes and the associated dual problem.Our problem formulation is mainly motivated by the psychological viewpoint that the con-sumer’s satisfaction level and risk tolerance sometimes depend on recent changes instead of theabsolute rate. Some large amount of expenditures, such as purchasing a car, a house or some lux-ury goods, not only spur some long term continuing spending for maintenance and repair, but alsolift up the investor’s standard of living gradually. A striking decline in future consumption planmay result in intolerable disappointment and discomfort. To depict the quantitative influence ofthe relative change towards the investor’s preference, it makes good sense to introduce the utility tomeasure the distance between the consumption control and a proportion of the past consumptionpeak. On the other hand, during some economic recession periods such as recent global economybattered by Covid-19, it is unrealistic to mandate that the investor needs to catch up with thepast spending maximum all the time. To capture the possibility that the investor may strategi-cally decrease the consumption budget to fall below the benchmark so that more wealth can beaccumulated from the financial market to meet future higher consumption plan, we choose to workwith the exponential utility instead of Isoelastic utility that is defined on the positive real line. Asa direct consequence, the investor can bear a negative gap between the consumption control andthe reference level. The flexibility to compromise the consumption plan below the reference pointfrom time to time makes the model suitable to accommodate more versatile market environmentsand mathematically unique and interesting.Utility maximization with a reference point has become an important topic in the research ofprospect theory and behavioral finance, see [21], [15], [14] and [13] on portfolio management witheither a fixed or an adaptive reference level. Our paper differs from the previous work as we donot distinguish the utility on gain and loss separately and our reference level has path-dependentnature and is dynamically updated by the control itself. The impact of the reference to the pastconsumption maximum becomes highly implicit in our setting, which makes the problem appealing2nd challenging. On the other hand, our formulation is closely related to the so-called consumptionhabit formation preference, which measures the deviation of the consumption from the standard ofliving conventionally defined as the average of the accumulative consumption. See some previouswork on addictive consumption habit formation in [4], [8], [20], [19], [11], [24], [25], [23] and non-addictive consumption habit formation in [7]. Recently, there are also some emerging research onthe combination of the reference point and the consumption habit formation, see for instance [6]and [3], in which the reference point is generated by the habit formation process and differentutility functions are equipped when the consumption is above the habit and when the consumptionis below the habit. It will be an interesting future work for us to also consider this S-shaped utilitydefined on the difference between the consumption and the consumption peak reference level andinvestigate the structure of the optimal consumption. Among the aforementioned work, it is worthnoting that [7] considers the utility defined on the whole real line and also permits the admissibleconsumption to fall below the habit level from time to time, namely the consumption habit isnot addictive. [7] extends the martingale method in [5] by using the adjusted state price densityprocess, which produces a nice construction of the optimal consumption in the complete marketmodel. However, our running maximum process in the utility function differs substantially from[7] and the duality approach is again not applicable.One of the main contributions of the present paper is to show that our path-dependent controlproblem with consumption constraint can be solved under the umbrella of dynamic programmingand PDE approach. The optimal consumption and portfolio can be obtained in piecewise feedbackforms across different regions. Furthermore, all free boundary curves to separate these regions,albeit complicated, can be fully explicitly characterized. Comparing with Merton problem withexponential utility, our value function and feedback optimal controls have distinctive and moreinteresting features. On the other hand, in terms of the control problem and the associated HJBequation, it is worth noting that [2], [12] and [1] are technically close to the present paper. However,[12] studies the optimal consumption under a Cobb-Douglas utility that is defined on the ratio ofthe consumption rate and the consumption running maximum, and [2] and [1] considers an optimalconsumption and dividend control problem respectively with a standard power utility and thedrawdown constraint is only mandated on the control and does not appear in the utility. As opposedto [12] and [1], our utility measures the difference between the control and its running maximumand the non-negativity constraint on consumption is actively imposed under the exponential utilityfunction. Mathematically speaking, the change of variable and dimension reduction in [12] and [1]can not be exercised in the present framework and we confront a two dimensional value functionand its associated nonlinear PDE problem. Despite of its complex structure and the non-negativityconstraint on optimal control, it is revealed in the present paper that the existence of the classicalsolution to the associated HJB equation can be obtained in the analytic form with the aid of thedual transform, the smooth-fit principle and other novel arguments.In summary, by noting that the consumption control is restricted between 0 and the peaklevel, we first heuristically derive the HJB equation in different forms based on the decompositionof the domain { ( x, h ) ∈ R + × R + } into disjoint regions of ( x, h ) such that the feedback optimalconsumption satisfies (i) c ∗ ( x, h ) = 0; (ii) 0 < c ∗ ( x, h ) < h ; (iii) c ∗ ( x, h ) = h . To overcomethe obstacle from nonlinearity, we apply the dual transformation only with respect to the statevariable x and treat h as the parameter that is involved in some free boundary conditions. Thelinearized dual PDE in different regions can be handled as ODE problem with the parameter h . Byusing smooth-fit principle and some intrinsic boundary conditions from the nature of the problem,we successfully obtain the explicit solution of the dual ODE problem that eventually enables usto express the value function, the feedback optimal investment and consumption in terms of theprimal variables after the inverse transform. Unlike [2], [12] and [1], taking the weight parameter3 < λ < x ( h ), ˘ x ( h ), x ( h ) and x ( h )for the wealth variable x as sophisticated nonlinear functions of the variable h such that we canprovide the feedback optimal consumption in the way that: (i) c ∗ ( x, h ) = 0 when x ≤ x ( h ); (ii)0 < c ∗ ( x, h ) < λh when x ( h ) < x < ˘ x ( h ); (iii) λh ≤ c ∗ ( x, h ) < h when ˘ x ( h ) ≤ x < x ( h ); (iv) c ∗ ( x, h ) = h but h is a previously attained maximum level when x ( h ) ≤ x < x ( h ); (v) c ∗ ( x, h ) = h and the instant c ∗ ( x, h ) creates a new historical maximum level when x = x ( h ). Two extreme cases λ = 0 and λ = 1 are also discussed separately. In particular, λ = 0 corresponds to Merton problemwith non-negativity constraint and we recover the result in [5] using PDE approach. When λ = 1,it is interesting to observe that the value function is not strictly concave any more so that we needto apply the dual transform in a restricted domain. Moreover, we reveal an interesting observationthat there is no need to consider the singular consumption that increases its running maximumprocess, which differs from the case 0 < λ <
1. At last, the complete proof of the verificationtheorem is rigorously established.Building upon the explicit value function and the feedback optimal controls, some quantitativeproperties and numerical examples are presented. The impacts of the variable h and the referenceweight parameter on the boundary curves x ( h ), ˘ x ( h ), x ( h ) and x ( h ) can be numerically illus-trated and the financial insights are observed. We also perform some sensitivity analysis on thevalue function, the optimal consumption and portfolio with respect to the reference weight para-meter, the drift of the risky asset, the volatility of the risky asset and the risk aversion parameterrespectively and conclude some interesting financial implications.The remainder of the paper is organized as follows. Section 2 introduces the market model andformulates the stochastic control problem under the utility with the reference to consumption peak.Section 3 presents the associated HJB equation and our technical computations to obtain the fullyexplicit solution using dual transform, smooth-fit principle and some intrinsic boundary conditions.Some numerical sensitivity analysis are presented in Section 4. At last, Section 5 provides therigorous proof of the verification theorem and other main results in the previous sections. Let (Ω , F , F , P ) be a filtered probability space, in which F = ( F t ) t ≥ satisfies the usual conditions.We consider a financial market consisting of one riskless asset and one risky asset. The risklessasset price satisfies dB t = rB t dt where r ≥ dS t = S t µdt + S t σdW t , where W is an F -adapted Brownian motion and both the mean return µ and volatility σ > κ := µ − rσ . It is worth noting that ourmathematical arguments and all conclusions can be readily generalized to the model with multiplerisky assets as long as the market is complete. For the sake of simple presentation, we shall onlyfocus on the model with a single risky asset. It is assumed that κ > µ > r that the return of the risky asset is higher than the interest rate.Let ( π t ) t ≥ represent the dynamic amount that the investor allocates in the risky asset and( c t ) t ≥ denote the dynamic consumption rate of the investor. The resulting self-financing wealthprocess ( X t ) t ≥ satisfies dX t = rX t dt + π t ( µ − r ) dt + π t σdW t − c t dt, t ≥ , with the initial wealth X = x ≥
0. 4he consumption-portfolio pair ( c, π ) is said to be admissible , denoted by ( c, π ) ∈ A ( x ), if theconsumption rate maintains non-negative, i.e. c t ≥ t ≥ c is F -predictable and π is F -progressively measurable and both satisfy the integrability condition R ∞ ( c t + π t ) dt < ∞ a.s.Moreover, no bankruptcy of the investor is allowed in the sense that X t ≥ t ≥ U ( x ) = − β e − βx in the present paper with β > x ∈ R . We are interested in the following infinite time utility maximization defined on thedifference between the current consumption rate and its historical running maximum that u ( x, h ) = sup ( π,c ) ∈A ( x ) E (cid:20)Z ∞ e − ρt U ( c t − λH t ) dt (cid:21) , (2.1)where we define H t = max { h, sup s ≤ t c s } , H = h ≥ , and the proportional constant 0 ≤ λ ≤ ρ = r to simplify some future computations. Here, H = h ≥ H . To achieve the value function, it is not necessary for the optimal consumptioncontrol to exceed the reference level at any time. The lifetime average of the outperformancebetween the consumption and the reference level plays the key role. Meanwhile, as c t − λH t can benegative sometimes, the non-negativity constraint c t ≥ t ≥ c t represents the consumption rate in the conventional sense. This control constraint spurssome new mathematical challenges when we handle the associated HJB equation using dynamicprogramming arguments in subsequent sections. Remark 2.1.
The problem (2.1) stems from some psychological consumption behavior that theinvestor sometimes can be very sensitive to the deviation from the past consumption pattern. Inparticular, the investor may have a strong memory of the past large amount of expenditures suchas to purchase a large house or a fancy car, which may overturn the investor’s living environmentand future budget plan. In response to the psychological consistency on consumption stream, theinvestor can be prone to aggressively catch up with the past consumption peak to some extent, whichmotivates us to consider the preference that measures the distance between the current consumptionchoice and a proportion of the historical maximum level.The problem (2.1) in the extreme case λ = 0 is reduced to the standard Merton problem onconsumption rate under exponential utility with non-negativity constraint that has been studied in[5] in the complete market model. In particular, to handle the constraint that the consumption rate isnon-negative, [5] applied the martingale method and formulated the control problem with constraintinto a relaxed form by introducing the Lagrange multipliers. By using the dual representation, theoptimal non-negative consumption can be expressed in a technical integral form involving the uniquestate price density process. In the same complete market framework, this martingale method hasbeen further refined by [7] to study the optimal consumption under non-addictive habit formationwhen the utility is generated by the difference between the non-negative consumption rate and theaccumulative integral of past consumption control. By introducing the adjusted state price densityprocess and the stochastic Lagrange multiplier process, the duality gap can be closed and the optimalnon-negative consumption can be constructed and verified using the dual representation. Contrary o [5] and [7], the presence of consumption running maximum inside the utility invalidates themartingale method because it becomes very complicated to construct the adjusted dual process anddual problem.On the other hand, the problem (2.1) in the extreme case λ = 1 is related to the so-called ratch-eting consumption behavior studied in the seminal paper [9] and several subsequent work. In [9] withpower utility function, the ratcheting constraint that the consumption rate is non-decreasing, i.e. c t ≥ sup s ≤ t c s is mandated purely in the definition of admissible strategies while its utility functionis defined on the consumption rate in a conventional way. In the same framework with power utility,[2] further generalizes the ratcheting constraint in [9] to a drawdown type constraint on consumptionin the sense that c t ≥ λ sup s ≤ t c s for some λ ∈ [0 , . Our present formulation differs substantiallyfrom [9] and [2]. The outperformed difference between the current consumption rate and a fractionof the benchmark process is now chosen as the metric to measure the satisfaction of the investor.Moreover, we choose to work with exponential utility instead of power utility so that the ratchetingor drawdown constraint is no longer strictly enforced. The investor can strategically suppress thecurrent consumption to some subsistence level that is below the reference level sometimes, which inturn may benefit the investor to attain a larger future consumption rate that is beyond the referencelevel with a higher probability and longer time periods. To embed the control problem into the Markovian framework and derive the associated HJB equa-tion using dynamic programming arguments, we treat both X t and H t as the controlled stateprocesses given the control policy ( c, π ). The value function u ( x, h ) becomes two dimensional de-pending on variables x ≥ h ≥
0, namely the initial wealth and the initial reference level forconsumption. Let us consider the processΓ t := e − rt u ( X t , H t ) + Z t e − rs U ( c s − λH s ) ds. (3.1)The martingale optimality principle implies that (Γ t ) t ≥ is a local supermartingale under all ad-missible controls and (Γ t ) t ≥ is a local martingale given the optimal control (if it exists).If the function u ( x, h ) is smooth enough, by applying Itˆo’s formula to the process (Γ t ) t ≥ , wecan derive that e rt d Γ t = (cid:20) − ru + u x ( rX t + π ( µ − r ) − c t ) + 12 σ π u xx + U ( c t − λH t ) (cid:21) dt + u h dH t + u x πσdW t , which heuristically leads to the associated HJB variational inequality sup c ∈ [0 ,h ] ,π ∈ R h − ru + u x ( rx + π ( µ − r ) − c ) + σ π u xx − β e β ( λh − c ) i = 0 ,u h ( x, h ) ≤ , (3.2)for x ≥ h ≥
0. To guarantee the local martingale property of u ( X ∗ t , H ∗ t ) under the optimalportfolio π ∗ t and consumption control c ∗ t , we have to require that u h ( X ∗ t , H ∗ t ) = 0 whenever H ∗ t increases for some ω , i.e., the current consumption rate c ∗ t creates the new historical maximum levelthat H ∗ t = c ∗ t and c ∗ t > H ∗ s for s < t . This motivates us to mandate an important free boundarycondition that u h ( x, h ) = 0 on some set of ( x, h ) that will be determined explicitly later in (3.20)in the section when we derive and analyze the associated HJB equation.6n the present paper, we aim to find some deterministic functions π ∗ ( x, h ) and c ∗ ( x, h ) to providethe feedback form of the optimal portfolio and consumption strategy. To this end, if u ( x, · ) is C w.r.t the variable x , the first order condition gives the optimal portfolio in a feedback form by π ∗ ( x, h ) = − µ − rσ u x u xx . The previous HJB variational inequality (3.2) can first be written assup c ∈ [0 ,h ] (cid:20) − β e β ( λh − c ) − cu x (cid:21) − ru + rxu x − κ u x u xx = 0 , and u h ≤ , ∀ x ≥ , h ≥ , (3.3)together with the free boundary condition u h = 0 on some set of ( x, h ) ∈ R + × R + that will becharacterized later. To handle the control constraint 0 ≤ c ≤ h , we consider two extreme casesthat λ = 0 and λ = 1 and the more interesting case 0 < λ < λ = 0 To tackle the HJB equation (3.3), let us first consider the extreme case without the reference toits historical maximum, i.e. λ = 0. Recall that this case corresponds to the standard optimalconsumption under exponential utility with non-negativity constraint that has been studied in [5].Instead of using the martingale method as in [5], we provide the solution in a more explicit mannerbased on the analysis of the HJB equation.In this case, the value function u actually does not depend on h , and we can simply write it as u ( x ). The free boundary condition can be ignored and some results in this extreme case will beused later in the problem when λ > λ = 0, the HJB variational inequality (3.3) can be simplified into a standard ODE problemwithout worrying about u h ( x, h ) ≤
0. The first order condition without the non-negativity con-straint gives the auxiliary feedback control ˆ c ( x ) := − β ln u x , and we need to distinguish two casesbased on the value of ˆ c ( x ) as below. Region I: on the set { x ∈ R + : u x ( x ) ≥ } , we have ˆ c ( x ) ≤
0. The optimal consumption istherefore c ∗ ( x ) = 0 and the ODE (3.3) is simplified to − β − ru + rxu x − κ u x u xx = 0 . (3.4) Region II: on the set { x ∈ R + : u x ( x ) < } , we have ˆ c ( x ) >
0. The optimal consumption is then c ∗ ( x ) = − β ln u x > − β u x + 1 β u x ln u x − ru + rxu x − κ u x u xx = 0 . (3.5)To guarantee the global regularity of the solution, we need to impose the smooth-fit conditionalong the free boundary { x ∈ R + : u x ( x ) = 1 } . Moreover, with the aid of some boundary conditionsat x = 0, we can actually determine its solution explicitly. To be precise, we observe that as thewealth level x declines to zero, the consumption rate c will first turn to zero at some point x ∗ (to bedetermined later), then when x continues to tend to 0, the optimal investment π should also go to0. Otherwise, we will confront the risk of bankruptcy by keeping trading with the nearly 0 wealth.Using the optimal portfolio π ∗ ( x ) = − µ − rσ u x u xx , the boundary condition becomeslim x → u x ( x ) u xx ( x ) = 0 . (3.6)7n addition, note that if we start with 0 initial wealth, the wealth level will never change as there isno trading according to the previous condition, and the consumption rate should be 0 all the timeconsequently. Therefore, we can conclude thatlim x → u ( x ) = Z + ∞ − β e e − rt dt = − rβ . (3.7)On the other hand, as the wealth tends to infinite, one can consume as much as possible that leadsto infinitely large admissible consumption rate and also a small variation in the wealth has thenegligible effect on the change of the value function. It thus follows thatlim x → + ∞ u ( x ) = 0 and lim x → + ∞ u x ( x ) = 0 . (3.8)To handle the nonlinear terms in the HJB equation (3.4) and (3.5), we employ the dual transformof the function u ( x ) that is defined by v ( y ) := sup x ≥ ( u ( x ) − xy ), y >
0. For the given x , we considerthe variable y := u x ( x ) and it holds that u ( x ) = v ( y ) + xy . We can further deduce that x = − v y ( y ) , u ( x ) = v ( y ) − yv y ( y ) and u xx ( x ) = − v yy ( y ) . The nonlinear ODE (3.4) and (3.5) can be linearized as κ y v yy − rv = β , if y ≥ , β y − β y ln y, if y < , (3.9)and the free boundary condition is transformed to the point y = 1. Note that y ≥ c ∗ ( x ) = 0 and y < c ∗ ( x ) > y → v y ( y ) = −∞ and lim y → ( v ( y ) − yv y ( y )) = 0 . (3.10)Using the duality transform again, the boundary conditions (3.6) and (3.7) at x = 0 can bereformulated into free boundary conditions that yv yy ( y ) → v ( y ) − yv y ( y ) → − rβ as v y ( y ) → . (3.11)The next result gives the explicit solution to the dual ODE problem (3.9) and its proof is givenin Section 5.2. Proposition 3.1.
Given the boundary conditions in (3.10) , the free boundary conditions in (3.11) and also the smooth-fit condition at y = 1 , The ODE (3.9) admits the unique solution given explicitlyby v ( y ) = C y r − rβ , if y ≥ ,C y r + yrβ (ln y + κ r − , if y < , here constants C and C are given by C := r − r − r κ r β > ,C := r − r − r κ r β < , (3.12) in which the constants r > and r < are two roots of the algebraic equation z − z − rκ = 0 , which are given by r , = 12 (cid:16) ± r rκ (cid:17) . (3.13)By using the dual value function v ( y ) in Proposition 3.1, the optimal consumption and invest-ment c ∗ and π ∗ can be expressed in terms of the dual value function and dual variable in thefeedback form for y > Theorem 3.1.
Let x ≥ be the initial wealth. We consider the process Y t := y ∗ e rt M t , where M t := e − ( r + κ ) t − κW t is the discounted state price density process, where y ∗ = y ∗ ( x ) is the uniquesolution to the budget constraint E [ R ∞ c ∗ ( Y t ) M t dt ] = x . The optimal consumption c ∗ t = c ∗ ( Y t ) andportfolio π ∗ t = π ∗ ( Y t ) in the problem (2.1) for λ = 0 are given by π ∗ ( y ) = µ − rσ yv yy ( y ) = µ − rσ r ( r − C y r − = 2 rκ C y r − , if y ≥ ,r ( r − C y r − + 1 rβ = 2 rκ C y r − + 1 rβ , if y < ,c ∗ ( y ) = , if y ≥ , − β ln y, if y < , where we used the fact that r ( r −
1) = r ( r −
1) = rκ . Note that as it is assumed that µ > r , wealways have π ∗ ( y ) > be the definition of C and C . Actually, we can further rewrite the optimal controls in terms of the primal variables using theinverse transform. To this end, let us denote k ( x ) := u ′ ( x ) and the duality relationship implies that u ( x ) = v ( k ( x )) + xk ( x ). Moreover, k ( x ) has two different expressions k ( x ) and k ( x ) dependingthe value of x . Using the dual relationship, we can obtain that x = − C r ( k ( x )) r − , if k ( x ) ≥ , − C r ( k ( x )) r − − rβ (cid:18) ln k ( x ) + κ r (cid:19) , if k ( x ) < . Therefore, the free boundary point x ∗ dividing the two regions is given by x ∗ = − C r > , with C given in (3.12). For x > − C r , the function k ( x ) is uniquely determined by the im-plicit equation that x = − C r ( k ( x )) r − − rβ (cid:16) ln k ( x ) + κ r (cid:17) because the function G ( y ) :=9 C r y r − − rβ (cid:16) ln y + κ r (cid:17) is decreasing and lim y → G ( y ) = + ∞ and lim y → G ( y ) = − C r − κ r β = − C r . For x ≤ − C r , we obtain that k ( x ) = (cid:16) − xC r (cid:17) r − . The next result followsdirectly from Theorem 3.1 and the arguments above. Corollary 3.1.
For initial wealth x ≥ , we can express the value function and feedback optimalconsumption and portfolio by: u ( x ) = C (cid:18) − xC r (cid:19) r r − − rβ + x (cid:18) − xC r (cid:19) r − , if x ≤ − C r ,C ( k ( x )) r + k ( x ) rβ (cid:20) ln k ( x ) + κ r − xrβ (cid:21) , if x > − C r . (3.14) The optimal strategy c ∗ and π ∗ can therefore be written in the feedback form using x ≥ by π ∗ ( x ) = µ − rσ (1 − r ) x, if x ≤ − C r , rκ C k r − ( x ) + 1 rβ , if x > − C r , (3.15) c ∗ ( x ) = , if x ≤ − C r , − β ln k ( x ) , if x > − C r . (3.16)Here, we choose market parameters r = 0 . µ = 0 . σ = 0 .
25 and β = 1 and graph the valuefunction u ( x ) in Figure-1, the optimal consumption c ∗ ( x ) in Figure-2, and the optimal investment π ∗ ( x ) in Figure-3. In particular, we use the vertical dot line to highlight the free boundary point x = − C r in all figures to separate the domain of x . Wealth x u ( x ) -18-16-14-12-10-8-6-4-20 Value function
Value functionx free
Wealth x c ( x ) Consumption c(x)x free
Wealth x π ( x ) Portfolio π (x)x free Figure 1 Figure 2 Figure 3 < λ < We next consider the original control problem binding with the reference to the historical consump-tion peak when 0 < λ < c t ≥ ≤ c t ≤ H t , we first need to decompose the domain ( x, h ) ∈ R + × R + into threedifferent regions such that the feedback optimal consumption strategy satisfies: (1) c ∗ ( x, h ) = 0;(2) 0 < c ∗ ( x, h ) < h ; (3) c ∗ ( x, h ) = h . Let us denote the auxiliary control ˆ c ( x, h ) := − β ln u x + λh ,which is simply derived by the first order condition in the HJB equation (3.3). We need to separatethe following regions: 10 egion I : on the set R := (cid:8) ( x, h ) ∈ R + × R + : u x ( x, h ) ≥ e λβh (cid:9) , we have ˆ c ( x, h ) ≤
0, and thereforethe optimal consumption rate is c ∗ ( x, h ) = 0 and the HJB variational inequality becomes − β e λβh − ru + rxu x − κ u x u xx = 0 , and u h ≤ . (3.17) Region II : on the set R := (cid:8) ( x, h ) ∈ R + × R + : e − (1 − λ ) βh < u x ( x, h ) < e λβh (cid:9) , we have that 0 < ˆ c ( x, h ) < h , and therefore the optimal consumption rate is c ∗ = − β ln u x + λh . The HJB variationalinequality can be written as − β u x + u x ( 1 β ln u x − λh ) − ru + rxu x − κ u x u xx = 0 , and u h ≤ . (3.18) Remark 3.1.
As pointed out in Remark 2.1, the main reason for us to consider the exponentialutility resides in the flexibility that the optimal consumption c ∗ can fall below the reference level λH ∗ ,which matches better with the real life situation that the investor can bear unfulfilling consumptionduring the economic recession periods. Based on the feedback form of the optimal consumption c ∗ = − β ln u x + λh in Region II, we can characterize the domain of ( x, h ) such that the investorlowers the consumption rate below the reference, i.e. c ∗ t < λH ∗ t if and only if ( x, h ) is in the subset (cid:8) ( x, h ) ∈ R + × R + : 1 < u x ( x, h ) < e λβh (cid:9) . This subset will be further characterized explicitly inRemark 3.4 as a threshold (depending on h ) of the wealth level x .Region III : on the set R := (cid:8) ( x, h ) ∈ R + × R + : u x ( x, h ) ≤ e − (1 − λ ) βh (cid:9) , we have ˆ c ( x, h ) ≥ h andthe optimal consumption rate is c ∗ ( x, h ) = h that implies the instant consumption rate c ∗ t coincideswith the running maximum process H ∗ t . However, two subtle cases may occur that motivate us tosplit this region further.(i) In a certain region (to be determined), the historical maximum level is already attained atsome previous time s before time t and the current optimal consumption rate is either torevisit this maximum level from below or to sit on the same maximum level. This is the casethat the running maximum process H t keeps flat from time s to time t , and the feedbackform of c ∗ t = H ∗ t = c ∗ s for some time s < t . In this case, it is very natural to treat H ∗ t as thestate process and plug it to the feedback form c ∗ ( x, h ) = h .(ii) In the complementary region, the optimal consumption rate creates a new record of themaximum level that is strictly larger than its past consumption, and the running maximumprocess H t is strictly increasing at the instant t . This corresponds to the case that c ∗ t = H ∗ t isa singular control and c ∗ t > H ∗ s for s < t and we have to mandate the free boundary condition u h ( x, h ) = 0 from the martingale optimality condition. In this region, the feedback form c ∗ ( x, h ) = h is useless because H ∗ t is updated by c ∗ t itself, which can not provide any effectiveinformation.Restricted to the set (cid:8) ( x, h ) ∈ R + × R + : u x ( x, h ) > e λβh (cid:9) , the case ( ii ) that H ∗ t increases andis updated by the singular control c ∗ t suggests us to treat the H ∗ t = c ∗ t as a singular control instead ofthe state process. That is, the dimension of the problem can be reduced and we can first substitute h = c in (3.3) and then apply the first order condition to − β e β ( λc − c ) − cu x with respect to c . Underthe condition that λ <
1, we can obtain the auxiliary singular control ˆ c ( x ) := β ( λ − ln( u x − λ ), whichis the feedback form depending only on X t . It then becomes convenient to see that c ∗ t can update H ∗ t to a new level if and only if the feedback control c ∗ t = ˆ c ( X ∗ t ) ≥ H ∗ t so that H ∗ t is instantlyincreasing. We can then separate Region III into three subsets:11 egion III-(i) : on the set D := { ( x, h ) ∈ R + × R + : (1 − λ ) e − (1 − λ ) βh < u x ( x, h ) ≤ e − (1 − λ ) βh } ,we have a contradiction that ˆ c ( x ) < h , and therefore c ∗ t is not a singular control. We still needto follow the previous feedback form c ∗ ( x, h ) = h , in which h is a previously attained maximumlevel. The corresponding running maximum process remains flat at the instant time. In this regionof ( x, h ), we only know that u h ( x, h ) ≤ dH t = 0. The HJB variational inequality iswritten as − β e β ( λh − h ) − hu x − ru + rxu x − κ u x u xx = 0 , and u h ≤ . (3.19) Region III-(ii) : on the set D := (cid:8) ( x, h ) ∈ R + × R + : u x ( x, h ) = (1 − λ ) e − (1 − λ ) βh (cid:9) , we get ˆ c ( x ) = h and the feedback optimal consumption is c ∗ ( x, h ) = β ( λ − ln( u x − λ ) = h . This corresponds to thesingular control c ∗ t that creates a new peak for the whole path and H ∗ t = c ∗ t = β ( λ − ln( u x ( X ∗ t ,H ∗ t )1 − λ ) isstrictly increasing at the instant time so that H ∗ t > H ∗ s for s < t and we must require the followingfree boundary condition that u h ( x, h ) = 0 on n ( x, h ) ∈ R + × R + : u x ( x, h ) = (1 − λ ) e − (1 − λ ) βh o . (3.20)In this region, it is noted that c ∗ ( x, h ) = h = β ( λ − ln( u x − λ ). Therefore, the HJB equation followsthe same PDE (3.19) but together with the new free boundary condition (3.20). Region III-(iii) : on the set D := (cid:8) ( x, h ) ∈ R + × R + : u x ( x, h ) < (1 − λ ) e − (1 − λ ) βh (cid:9) , we get ˆ c ( x ) >h . This indicates that the initial reference level h is below the feedback control ˆ c ( x ), and theoptimal consumption is again a singular control c ∗ ( x, h ) = β ( λ − ln( u x − λ ), which creates a newconsumption peak. As the running maximum process H ∗ t is updated immediately by c ∗ t , the feed-back optimal consumption pulls the associated H ∗ t − upward from its original value to the new value β ( λ − ln( u x ( X ∗ t ,H ∗ t )1 − λ ) in the direction of h and X ∗ t remains the same, in which u ( x, h ) is the solutionof the HJB equation (3.19) on the set D . This suggests that for any given initial value ( x, h ) in theset D , the feedback control c ∗ ( x, h ) pushes the value function jumping immediately to the point( x, ˆ h ) on the boundary set D where ˆ h = β ( λ − ln( u x ( x, ˆ h )1 − λ ) for the given level of x .Therefore, it is sufficient for us to only concentrate ( x, h ) on the effective domain of the originalstochastic control problem that C := n ( x, h ) ∈ R + × R + : u x ( x, h ) ≥ (1 − λ ) e − (1 − λ ) βh o , (3.21)equivalently C = R ∪ R ∪ D ∪ D ⊂ R . The only possibility for ( x, h ) ∈ D = C c occurs atthe initial time t = 0, and the value function is just equivalent to the value function of ( x, ˆ h ) onthe boundary D with the same x . In other words, if the controlled process ( X ∗ , H ∗ ) starts from( x, h ) in the region C , then ( X ∗ t , H ∗ t ) will always stay inside the region C and will either reflect atthe boundary or move along the boundary D whenever it hits the boundary D (but will never goacross the boundary). On the other hand, if the process ( X ∗ , H ∗ ) starts from the value ( x, h ) insidethe region D , the optimal control enforces an instant jump (and the only jump) of the process H from H − = h to H = ˆ h on the boundary D and both processes X t and H t become continuousprocesses diffusing inside the effective domain C afterwards for t > x, h ) such that u x ( x, h ) = e λβh , u x ( x, h ) = e − (1 − λ ) βh , which separate the different regions that we discussed above.12imilar to the case when λ = 0, we can again employ the dual transform of the value functionto linearize the HJB equation. In particular, we choose the dual transform only with respect tothe variable x and treat the variable h as a parameter. Let h ≥ x ≥ x, h ) ∈ C and define the dual function on the domain y ≥ (1 − λ ) e − (1 − λ ) βh that v ( y, h ) := sup ( x,h ) ∈C ,x ≥ [ u ( x, h ) − xy ] , y ≥ (1 − λ ) e − (1 − λ ) βh . For the given ( x, h ), let us define ˆ y ( x, h ) := u x ( x, h ) (short as ˆ y ), the dual representation implies u ( x, h ) = v (ˆ y, h ) + x ˆ y as well as v y (ˆ y, h ) = − x . We then have u h ( x, h ) = ∂∂h ( v (ˆ y, h ) + x ˆ y ) = v h (ˆ y, h ) + ( v y (ˆ y, h ) + x ) d ˆ ydh = v h (ˆ y, h ) . In view of the free boundary condition (3.20), we obtain the boundary condition v h ( y, h ) = 0 on the set n ( y, h ) ∈ (0 , + ∞ ) × R + : y = (1 − λ ) e ( λ − βh o . (3.22)To align with nonlinear HJB variational inequality (3.17), (3.18), (3.19) in three different re-gions, the transformed dual variational inequality can be written as κ y v yy − rv = β e λβh , if y ≥ e λβh , β y − y (cid:18) β ln y − λh (cid:19) , if e ( λ − βh < y < e λβh , β e ( λ − βh + hy, if (1 − λ ) e ( λ − βh ≤ y ≤ e ( λ − βh , (3.23)together with the free boundary condition (3.22). As we regard h as a parameter from this pointonwards, we can fix h and study the above equation as the ODE problem of the variable y .Similar to the case when λ = 0, after the dual transform, the boundary condition (3.8) giveslim y → v y ( y, h ) = −∞ and lim y → ( v ( y, h ) − yv y ( y, h )) = 0 , (3.24)and the boundary conditions (3.6) and (3.7) at x = 0 is equivalent to yv yy ( y, h ) → v ( y, h ) − yv y ( y, h ) → − rβ e − λβh as v y ( y, h ) → . (3.25)By using the previous conditions, we can solve the dual ODE (3.23) fully explicitly and its proofis provided in Section 5.2. Proposition 3.2.
Let h ≥ be a given parameter. Given the boundary conditions in (3.24) ,free boundary conditions (3.25) and free boundary condition (3.22) , the smooth-fit conditions withrespect to y at free boundary points y = e λβh and y = e ( λ − βh , the ODE (3.23) in the domain y ≥ (1 − λ ) e ( λ − βh admits the unique solution given explicitly by v ( y, h ) = C ( h ) y r − rβ e λβh , if y ≥ e λβh ,C ( h ) y r + C ( h ) y r − yrβ + yrβ (cid:18) ln y − λβh + κ r (cid:19) , if e ( λ − βh < y < e λβh ,C ( h ) y r + C ( h ) y r − r hy − rβ e ( λ − βh , if (1 − λ ) e ( λ − βh ≤ y ≤ e ( λ − βh , (3.26)13 here functions C ( h ) , C ( h ) , C ( h ) , C ( h ) and C ( h ) are given explicitly in (3.27) , (3.28) , (3.29) , (3.30) and (3.31) respectively that C ( h ) := (1 − λ ) r − r ( r − κ r − r ) βr (cid:20) − r − r e ( λ − − r ) βh − λ (1 − r ) λ (1 − r ) − ( r − r ) e [ λ (1 − r ) − ( r − r )] βh (cid:21) + (1 − r ) κ r − r ) βr h e ( λ − − r ) βh − e λ (1 − r ) βh i ; (3.27) C ( h ) := ( r − κ r − r ) βr e λ (1 − r ) βh ; (3.28) C ( h ) := (1 − λ ) r − r ( r − κ r − r ) βr (cid:20) − r − r e ( λ − − r ) βh − λ (1 − r ) λ (1 − r ) − ( r − r ) e [ λ (1 − r ) − ( r − r )] βh (cid:21) + (1 − r ) κ r − r ) βr e ( λ − − r ) βh ; (3.29) C ( h ) := (1 − r ) κ r − r ) βr h e ( λ − − r ) βh − e λ (1 − r ) βh i ; (3.30) C ( h ) := (1 − λ ) r − r ( r − κ r − r ) βr (cid:20) − r − r e ( λ − − r ) βh − λ (1 − r ) λ (1 − r ) − ( r − r ) e [ λ (1 − r ) − ( r − r )] βh (cid:21) . (3.31) Here, constants r , are given previously in (3.13) . Remark 3.2.
Based on explicit forms in (3.27) , (3.28) and (3.29) , let us note the following asymp-totic results of the coefficients that C ( h ) = O (cid:16) e ( λ − − r ) βh (cid:17) + O (cid:16) e [ λ (1 − r ) − ( r − r )] βh (cid:17) + O (cid:16) e λ (1 − r ) βh (cid:17) ,C ( h ) = O (cid:16) e λβh (1 − r ) (cid:17) ,C ( h ) = O (cid:16) e ( λ − − r ) βh (cid:17) + O (cid:16) e [ λ (1 − r ) − ( r − r )] βh (cid:17) ,C ( h ) = O (cid:16) e λβh (1 − r ) (cid:17) + O (cid:16) e ( λ − − r ) βh (cid:17) ,C ( h ) = O (cid:16) e ( λ − − r ) βh (cid:17) + O (cid:16) e [ λ (1 − r ) − ( r − r )] βh (cid:17) , which will be used in later proofs. We can now present the main result of this paper, which provides the optimal investment andconsumption in the feedback form explicitly using the dual variables for 0 < λ <
1. The completeproof is deferred to Section 5.1.
Theorem 3.2 (Verification Theorem) . Let ( x, h ) ∈ C and < λ < , where x is the initial wealthand h ≥ is the initial reference level and C stands for the effective domain (3.21) . We considerthe process Y t := y ∗ e rt M t , where M t := e − ( r + κ ) t − κW t is the discounted state price density processand H ∗ t = h ∨ sup s ≤ t c ∗ ( Y s , H ∗ s ) is the reference process under the optimal control, and the constant y ∗ = y ∗ ( x, h ) is the unique solution to the budget constraint E (cid:2)R ∞ c ∗ ( Y t , H ∗ t ) M t dt (cid:3) = x . The valuefunction u ( x, h ) can be attained by employing the optimal consumption and portfolio strategies inthe feedback form that c ∗ t = c ∗ ( Y t , H ∗ t ) and π ∗ t = π ∗ ( Y t , H ∗ t ) , t ≥ , which are given by: ∗ ( y, h ) = , if y ≥ e λβh , − β ln y + λh, if e ( λ − βh < y < e λβh ,h, if (1 − λ ) e ( λ − βh < y ≤ e ( λ − βh , λ − β ln (cid:16) − λ y (cid:17) , if y = (1 − λ ) e ( λ − βh , (3.32) π ∗ ( y, h ) = µ − rσ yv yy ( y, h )= µ − rσ rκ C ( h ) y r − , if y ≥ e λβh , rκ C ( h ) y r − + 2 rκ C ( h ) y r − + 1 rβ , if e ( λ − βh < y < e λβh , rκ C ( h ) y r − + 2 rκ C ( h ) y r − , if (1 − λ ) e ( λ − βh ≤ y ≤ e ( λ − βh , (3.33) In particular, the running maximum process H ∗ t is strictly increasing such that H ∗ t = c ∗ t > c ∗ s for any time s < t if and only if Y t = (1 − λ ) e ( λ − βH ∗ t and its feedback optimal consumptionis c ∗ t = λ − β ln (cid:16) − λ Y t (cid:17) . If we have y ∗ ( x, h ) < (1 − λ ) e ( λ − βh at the initial time, the optimalconsumption creates a new peak and brings H ∗ − = h jumping immediately to a higher level H ∗ = λ − β ln (cid:16) − λ y ∗ ( x, h ) (cid:17) such that t = 0 becomes the only jump time of H ∗ t .Moreover, for any initial data ( X ∗ , H ∗ ) = ( x, h ) ∈ C , the stochastic differential equation dX ∗ t = rX ∗ t dt + π ∗ t ( µ − r ) dt + π ∗ t σdW t − c ∗ t dt (3.34) has a unique strong solution given the optimal feedback control ( c ∗ , π ∗ ) as above. Remark 3.3.
Note that the feedback optimal consumption c ∗ in (3.53) is predictable. Indeed, if Y t > (1 − λ ) e ( λ − βH t , the optimal consumption at time t is determined by the continuous process Y t and the past consumption maximum right before t , i.e. H t − , which is predictable. In this case,the current consumption does not create the new maximum level. When Y t = (1 − λ ) e ( λ − βH t , theoptimal consumption is determined directly by the continuous process Y t , which is again predictable. In Theorem 3.2, the feedback forms of the optimal investment and consumption are givenexplicitly in terms of the dual value function and the dual variables. We can also conduct theinverse dual transformation and express the primal value function u ( x, h ) and the feedback controlsin terms of x and h , albeit in more complicated forms. In the main body of the proof of Theorem3.2, we will take full advantage of the simplicity in the dual feedback formulas and verify theoptimality of the feedback controls using the duality relationship and some estimations based onthe dual process Y t = y ∗ e rt M t . However, to show the existence of a unique strong solution of SDE(3.34), we have to derive the feedback controls in terms of X ∗ t and H ∗ t and the step of inverse dualtransform becomes necessary, which will be established as follows.By using the dual relationship between u and v , we have that the optimal x = g ( · , h ) := − v y ( · , h ) . (3.35)Defining f ( · , h ) as the inverse of g ( · , h ), we have that u ( x, h ) = v ◦ ( f ( x, h ) , h ) + xf ( x, h ) . (3.36)15ote that v has different expressions in the regions c = 0, 0 < c < h and c = h , the function f should also have three different expressions in these regions and we denote them respectively by f , f and f .By the definition of g in (3.35), the invertibility of the map x g ( x, h ) is guaranteed by thefollowing important result and its proof is deferred to Section 5.2. Lemma 3.1.
In all three regions, we have that v yy ( y, h ) > , ∀ h > and the inverse Legendretransform u ( x, h ) = inf y ≥ (1 − λ ) e − (1 − λ ) βh [ v ( y, h ) + xy ] is well defined. Moreover, this implies that thefeedback optimal portfolio π ∗ ( y, h ) > always holds. Using (3.35) and Proposition 3.2, the function f is implicitly determined in different regions bythe following equations:(i) If f ( x, h ) ≥ e λβh , f ( x, h ) can be determined by x = − C ( h ) r ( f ( x, h )) r − . (3.37)(ii) If e ( λ − βh < f ( x, h ) < e λβh , Lemma 3.1 implies that v y ( y, h ) is strictly increasing in y and f ( x, h ) can be uniquely determined by x = − C ( h ) r ( f ( x, h )) r − − C ( h ) r ( f ( x, h )) r − − rβ (cid:18) ln f ( x, h ) − λβh + κ r (cid:19) . (3.38)(iii) If (1 − λ ) e ( λ − βh ≤ f ( x, h ) ≤ e ( λ − βh , Lemma 3.1 implies that v y ( y, h ) is strictly increasingin y and f ( x, h ) can be uniquely determined by x = − C ( h ) r ( f ( x, h )) r − − C ( h ) r ( f ( x, h )) r − + hr . (3.39).In region R , we can obtain the explicit form of f ( x, h ) = (cid:16) − xC ( h ) r (cid:17) r − . The condition f ( x, h ) ≥ e λβh gives us that this is valid when x ≤ x ( h ), where we define the free boundary by x ( h ) := − e λβh ( r − C ( h ) r . (3.40)In region R , the function f is uniquely determined implicitly by (3.38) when x ( h ) < x In addition, as in Remark 3.1, we know that the optimal consumption falls below thereference level if and only if < f ( x, h ) < e λβh . Using (3.38) again, we can determine the criticalpoint ˘ x ( h ) by ˘ x ( h ) := − C ( h ) r − C ( h ) r + λ hr − κ r β . (3.42) It then follows that if and only if the wealth level x is sufficiently small that satisfies x ( h ) < x < ˘ x ( h ) , the optimal consumption rate meets the compromised plan < c ∗ t ( x, h ) < λH ∗ t ( x, h ) . 16n region D ∪ D , the expression of f is uniquely defined implicitly by the equation (3.39).This expression of f holds when x ( h ) ≤ x ≤ x ( h ), where x is the solution of f ( x, h ) = (1 − λ ) e ( λ − βh . It follows from (3.39) that the free boundary point x ( h ) is explicitly given by x ( h ) := − C ( h ) r (1 − λ ) r − e ( λ − r − βh − C ( h ) r (1 − λ ) r − e ( λ − r − βh + hr . (3.43)Moreover, in view of definitions of C ( h ) and C ( h ) in (3.30) and (3.31), one can check that x ( h )is strictly increasing in h and hence we can define the inverse function˜ h ( x ) := ( x ) − ( x ) , x ≥ . (3.44)Therefore, along the free boundary x = x ( h ), we can write the feedback form of the optimalconsumption in (3.53) for y = (1 − λ ) e ( λ − βh by c ∗ ( x ) = λ − β ln (cid:16) − λ f ( x, ˜ h ( x )) (cid:17) , which onlydepends on the variable x . That is, the optimal consumption can be determined by the currentwealth process X ∗ t and the associated running maximum process H ∗ t is instantly increasing.In what follows, for some parameters r = 0 . µ = 0 . σ = 0 . β = 1, λ = 0 . 5, we graphall free boundary curves x ( h ), ˘ x ( h ), x ( h ) and x ( h ) as functions of h ≥ h = 1and plot all boundary curves in terms of the parameter λ ∈ [0 . , . 98] (recall that each function C i ( h ; λ ) depends on λ ). h x ( h ) Free boundary curves x (h)x (h)x (h)\breve{x}(h) λ x ( ; λ ) λ - x x (1; λ )x (1; λ )x (1; λ )\breve{x}(1;\lambda) Figure 4 Figure 5 Although x ( h ), ˘ x ( h ), x ( h ) and x ( h ) are all complicated nonlinear functions of h , from Figure4, we note that all free boundary curves are increasing in the variable h with the given parameters.The graphs are consistent with the intuition that if the past reference level is higher, the investorwould expect larger wealth thresholds to trigger the change of consumption from 0 to c ∗ > c ∗ < H ∗ to the historical maximum c ∗ = H ∗ . We also recall that we only consider theeffective domain that is the region below the boundary curve x ( h ) (including the boundary curve x ( h )). It is interesting to observe from Figure 5 that x ( h ; λ ) and x ( h ; λ ) are both decreasingin 0 . ≤ λ ≤ . 98, while ˘ x ( h ; λ ) and x ( h ) are both increasing in 0 . ≤ λ ≤ . 98. That is, ifthe investor clings to a larger proportion of the past spending maximum, it is more likely that theinvestor will switch from zero consumption to positive consumption and from a lower consumption c ∗ < H ∗ t to the past maximum level H ∗ t , which matches with the real life situations. On the other17and, with a higher proportion λ towards the consumption peak, the investor needs to accumulatelarger wealth to consume at the reference level c ∗ = λH ∗ t or consume at the peak to create a newhistorical maximum record that H ∗ t = c ∗ t > H ∗ s for s < t .In particular, Figure 5 illustrates that ˘ x ( h ; λ ) is increasing in terms of λ , which indicates thatif the investor adheres more to the past consumption peak with a larger proportion λ , it is morelikely that the investor will suppress the optimal consumption rate c ∗ t below λH ∗ t due to the largerthreshold ˘ x ( h ; λ ) for the wealth level. That is, the more the investor cares about the past con-sumption peak H ∗ t , the more conservative the investor will become by comparing c ∗ t and λH ∗ t . Thisobservation can partially explain the real life situations that the constantly aggressive consumptionbehavior may not lead to a long term happiness. A high consumption plan also creates a highlevel of psychological competition with the past pattern such that this aggressive consumptionbehavior may not be sustainable for the whole lifetime. A wise investor who takes into accountthe past reference will strategically lower the consumption rate from time to time (triggered by awealth threshold) below the target reference such that the reference process can be maintained ata reasonable level and the overall lifetime performance can eventually become a win.Plugging all different pieces of f back into equation (3.36), we can readily get the followingresult, in which the primal value function u and optimal feedback controls are all given in terms ofthe primal variables x and h . Corollary 3.2. For ( x, h ) ∈ C and < λ < , the value function u ( x, h ) of the control problem in (2.1) can be explicitly expressed in a piecewise manner by u ( x, h )= C ( h ) f ( x, h ) r − rβ e λβh + xf ( x, h ) , if x ≤ x ( h ) ,C ( h )( f ( x, h )) r + C ( h )( f ( x, h )) r + f ( x, h ) rβ (cid:20) ln f ( x, h ) − λβh + κ r − xrβ (cid:21) , if x ( h ) < x < x ( h ) ,C ( h )( f ( x, h )) r + C ( h )( f ( x, h )) r − r hf ( x, h ) − rβ e ( λ − βh + xf ( x, h ) , if x ( h ) ≤ x ≤ x ( h ) , (3.45) where the free boundaries x ( h ) , x ( h ) and x ( h ) are given explicitly in (3.40) , (3.41) and (3.43) respectively. Moreover, the feedback optimal consumption and portfolio can also be given in termsof primal variables ( x, h ) accordingly: c ∗ ( x, h ) = , if x ≤ x ( h ) , − β ln f ( x, h ) + λh, if x ( h ) < x < x ( h ) ,h, if x ( h ) ≤ x < x ( h ) , λ − β ln (cid:16) − λ f ( x, ˜ h ( x )) (cid:17) , if x = x ( h ) , (3.46)18 here ˜ h ( x ) is given in (3.44) , and π ∗ ( x, h )= µ − rσ (1 − r ) x, if x ≤ x ( h ) , rκ C ( h ) f r − ( x, h ) + 2 rκ C ( h ) f r − ( x, h ) + 1 rβ , if x ( h ) < x < x ( h ) , rκ C ( h ) f r − ( x, h ) + 2 rκ C ( h ) f r − ( x, h ) , if x ( h ) ≤ x ≤ x ( h ) . (3.47) We also have that < c ∗ t ( x, h ) < λH ∗ t ( x, h ) if and only if x ( h ) < x < ˘ x ( h ) where the threshold ˘ x ( h ) is given by (3.42) . Remark 3.5. In all referred work [2], [12] and [1], the domain of ( x, h ) is split into several regionsby linear free boundaries such as ν ≤ xh ≤ ν for some constants ν , , in which different optimalconsumption policies (or dividends) need to be followed. In contrary, our free boundary curves x ( h ) , x ( h ) and x ( h ) can be explicitly characterized by (3.40) , (3.41) and (3.43) (see graphs inFigure 4), which are nonlinear functions of the variable h . The sophisticated and more interestingdecomposition of the domain results from both the exponential utility function with non-negativityconstraint and the presence of the consumption running maximum inside the utility function. λ = 1 At last, we present some main results for the extreme case λ = 1. We separate this subsectionfrom the previous case 0 < λ < λ = 1. Solving the HJB equation essentially follows the same arguments inthe case 0 < λ < 1. However, the effective domain C defined in (3.21) needs to be modified to C := { ( x, h ) ∈ R + × R + : u x ( x, h ) ≥ } . (3.48)Equivalently, C = R ∪ R ∪ R = R , where R , R are defined the same as in the previoussubsection for 0 < λ < R = (cid:8) ( x, h ) ∈ R + × R + : u x ( x, h ) ≥ e βh (cid:9) , in which the optimalconsumption c ∗ t = 0. We also have R = (cid:8) ( x, h ) ∈ R + × R + : 1 < u x ( x, h ) < e βh (cid:9) , in which thefeedback optimal consumption c ∗ t = − β ln u x + h . As opposed to the case 0 < λ < 1, we nowconsider R = { ( x, h ) ∈ R + × R + : 0 ≤ u x ( x, h ) ≤ } and note that the previous auxiliary singularcontrol ˆ c ( x ) = β ( λ − ln( u x − λ ) to further split the region R for the case 0 < λ < λ = 1.In fact, in the extreme case λ = 1, there is no need to consider the singular optimal consumptionthat excesses the previous maximum level h . In the whole region R , the optimal consumption isno longer unique, but one feedback optimal consumption c ∗ ( x, h ) = h is to follow the previouslyattained maximum level, which is the initial level H ∗ = h and c ∗ ( x, h ) ≤ h for any x ≥ 0. Comparingwith other work [2], [12] and [1], this unique and interesting phenomenon that we can only focuson the optimal control such that H ∗ t will never increase for λ = 1 results from the nature of theformulation U ( c t − H t ) where the utility is defined on the difference. For the case 0 < λ < 1, theutility U ( c t − λH t ) allows the investor to gain positive outperformance c t − λH t > c t to increase H t . On the other hand, for the case λ = 1, the investor can only obtain0 = c t − H t by choosing to consume more than the past maximum. However, the investor canalso easily achieve the same goal of zero difference c t − H t by following the previously attainedmaximum level without creating any new record. Therefore, to achieve the largest gap c t − H t = 0,19ne equivalent optimal way is to sit on the previous consumption peak and the investor has noincentives to switch to a singular control to increase the reference process H t at any time. Evenif the initial wealth is sufficiently large, the investor will choose to consume the constant initiallevel h = H such that c t = H t as long as it is sustainable but never to excess this reference levelduring the life time. Consequently, in this subsection, we shall only adopt the feedback control c ∗ ( x, h ) = h in the region R .Based on the observations above, if the wealth x is larger than or equal to the subsistence level x ≥ x ∗ := hr , the investor can always choose to invest zero amount π ∗ t ≡ x ∗ in the bank account such that the interest rate can support the constantconsumption at the initial reference level c ∗ t = H = h , t ≥ 0. That is, we have c ∗ t − H ∗ t = 0 for t ≥ 0. Note that x ∗ = hr can also be obtained by x ( h ) in (3.43) by setting λ = 1. As a consequence,the value function defined in (2.1) attains its maximum value u ( x, h ) = − rβ for x ≥ hr . That is, theprimal value function u ( x, h ) for λ = 1 is no longer strictly concave and u ( x, h ) remains constant(and u x ( x, h ) = 0) for x ≥ hr , which differs substantially from the case 0 < λ < 1. Therefore, wehave the asymptotic conditions thatlim x → hr u x ( x, h ) = 0 , and lim x → hr u ( x, h ) = − rβ . (3.49)For each h ≥ 0, we expect that the value function x u ( x, h ) is strictly concave for 0 ≤ x < hr and the dual transform method in the previous sections can still be applied on this interval [0 , hr ).In view of the set C when λ = 1, we will now consider y > v ( y, h ) := sup ≤ x< hr [ u ( x, h ) − xy ] , y > , As a consequence of (3.49), we have the asymptotic conditions thatlim y → v y ( y, h ) = − hr and lim y → ( v ( y, h ) − yv y ( y, h )) = − rβ , (3.50)which are completely different from the boundary condition (3.24) for 0 < λ < < λ < 1, we can write down the linear dual ODE forthe case λ = 1 as κ y v yy − rv = β e βh , if y ≥ e βh , β y − y (cid:18) β ln y − h (cid:19) , if 1 < y < e βh , β + hy, if 0 < y ≤ , (3.51)By following the arguments of Proposition 3.2, and replacing the free boundary condition (3.22)now by the new boundary condition (3.50) as y → Proposition 3.3. Let h ≥ be a given parameter, the ODE (3.51) admits the unique solution xplicitly by v ( y, h ) = C ( h ) y r − rβ e βh , if y ≥ e βh ,C ( h ) y r + C ( h ) y r − yrβ + yrβ (cid:18) ln y − βh + κ r (cid:19) , if < y < e βh ,C ( h ) y r − r hy − rβ , if < y ≤ , (3.52) where C i ( h ) , i = 2 , , , are defined in (3.27) , (3.28) , (3.29) and (3.30) in Proposition 3.2 bysetting λ = 1 . We can similarly present the result of the verification theorem when λ = 1 as below. Theorem 3.3. Let h ≥ and ≤ x < h/r . We consider the process Y t := y ∗ e rt M t , where M t := e − ( r + κ ) t − κW t is the discounted state price density process and H ∗ t ≡ H ∗ = h is the constantreference process under the optimal control, and the constant y ∗ = y ∗ ( x, h ) is the unique solutionto the budget constraint E (cid:2)R ∞ c ∗ ( Y t , H ∗ t ) M t dt (cid:3) = x . The value function u ( x, h ) can be attained byemploying the optimal consumption and portfolio strategies in the feedback form that c ∗ t = c ∗ ( Y t , H ∗ t ) and π ∗ t = π ∗ ( Y t , H ∗ t ) , t ≥ , which are given by: c ∗ ( y, h ) = , if y ≥ e βh , − β ln y + h, if < y < e βh ,h, if < y ≤ , (3.53) π ∗ ( y, h ) = µ − rσ rκ C ( h ) y r − , if y ≥ e βh , rκ C ( h ) y r − + 2 rκ C ( h ) y r − + 1 rβ , if < y < e βh , rκ C ( h ) y r − , if < y ≤ , (3.54)Following the same inverse dual transform arguments in the previous subsection that u ( x, h ) =inf y> [ v ( y, h ) + xy ] for 0 ≤ x < hr and u ( x, h ) = − rβ for x ≥ hr , we can also obtain the functions¯ f i ( x ) = u x ( x, h ) in different regions that:(i) ¯ f ( x, h ) = (cid:16) − xC ( h ) r (cid:17) r − for 0 ≤ x ≤ ¯ x ( h ), where we define¯ x ( h ) := − e βh ( r − C ( h ) r . (3.55)(ii) ¯ f ( x, h ) that is uniquely determined by x = − C ( h ) r ( ¯ f ( x, h )) r − − C ( h ) r ( ¯ f ( x, h )) r − − rβ (cid:18) ln ¯ f ( x, h ) − βh + κ r (cid:19) , (3.56)21or ¯ x ( h ) < x < ¯ x ( h ) where¯ x ( h ) := − C ( h ) r − C ( h ) r + hr − κ r β . (3.57)(iii) ¯ f ( x, h ) = (cid:18) hr − xC ( h ) r (cid:19) r − for ¯ x ( h ) ≤ x < hr .We can conclude the corollary below on the value function and feedback optimal controls in thewhole domain. Corollary 3.3. For ( x, h ) ∈ R and λ = 1 , the value function u ( x, h ) of the control problem in (2.1) can be explicitly expressed by u ( x, h )= C ( h ) (cid:18) − xC ( h ) r (cid:19) r r − − rβ e βh + x (cid:18) − xC ( h ) r (cid:19) r − , if x ≤ ¯ x ( h ) ,C ( h )( ¯ f ( x, h )) r + C ( h )( ¯ f ( x, h )) r + ¯ f ( x, h ) rβ (cid:20) ln ¯ f ( x, h ) − βh + κ r − xrβ (cid:21) , if ¯ x ( h ) < x < ¯ x ( h ) ,C ( h ) hr − xC ( h ) r ! r r − − r h hr − xC ( h ) r ! r − − rβ + x hr − xC ( h ) r ! r − , if ¯ x ( h ) ≤ x < hr , − rβ , if hr ≤ x, where the free boundaries ¯ x ( h ) and ¯ x ( h ) are given explicitly in (3.55) and (3.57) respectively and ¯ f ( x, h ) is given implicitly by (3.56) . The feedback optimal consumption and portfolio are given by: c ∗ ( x, h ) = , if x ≤ ¯ x ( h ) , − β ln ¯ f ( x, h ) + h, if ¯ x ( h ) < x < ¯ x ( h ) ,h, if ¯ x ( h ) ≤ x, (3.58) and π ∗ ( x, h )= µ − rσ (1 − r ) x, if x ≤ ¯ x ( h ) , rκ C ( h ) ¯ f r − ( x, h ) + 2 rκ C ( h ) ¯ f r − ( x, h ) + 1 rβ , if ¯ x ( h ) < x < ¯ x ( h ) , rκ r (cid:18) hr − x (cid:19) , if ¯ x ( h ) ≤ x < hr , , if hr ≤ x, (3.59) and the resulting consumption running maximum process is constant that H ∗ t = H ∗ = h for t > . 22t last, based on Corollaries 3.1, 3.2 and 3.3, we present the result of the asymptotic behaviorof the optimal consumption-wealth ratio c ∗ t X ∗ t and the investment amount π ∗ t when the wealth issufficiently large and its proof is given in Section 5.2. Corollary 3.4. For λ = 0 , we have lim x → + ∞ c ∗ ( x ) x = r and lim x → + ∞ π ∗ ( x ) = µ − rrβσ . For < λ < ,as x ≤ x ( h ) , we consider the asymptotic behavior along the boundary curve x ( h ) as x, h → + ∞ ,and we have lim x → + ∞ , ( x,h ) ∈ x ( h ) c ∗ ( x, h ) x = r, lim x → + ∞ , ( x,h ) ∈ x ( h ) π ∗ ( x, h ) = ( µ − r )(1 − λ ) r − rβσ . As the wealth level gets sufficiently large, the optimal consumption is asymptotically proportionalto the wealth level that c ∗ t ≈ rX ∗ t and the optimal investment converges to a constant level that π ∗ t ≈ (1 − λ ) r − rβ for (0 ≤ λ < . That is, the investor will only allocate constant amount of wealthinto the risky asset and save most of the wealth into the bank account. For λ = 1 , the investor willstop the investment in the risky asset when the wealth exceeds the constant level hr and one optimalconsumption is to constantly spend the initial reference amount c ∗ t = h for t ≥ . This section reports some numerical examples of sensitivity analysis using the previous explicitvalue function and feedback optimal controls in Corollary 3.2.We first present the 3-dimensional graphs of the value function u ( x, h ), the optimal consumption c ∗ ( x, h ) and optimal portfolio π ∗ ( x, h ) in the next three figures. In particular, we choose the marketparameters that r = 0 . µ = 0 . σ = 0 . β = 1, λ = 0 . h ∈ [0 . , . 5] and x ∈ [0 , Wealth x Value function Spending maximum u Wealth x Consumption Spending maximum c Wealth x Portfolio Spending maximum π Figure 6 Figure 7 Figure 8 We first perform the sensitivity analysis by plotting graphs of the value function, the feedbackoptimal consumption and the feedback optimal portfolio for different values of the reference weightparameter λ = 0 . , . , ..., . , . 9. Here, we choose the market parameters that r = 0 . µ = 0 . = 0 . β = 1 and fix the variable h = 1 and plot all graphs as functions of x for 0 ≤ x ≤ x (1). Wealth x u ( x , ) -50-45-40-35-30-25-20-15-10-5 Value function λ =0.1 λ =0.2 λ =0.3 λ =0.4 λ =0.5 λ =0.6 λ =0.7 λ =0.8 λ =0.9 Wealth x c ( x , ) Consumption λ =0.1 λ =0.2 λ =0.3 λ =0.4 λ =0.5 λ =0.6 λ =0.7 λ =0.8 λ =0.9 Wealth x π ( x , ) Portfolio λ =0.1 λ =0.2 λ =0.3 λ =0.4 λ =0.5 λ =0.6 λ =0.7 λ =0.8 λ =0.9 Figure 9 Figure 10 Figure 11 From Figure 10, for x (1) < x < x (1), we can see that the feedback consumption c ∗ ( x, 1) isincreasing in x . More importantly, for the fixed 0 < x < x (1), the feedback optimal consumption c ∗ ( x, λ ) is increasing in the parameter λ ∈ (0 , x ≤ x (1), the feedback π ∗ ( x, 1) is increasing andlinear in x ; and for x (1) < x < x (1), the feedback π ∗ ( x, 1) is increasing and concave in x ; andfor x (1) ≤ x ≤ x (1), the feedback π ∗ ( x, 1) is increasing and convex in x , Moreover, for thefixed 0 < x < x (1), π ∗ ( x, λ ) is decreasing in the parameter λ ∈ (0 , λ . Theinvestor may strategically invest less in the market to save enough cash for higher consumptionplan influenced by λ . From Figure 9, for each 0 < x < x (1), the graphs illustrate that the valuefunction u ( x, h ; λ ) is decreasing in λ ∈ (0 , λ , it does not necessarily imply that the value function alsoincreases. In our preference formulation, the utility function is defined on the difference betweenthe consumption rate c ∗ t and the reference process λH ∗ t . As both λ and c ∗ t increase, the referenceprocess λH ∗ t increases as well. From Figure 9, we can see that λH ∗ t actually increases faster thanthe consumption rate c t when λ increases, which leads to a drop of the difference c ∗ t − λH ∗ t so thatthe resulting value function actually decreases.We next present the impact of the drift parameter µ on the value function, the feedback optimalconsumption and the feedback optimal portfolio by considering µ = 0 . , . , . , . , . 18. Weagain fix marker parameters r = 0 . σ = 0 . λ = 0 . β = 1 and the maximum reference variable h = 1 and plot the graphs as functions of x for 0 ≤ x ≤ x (1). Wealth x u ( x , ) -35-30-25-20-15-10-5 Value function µ =0.1 µ =0.12 µ =0.14 µ =0.16 µ =0.18 Wealth x c ( x , ) Consumption µ =0.1 µ =0.12 µ =0.14 µ =0.16 µ =0.18 Wealth x π ( x , ) Portfolio µ =0.1 µ =0.12 µ =0.14 µ =0.16 µ =0.18 Figure 12 Figure 13 Figure 14 µ . This implies that if the market performance is getting better asthe return of the risky asset increases, the investor can accumulate more wealth from the financialmarket to support a higher consumption plan. Likewise, Figure 14 illustrates that the investor’soptimal portfolio in the financial market increases as the stock return increases. Figure 12 showsthat the primal value function is increasing with respect to the drift parameter µ . It illustrates thatwhen the return parameter µ increases, the increase in optimal consumption rate c ∗ t dominates theincrease in the running maximum process H ∗ t so that the life time value function is lifted up.Similarly, for the market parameters r = 0 . µ = 0 . β = 1, λ = 0 . h = 1, we continue to present the sensitivity analysis of the value function u ( x, c ∗ ( x, 1) and π ∗ ( x, 1) with respect to different volatility parameters σ = 0 . , . , . , . , . Wealth x u ( x , ) -35-30-25-20-15-10-5 Value function σ =0.1 σ =0.2 σ =0.4 σ =0.6 σ =0.8 Wealth x c ( x , ) Consumption σ =0.1 σ =0.2 σ =0.4 σ =0.6 σ =0.8 Wealth x π ( x , ) Portfolio σ =0.1 σ =0.2 σ =0.4 σ =0.6 σ =0.8 Figure 15 Figure 16 Figure 17 From Figure 16, we observe that the monotonicity of the optimal consumption c ∗ ( x, 1) on thevolatility σ is not guaranteed and the dependence becomes much subtle and complicated. We canalso see that the consumption c ∗ ( x, 1) is not simply concave or convex in the variable x for theregion x (1) < x < x (1), which depends on the volatility σ and other market parameters. In thisexample, we can observe that the threshold x (1; σ ) is increasing in σ but the dependence of thethreshold x (1; σ ) on σ is unclear. Figure 15 and Figure 17 show that both the value function andthe optimal portfolio are decreasing in the volatility σ . These graphs are consistent with the reallife situation and are similar to some classical models such as the Merton problem that if the riskyasset has a higher volatility, the investor will allocate less wealth in the risky asset and the valuefunction on consumption also becomes lower.At last, for the market parameters r = 0 . µ = 0 . σ = 0 . λ = 0 . h = 1,we plot the sensitivity analysis of the value function u ( x, β ), the feedback controls c ∗ ( x, β ) and π ∗ ( x, β ) with respect to the risk aversion parameter β = 0 . , . , , , ≤ x ≤ x (1). We can see from Figure 19 that the threshold x (1; β ) is decreasing in the riskaversion parameter β and x ( h ) is increasing in β . That is, for a more risk averse investor, it ismore difficult to start a positive consumption c ∗ > c ∗ ( x, β ) has a very complicated depen-dence on the parameter β . Within this numerical example, the optimal portfolio π ∗ ( x, β ) is stilldecreasing in β from Figure 20, but the impact of β on the value function u ( x, β ) is no longermonotone. Only when the wealth is sufficiently large, the value function behaves increasing withhigher risk aversion β . 25 ealth x u ( x , ) -120-100-80-60-40-200 Value function β =0.2 β =0.5 β =1 β =2 β =5 Wealth x c ( x , ) Consumption β =0.5 β =1 β =1.5 β =2 β =5 Wealth x π ( x , ) Portfolio β =0.5 β =1 β =1.5 β =2 β =5 Figure 18 Figure 19 Figure 20 In this subsection, we only provide the complete proof of Theorem 3.2 for the case 0 < λ < Proof of Theorem 3.2. The proof of the verification theorem boils down to show that the solutionof the PDE indeed coincides with the value function, i.e. there exists ( π ∗ , c ∗ ) ∈ A ( x ) such that u ( x, h ) = E (cid:20)Z ∞ e − rt U ( c ∗ t − λH ∗ t ) dt (cid:21) . Taking into account the definition of H ∗ t = h ∨ sup s ≤ t c ∗ s , let us further defineˆ H t ( y ) := h ∨ (cid:18) λ − β ln (cid:20) − λ inf s ≤ t Y s ( y ) (cid:21)(cid:19) , (5.1)where Y t ( y ) = ye rt M t is the discounted martingale measure density process. For any admissiblestrategy ( π, c ) ∈ A ( x ), similar to the standard proof of Lemma 1 in [2], we have E (cid:20)Z ∞ c t M t dt (cid:21) ≤ x. (5.2)Regarding ( λ, h ) as fixed parameters, let us consider the dual transform of U with respect to c in the constrained domain that V ( y, h ) := sup ≤ c ≤ h [ U ( c − λh ) − cy ] = − β e λβh , if y ≥ e λβh , − β y + y ( 1 β ln y − λh ) , if e ( λ − βh < y < e λβh , − β e ( λ − βh − hy, if (1 − λ ) e ( λ − βh ≤ y ≤ e ( λ − βh . 26e remark that when λ = 0, V ( y, h ) is independent of h . Moreover, V ( y, h ) can be attained bythe construction of the feedback function c ∗ ( y, h ) given in (3.53).For any admissible ( π, c ) ∈ A ( x ), recall its resulting reference process H t = h ∨ sup s ≤ t c s , andfor all y > 0, we see that E (cid:20)Z ∞ e − rt U ( c t − λH t ) dt (cid:21) = E (cid:20)Z ∞ e − rt ( U ( c t − λH t ) − Y t ( y ) c t ) dt (cid:21) + y E (cid:20)Z ∞ c t M t dt (cid:21) ≤ E (cid:20)Z ∞ e − rt V ( Y t ( y ) , H ∗ t ) dt (cid:21) + yx (5.3)= E (cid:20)Z ∞ e − rt V ( Y t ( y ) , ˆ H t ( y )) dt (cid:21) + yx = v ( y, h ) + yx. where the second line follows by Lemma 5.3, the third line holds thanks to Lemma 5.2 below,and the last line is consequent on Lemma 5.1. In addition, in view of Lemma 5.3, the inequalitybecomes equality with the choice of c ∗ t = c ∗ t ( Y t ( y ∗ ) , H ∗ t ( y ∗ )), in which y ∗ is the unique solution tothe equation E (cid:2)R ∞ c ∗ ( Y t ( y ∗ ) , H ∗ t ( y ∗ )) M t dt (cid:3) = x for the given x > h ≥ ( π,c ) ∈A ( x ) E h Z ∞ e − rt U ( c t − λH t ) dt i = inf y> ( v ( y, h ) + yx ) = u ( x, h ) , which completes the proof of verification theorem.We then proceed to prove some auxiliary results that have been used to support the previousproof of the main theorem. Lemma 5.1. v ( y, h ) = E (cid:20)Z ∞ e − rt V ( Y t ( y ) , ˆ H t ( y )) dt (cid:21) . Proof. Note that the martingale measure density process M t satisfies the equation dM t = M t ( − rdt − κdW t ) . By (3.9) and (3.23), v ( y, h ) satisfies the ODE κ y v yy − rv + V ( y, h ) = 0 . By Itˆo’s formula, we have that d (cid:16) e − rt v ( Y t ( y ) , ˆ H t ( y )) (cid:17) = − e − rt V ( Y t ( y ) , ˆ H t ( y )) dt − κe − rt v y ( Y t ( y ) , ˆ H t ( y )) Y t ( y ) dW t + e − rt v h ( Y t ( y ) , ˆ H t ( y )) d ˆ H t ( y ) . Let us define the stopping time τ n := inf (cid:26) t ≥ (cid:12)(cid:12)(cid:12) Y t ( y ) ≥ n, ˆ H t ( y ) ≥ λ − β ln 1(1 − λ ) n (cid:27) . 27y integrating the above equation from 0 to T ∧ τ n and taking expectation on both sides, we havethat v ( y, h ) = E (cid:20)Z T ∧ τ n e − rt V ( Y t ( y ) ˆ H t ( y )) dt (cid:21) + E h e − r ( T ∧ τ n ) v ( Y T ∧ τ n ( y ) , ˆ H T ∧ τ n ( y )) i . (5.4)To wit, the integral term with respect to d ˆ H t ( y ) vanishes as ˆ H t ( y ) increases only if c ∗ t ( y ) = ˆ H t ( y )and we have v h ( Y t ( y ) , ˆ H t ( y )) = 0 by the free boundary condition. The expectation of the integralof dW t also vanishes as the local martingale Z T ∧ τ n κv y ( Y t ( y ) , ˆ H t ( y )) yM t dW t becomes a true martingale thanks to the definition of τ n and the fact that v is of class C .By passing to the limit as n → + ∞ , the first term in (5.4) tends to E hR T e − rt V ( Y t ( y ) ˆ H t ( y )) dt i by the monotone convergence theorem. Moreover, the second term in (5.4) can be written as E h e − r ( T ∧ τ n ) v ( Y T ∧ τ n ( y ) , ˆ H T ∧ τ n ( y )) i (5.5)= E h e − rT v ( Y T ( y ) , ˆ H T ( y )) { T ≤ τ n } i + E h e − rτ n v ( Y τ n ( y ) , ˆ H τ n ( y )) { T >τ n } i . As n → + ∞ , the first term in (5.5) clearly converges to E h e − rT v ( Y T ( y ) , ˆ H T ( y )) i . We will furthershow that the transversality condition holds in the sense that E h e − rT v ( Y T ( y ) , ˆ H T ( y )) i convergesto 0 as T → + ∞ in Lemma 5.4.We then claim that the second term in (5.5) also converges to 0 as T → + ∞ . To see this, itfollows by the definition of τ n that for all t ≤ τ n , we have inf s ≤ t Y s ( y ) ≥ n and Y t ( y ) ≤ n . Usingthe fact that when y is sufficiently large, v ( y, h ) is of order C ( h ) y r and Remark 3.2 gives that C ( h ) = O (cid:16) e ( λ − − r ) βh (cid:17) + O (cid:16) e [ λ (1 − r ) − ( r − r )] βh (cid:17) + O (cid:16) e λ (1 − r ) βh (cid:17) . We can then compare the order of v ( Y τ n ( y ) , ˆ H τ n ( y )) accordingly for the fixed τ n . First of all,we have Y t ( y ) r ≤ ( n ) r = n − r . Secondly, it is easy to see that O (cid:0) e ( λ − − r ) βh (cid:1) = O (cid:0) n r − (cid:1) , O (cid:0) e [ λ (1 − r ) − ( r − r )] βh (cid:1) = O (cid:18) n λ (1 − r − ( r − r − λ (cid:19) as well as O (cid:0) e λ (1 − r ) βh (cid:1) = O (cid:16) n λλ − (1 − r ) (cid:17) . Notethat all these three terms have an order smaller than O (1). Thirdly, similar to the proof of (A.25)in [12], we have that E [ { τ n ≤ T } ] ≤ n − κ (1 + y κ ) e CT , for any κ ≥ 1. Putting all pieces together, the desired claim holds thatlim T → + ∞ E h e − rτ n v ( Y τ n ( y ) , ˆ H τ n ( y )) { T >τ n } i = 0 . Lemma 5.2. E (cid:20)Z ∞ e − rt V ( Y t ( y ) , H ∗ t ) dt (cid:21) = E (cid:20)Z ∞ e − rt V ( Y t ( y ) , ˆ H t ( y )) dt (cid:21) . roof. The proof is similar to [12]. For the sake of completeness, we present the argument insketch. Suppose that H ∗ t is strictly increasing at t , the fact that H ∗ t = c ∗ t implies that the optimalconsumption is given by c ∗ t = λ − β ln( − λ Y t ( y )).Define I t := { s ≤ t : H ∗ is strictly increasing at s } . Then, for any s / ∈ I t , using the condition that Y s ( y ) ≤ (1 − λ ) e ( λ − βH ∗ s and the formula that c ∗ s = − β ln Y s ( y ) + λH ∗ s or c ∗ s = H ∗ s , we have that c ∗ s ≤ H ∗ s . Thus, we derive that H ∗ t = h ∨ sup s ∈I t c ∗ s = h ∨ sup s ∈I t λ − β ln (cid:18) − λ Y s ( y ) (cid:19) = h ∨ sup s ≤ t λ − β ln (cid:18) − λ Y s ( y ) (cid:19) = ˆ H t ( y ) . Lemma 5.3. The inequality (5.3) becomes an equality with the consumption control c ∗ t = c ∗ ( Y t ( y ∗ ) , ˆ H t ( y ∗ )) , t ≥ , with y ∗ = y ∗ ( x, h ) as the unique solution to E hR ∞ c ∗ ( Y t ( y ∗ ) , ˆ H t ( y ∗ )) M t dt i = x .Proof. The definition of V implies that for all ( π, c ) ∈ A ( x ), U ( c t − λH t ) − Y t ( y ) c t ≤ V ( Y t ( y ) , H t ).The inequality holds as an equality with the control c ∗ t . In other words, for any admissible ( c t ) ≤ t ≤ T ,we have that for all t ∈ [0 , T ], U ( c t − λH t ) − Y t ( y ) c t ≤ U ( c ∗ t − λH t ) − Y t ( y ) c ∗ t = V ( Y t ( y ) , H t ) . Multiplying both sides by e − rt and integrating from 0 to T , we have that Z ∞ e − rt ( U ( c t − λH t ) − Y t ( y ) c t ) dt ≤ Z ∞ e − rt V ( Y t ( y ) , H ∗ t ) dt. To turn (5.3) into an equality, the equality in (5.2) needs to be attained with some y to bedetermined later, and U ( c t − λH t ) − Y t ( y ) c t = V ( Y t ( y ) , H t ) (5.6)also needs to hold. Hence, we choose to employ c ∗ t ( y ) = c ∗ ( Y t ( y ) , ˆ H t ( y )) =: ˆ H t ( y ) F t ( y, Y t ( y )) , where we define F t ( y, z ) := { (1 − λ ) e − (1 − λ ) β ˆ Ht ( y ) ≤ z ≤ e − (1 − λ ) β ˆ Ht ( y ) } + ( − β ln z + λ ˆ H t ( y )) { e − (1 − λ ) β ˆ Ht ( y ) ≤ z ≤ e λβ ˆ Ht ( y ) } Note that the construction of c ∗ t ( y ) guarantees the validity of the equality (5.6).In view of the definition of ˆ H t ( y ) in (5.1), one can obtain that: (i) If y ↓ 0, then ˆ H t ( y ) ↑ + ∞ and F t ( y, Y t ( y )) > 0, which yields that E (cid:2)R ∞ M t c ∗ t ( y ) dt (cid:3) ↑ + ∞ ; (ii) If y ↑ + ∞ , then ˆ H t ( y ) ↓ h and F t ( y, Y t ( y )) ↓ 0, which yields that E (cid:2)R ∞ M t c ∗ t ( y ) dt (cid:3) ↓ 0. The existence of y ∗ satisfying the budgetconstraint (5.2) can be verified from the previous asymptotic behavior of ˆ H t ( y ) and F t ( y, Y t ( y )) bypassing to the limit y → y → + ∞ and the fact that E (cid:2)R ∞ M t c ∗ t ( y ) dt (cid:3) is continuous in thevariable y .We then prove the transversality condition, which is a key step in the proof of Lemma 5.1:29 emma 5.4. For all y > , the following transversality condition holds: lim T → + ∞ E h e − rT v ( Y T ( y ) , ˆ H T ( y )) i = 0 . Proof. Let us first recall thatˆ H t ( y ) = h ∨ (cid:18) λ − β ln (cid:20) − λ inf s ≤ t Y s ( y ) (cid:21)(cid:19) . From Proposition 3.2, in the interval e ( λ − βh < y < e λβh , which corresponds to the case0 < c t < H t , we have v ( y, h ) = C ( h ) y r + C ( h ) y r − yrβ + yrβ (cid:18) ln y − λβh + κ r (cid:19) . In the interval y ≥ e λβh , which corresponds to the case c t = 0, we have v ( y, h ) = C ( h ) y r − rβ e λβh , In the interval (1 − λ ) e ( λ − βh ≤ y ≤ e ( λ − βh , which corresponds to the case c t = h , we have v ( y, h ) = C ( h ) y r + C ( h ) y r − r hy − rβ e ( λ − βh , We first deal with the case 0 < c t < H t and check the asymptotic behavior of the followingexpectation E (cid:20) e − rT (cid:18) C ( ˆ H T ( y ))( Y T ( y )) r + C ( ˆ H T ( y ))( Y T ( y )) r − Y T ( y ) rβ + Y T ( y ) rβ (cid:18) ln Y T ( y ) − λβ ˆ H T ( y ) + κ r (cid:19)(cid:19)(cid:21) . We will consider its asymptotic behavior term by term.(i) Step 1 : Let us start by considering the asymptotic behavior of the third and fourth terms.For the third term, it is easy to see that E (cid:20) ye − ( r + κ ) T − κW T rβ (cid:21) = yrβ e − ( r + κ ) T E (cid:2) e − κW T (cid:3) = yrβ e − rT , (5.7)which converges to 0 as T → + ∞ .For the fourth term, we have that yM T rβ (cid:18) ln Y T ( y ) − λβ ˆ H T ( y ) + κ r (cid:19) = 1 rβ (cid:20) yM T (cid:18) rT + ln y + κ r (cid:19) + yM T ln M T − yM T λβ ˆ H T ( y ) (cid:21) . Similar to (5.7), we can show that E [ yM T ( rT + ln y + κ r )] converges to 0 and moreover we have E [ yM T ln M T ] = − ye − ( r + κ ) T (cid:18) E (cid:2) κW T e − κW T (cid:3) + (cid:18) r + κ (cid:19) T E (cid:2) e − κW T (cid:3)(cid:19) = − ye − rT (cid:18) r − κ (cid:19) T, which also converges to 0 as T → + ∞ . Furthermore, we can deduce that E [ yM T ˆ H T ( y )] ≤ E [ yM T h ] + E (cid:26) yM T λ − β ln (cid:20) − λ y inf s ≤ T ( e rs M s ) (cid:21)(cid:27) = O ( E [ yM T ]) + O e − rT E " e − κW T − κ T sup s ≤ T (cid:18) κW s + 12 κ s (cid:19) . T → + ∞ by repeating similar computations as above. As for thesecond term, we first know that E [ e − κW T sup s ≤ T W s ] = r T π − e κ T κT Φ( − κ √ T ) + e κ T κ h Φ( κ √ T ) − Φ( − κ √ T ) i . Let us define the equivalent measure Q under which W ( κ ) t := W t + κ t is a Brownian motion, withthe Randon-Nikodym derivative d Q d P (cid:12)(cid:12)(cid:12) F t := exp (cid:18) − κW t − κ t (cid:19) . It follows by Girsanov’s theorem that e − rT E " e − κW T − κ T sup s ≤ T (cid:18) κW s + 12 κ s (cid:19) = κe − rT E Q " e − κW ( κ T sup s ≤ T W ( κ ) s exp (cid:18) κW ( κ ) t − κ t (cid:19) = κe − rT (r T π e − κ T − κT Φ( − κ √ T ) + 1 κ (cid:20) Φ( 12 κ √ T ) − Φ( − κ √ T ) (cid:21)) , which clearly vanishes when T → + ∞ .(ii) Step 2 : Let us continue to consider the term with C ( h ). In view of the constraint Y T ( y ) 0, but we instead have that a + ζ = − κr + κ κ r ( − κ + 2 rκ > . 32e can show that the second upper bound ( a + b )( a + b +2 ζ )2 − r is strictly negative. FromCorollary A.7 of [12], the conditions under which this bound can be attained are a + b + ζ > a + b + 2 ζ > 0. Recall that κ > 0, we have that2 a + b + 2 ζ > ⇐⇒ − κr − κ (1 − r ) β + κ > ⇐⇒ β < − r − r . Now direct computations yield that( a + b )( a + b + 2 ζ ) − r = κ [ r + (1 − r ) β ][ r + (1 − r ) β − − r = κ (cid:8) r + (1 − r ) β + 2 r (1 − r ) β − r − (1 − r ) β (cid:9) − r = κ (cid:8) (1 − r ) β + 2 r (1 − r ) β − (1 − r ) β (cid:9) = κ (1 − r ) β [(1 − r ) β + 2 r − < . We can also derive that the upper bound ( a + b )( a + b +2 ζ )2 − r is strictly negative. Once againfrom Corollary A.7 of [12], the conditions under which this bound can be attained are a + b + ζ > a + b + 2 ζ > 0. As κ > 0, we equivalently need2 a + b + 2 ζ > ⇐⇒ − κr − κ λ (1 − r ) − ( r − r ) λ − β + κ > ⇐⇒ β < − r λ ∗ , where we define λ ∗ := ( r − r ) − λ (1 − r )1 − λ . Noting that λ ∗ > 0, we can show by straightforwardcomputations that ( a + b )( a + b + 2 ζ ) − r = κ ( r + λ ∗ β )( r + λ ∗ β − − r = κ ( r + λ ∗ β + 2 r λ ∗ β − r − λ ∗ β ) − r = κ ( λ ∗ β + 2 r λ ∗ β − λ ∗ β )= κ λ ∗ β ( λ ∗ β + 2 r − < . Putting all the pieces together, we conclude that Υ ( T ) → ( T ) → T → + ∞ .For the last term, we have that lim T → + ∞ E h e − rT ( Y T ( y )) r A c i = 0 from Lemma 5.5 below with a = − κr , b = 0 , η = rκr + κ .Let us now deal with the case c t = 0. Noting that in this case we need to calculate theorder of e − rT C ( ˆ H T ( y ))( Y T ( y )) r − e − rT rβ e λβ ˆ H T ( y ) when T → + ∞ , and C takes the order O (cid:0) e ( λ − − r ) βh (cid:1) + O (cid:0) e [ λ (1 − r ) − ( r − r )] βh (cid:1) + O (cid:0) e λ (1 − r ) βh (cid:1) . As the first two terms have beencalculated in the C term above, here we only need to focus on O (cid:0) e λβh (1 − r ) (cid:1) . By virtue of thecondition e λβ ˆ H T ( y ) < e rT yM T , it holds that e λβ ˆ H T ( y )(1 − r ) < ( e rT yM T ) − r . 33t follows that e − rT e λβ ˆ H T ( y )(1 − r ) ( e rT yM T ) r < yM T . The term e − rT rβ e λβ ˆ H T ( y ) is also bounded by rβ yM T using condition e λβ ˆ H T ( y ) < e rT yM T . It hasbeen shown that the expectation of the last term converges to 0 as T → + ∞ , which verifies theclaim in this case.We now turn to the proof of the case c t = h . Similar as before, we need to calculate theorder of e − rT h C ( ˆ H T ( y ))( Y T ( y )) r + C ( ˆ H T ( y ))( Y T ( y )) r − r ˆ H T ( y ) Y T ( y ) − rβ e ( λ − β ˆ H T ( y ) i when T → + ∞ . Remark 3.2 states that we have C ( h ) = O (cid:0) e λβh (1 − r ) (cid:1) + O (cid:0) e ( λ − − r ) βh (cid:1) and C ( h ) = O (cid:0) e ( λ − − r ) βh (cid:1) + O (cid:0) e [ λ (1 − r ) − ( r − r )] βh (cid:1) . Firstly, as C ( h ) has the same asymptotic behavior as C ( h ), the desired convergence result holds. For the term C ( h ), noting that its first term can bebounded using the same argument as the term C ( h ). For the second term, thanks to the condition e ( λ − β ˆ H T ( y ) ≤ Y T ( y ), we hence have that e ( λ − − r ) β ˆ H T ( y ) ≤ ( Y T ( y )) − r . Following the same computations for the term C ( h ), the desired result for C ( h ) also holds.The term e − rT rβ e ( λ − β ˆ H T ( y ) is first bounded in view of the condition e ( λ − β ˆ H T ( y ) < e rT yM T .Using similar arguments for the case c t = 0, we can obtain its convergence result. The last term r ˆ H T ( y ) Y T ( y ) term has already been handled in the proof for the case 0 < c t < h , which eventuallycompletes the whole proof.The following result has been used in the previous proof, which is essentially similar to CorollaryA.7 of [12]. We present it here for the completeness. Lemma 5.5. Let B ( ζ ) t = B t + ζt , where B is a standard Brownian motion, (cid:16) B ( ζ ) t (cid:17) ∗ be the runningmaximum of B ( ζ ) t . Then for any constant a, b, k with a + b + 2 ζ = 0 , k ≥ , we have E (cid:20) e aB ( ζ ) T + b (cid:16) B ( ζ ) T (cid:17) ∗ n(cid:16) B ( ζ ) T (cid:17) ∗ ≤ k o (cid:21) = 2( a + b + c )2 a + b + ζ exp (cid:26) ( a + b )( a + b + 2 ζ )2 T (cid:27) (cid:20) Φ (cid:16) ( a + b + ζ ) √ T (cid:17) − Φ (cid:18) ( a + b + ζ ) √ T − k √ T (cid:19)(cid:21) + 2( a + ζ )2 a + b + 2 ζ " exp (cid:26) a ( a + 2 ζ )2 T (cid:27) Φ (cid:16) − ( a + ζ ) √ T (cid:17) − exp (cid:26) (2 a + b + 2 ζ ) k + a ( a + 2 ζ )2 T (cid:27) Φ (cid:18) − ( a + ζ ) √ T − k √ T (cid:19) . In particular, we have that lim T → + ∞ E (cid:20) e aB ( ζ ) T + b (cid:16) B ( ζ ) T (cid:17) ∗ n(cid:16) B ( ζ ) T (cid:17) ∗ ≤ k o (cid:21) = 0 . At last, we turn to prove the existence of the unique strong solution to the SDE (3.34) for X ∗ t .First, we need to establish the following results concerning the regularity of the feedback functions c ∗ ( x, h ) and π ∗ ( x, h ).By the definition of g in (3.35) and the fact that f ( · , h ) is the inverse of g ( · , h ), we have thefollowing results of the function f . 34 emma 5.6. The function f is C within each of the subsets of R : x ≤ x ( h ) , x ( h ) < x < x ( h ) and x ( h ) ≤ x ≤ x ( h ) , and it is continuous at the boundary of x = x ( h ) and x = x ( h ) . Moreover,we have that: f x ( x, h ) = 1 g y ( f, h )= (cid:0) − C ( h ) r ( r − f ( x, h )) r − (cid:1) − , if x ≤ x ( h ) , − C ( h ) r ( r − f ( x, h )) r − − C ( h ) r ( r − f ( x, h )) r − − rβf ( x, h ) ! − , if x ( h ) < x < x ( h ) , (cid:18) − C ( h ) r ( f ( x, h )) r − − C ( h ) r ( f ( x, h )) r − + 1 r h (cid:19) − , if x ( h ) ≤ x ≤ x ( h ) , (5.8) and f h ( x, h ) = − g h ( f ( x, h ) , h ) · f x ( x, h ) . (5.9) Proof. The proof of the lemma is similar to lemma 6.1 of [10]. As the inverse of g , the function f satisfies that g ( f ( x, h ) , h ) = x, for ( x, h ) ∈ R . (5.10)By definition, the function g ( · , h ) and its inverse f ( · , h ) are C and decreasing, for any h > g in (3.35), we can calculate the partialderivative g h explicitly. As g h is clearly a continuous function in each of the closed intervals, it isbounded, i.e. ∃ a constant α > 0, such that g h ( x, h ) ≤ α, ∀ ( x, h ) ∈ R . Now in order to provethat f is C within each of the intervals x ≤ x ( h ), x ( h ) < x < x ( h ) and x ( h ) ≤ x ≤ x ( h ), wecan verify that f is differentiable in each variable with continuous partial derivative.First, let us prove that f ∈ C in each of the closed intervals, which implies that f x ∈ C ineach of the closed intervals. Indeed, for a pair ( x, h ) belonging to one of the intervals and a l smallenough, we have that g ( f ( x, h + l ) , h ) − x = g ( f ( x, h + l ) , h ) − g ( f ( x, h + l ) , h + l ) ≤ αl −→ l → . Now using the continuity of f ( · , h ), we obtain f ( x, h + l ) − f ( x, h ) = f ( g ( f ( x, h + l ) , h ) , h ) − f ( x, h ) −→ l → . Finally, for sufficiently small l , we have that f ( x + l , h + l ) − f ( x, h ) = f x ( x l , h + l ) l + f ( x, h + l ) − f ( x, h ) , which will tend to 0 when l , l tend to 0, and this shows that f is continuous at an arbitrary point( x, h ).Secondly, let us show that f is differentiable with respect to h with continuous partial derivatives.Let the pair ( x, h ) in a certain interval and l small enough such that ( x, h + l ) is in the same interval.We have that1 l { f ( x, h + l ) − f ( x, h ) } = 1 l { f ( x, h + l ) − f ( g ( f ( x, h ) , h + l ) , h + l ) } = f x ( x l , h + l ) 1 l { g ( f ( x, h ) , h ) − g ( f ( x, h ) , h + l ) } , x l ∈ [ x, x + g ( f ( x, h ) , h + l )]. Since f x ∈ C and g h ( f ( x, h ) , · ) is continuous, we obtain1 l { f ( x, h + l ) − f ( x, h ) } −→ l → − f x ( x, h ) g h ( f ( x, h ) , h ) , which indicates (5.9). Now the continuity of f h follows from (5.9) and the continuity of f . Lemma 5.7. The functions c ∗ and π ∗ are Lipschitz on C .Proof. By (3.53) and the dual fransformation relationship, we can express c ∗ ( x, h ) in terms of theprimal variables as in (3.58). Using Lemma 5.6 which implies the C regularity of f and (3.58),together with the continuity of f at the boundary between the three regions, we can draw theconclusion that c ∗ ( x, h ) is Lipschitz on C .Recall that from Proposition 3.2, the coefficients C ( h ), C ( h ), C ( h ), C ( h ) and C ( h ) are C in closed intervals and hence are Lipschitz. From Lemma 5.6, we get the Lipschitz property of f on C . Now using (3.59) in which π ∗ ( x, h ) is expressed in terms of the primal variables, we canconclude that π ∗ ( x, h ) is Lipschitz on C .We can proceed to prove the existence of strong solution with the optimal feedback. Proposition 5.1. The SDE (3.34) has a unique strong solution ( X ∗ t , H ∗ t ) for any initial condition ( x, h ) ∈ C .Proof. Let us introduce the functional G ( t, x ( t ) , h ( t )) := rx ( t ) + π ∗ t ( x ( t ) , h ( t ))( µ − r ) − c ∗ t ( x ( t ) , h ( t )) , and H ( t, x ( t ) , h ( t )) := π ∗ t ( x ( t ) , h ( t )) . By Lemma 5.7 which implies the Lipschitz property of c ∗ and π ∗ , we can easily derive that both G and H are Lipschitz functions. This justifies the existence of strong solution for the SDE (3.34). Proof of Proposition 3.1. It is easy to verify that the general solution of the equation (3.9) is givenby v ( y ) = C y r + C y r − rβ , if y ≥ ,C y r + C y r + yrβ (ln y + κ r − , if y < , (5.11)where C , C , C , C are some constants to be determined and r , are given in (3.13). Note thatwe have r > v y ( y ) → −∞ as y → C ≤ 0. Then v ( y ) − yv y ( y ) → y → C ≡ v y ( y ) → y → y for some y . It is obvious that y > x = 0 belongs to the region c ∗ ( x ) = 0. It thus follows that C r y r − = − C r y r − ,C (1 − r ) y r = − C (1 − r ) y r ,C r ( r − y r − = − C r ( r − y r − . C = 0 and C = 0, the first equation and the third equation lead to a obvious contradiction. If C = 0 and C = 0, we get another contradiction from the zero-order and first-order smooth-fittingcondition along the boundary y = 1. Noting r > r < 0, the only possible choice is y = + ∞ and C ≡ 0. By taking the inverse transform, we note that y = + ∞ and C = 0 actually implythe hidden boundary condition that u x → + ∞ as x → C = 0 and C = 0 back into the general solution (5.11) and using the smooth-fitting condition at y = 1, we get equations C − rβ = C − rβ + κ r β ,C r = C r + κ r β , which give constants C and C in (3.12). Proof of Proposition 3.2. We can first obtain the special solution v ( y, h ) = − rβ e λβh for the firstequation, v ( y, h ) = − yrβ + yrβ (ln y − λβh + κ r ) for the second equation, and v ( y, h ) = − r hy − rβ e ( λ − βh for the third equation in (3.23). Therefore, we can summarize the general solution ofthe ODE (3.23) by v ( y, h ) = C ( h ) y r + C ( h ) y r − rβ e λβh , if y ≥ e λβh ,C ( h ) y r + C ( h ) y r − yrβ + yrβ (cid:18) ln y − λβh + κ r (cid:19) , if e ( λ − βh < y < e λβh ,C ( h ) y r + C ( h ) y r − r hy − rβ e ( λ − βh , if (1 − λ ) e ( λ − βh ≤ y ≤ e ( λ − βh , (5.12)in which C i ( h ), i = 1 , ..., , are functions of h to be determined.By virtue of the explicit form of v ( y, h ) in (5.12) along the free boundary y = (1 − λ ) e ( λ − βh ,the condition v h ( y, h ) = 0 in (3.22) implies that C ′ ( h )(1 − λ ) r e ( λ − βhr + C ′ ( h )(1 − λ ) r e ( λ − βhr = (cid:16) r − rβ (cid:17) (1 − λ ) e ( λ − βh . (5.13)Similar to the case when λ = 0, the free boundary condition v y ( y, h ) → y → + ∞ . Together with free boundary conditions in (3.25) and the formula of v ( y, h ) in the region y ≥ e λβh , we deduce that C ( h ) ≡ 0. Moreover, it is easy to see that as h → + ∞ , we get y → − λ ) e ( λ − βh ≤ y ≤ e ( λ − βh and therefore the boundary conditions in (3.24)also implies the asymptotic condition that C ( h ) → h → + ∞ .To determine the left parameters, we apply the smooth-fit conditions with respect to the variable y at three boundary points y = e λβh and y = e ( λ − βh . After simple manipulations, we can deduce37he system of equations: C ( h ) e λβhr − rβ e λβh = C ( h ) e λβhr + C ( h ) e λβhr − rβ e λβh + 12 r β e λβh κ ,C ( h ) r e λβhr = C ( h ) r e λβhr + C ( h ) r e λβhr + 12 r β e λβh κ ,C ( h ) e ( λ − βhr + C ( h ) e ( λ − βhr + 12 r β e ( λ − βh κ = C ( h ) e ( λ − βhr + C ( h ) e ( λ − βhr − rβ e ( λ − βh ,C ( h ) r e ( λ − βhr + C ( h ) r e ( λ − βhr + 12 r β e ( λ − βh κ = C ( h ) r e ( λ − βhr + C ( h ) r e ( λ − βhr . (5.14)The system of equations above can be solved fully explicitly. To this end, the linear system canbe regarded as linear equations in terms of variables C ( h ), C ( h ) − C ( h ), C ( h ) − C ( h ) and C ( h ) − C ( h ). We can solve the first two equations and obtain C ( h ) explicitly in (3.28) and C ( h ) − C ( h ). By solving the last two equations, we also get C ( h ) − C ( h ), which yields C ( h ) in(3.30) by substituting the function C ( h ).Plugging the derivative C ′ ( h ) back into the boundary condition (5.13), we obtain that C ′ ( h )(1 − λ ) r e ( λ − βhr =(1 − λ ) r e ( λ − βhr ( r − κ r − r ) βr × h ( λ − − r ) e ( λ − − r ) βh − λ (1 − r ) e λ (1 − r ) βh i . By using the asymptotic condition that C ( h ) → h → + ∞ and λ (1 − r ) − ( r − r ) < C ( h ) explicitly in (3.31).Substituting C ( h ) back to( r − r )( C ( h ) − C ( h )) e ( λ − βhr = ( r − e ( λ − βh r β κ , we can get C ( h ) in (3.29). Substituting C ( h ) to the equation that( r − r )( C ( h ) − C ( h )) e λβhr = ( r − e λβh r β κ , we can at last obtain C ( h ) in (3.27). Proof of Lemma 3.1. We shall analyze each region separately.(i) In the region y ≥ e λβh , v yy ( y, h ) = r ( r − C ( h ) y r − , as r ( r − 1) = rκ > C ( h ) > − λ ) e ( λ − βh ≤ y ≤ e ( λ − βh , v yy ( y, h ) = r ( r − C ( h ) y r − + r ( r − C ( h ) y r − , the conclusion follows by the fact that C ( h ) > C ( h ) > r ( r − 1) = r ( r − 1) = rκ > 0. 38iii) In the region e ( λ − βh < y < e λβh , we can proceed in the following two steps: Step 1 : We can first equivalently check that yv yy ( y, h ) > e ( λ − βh and e λβh , i.e. 2 rκ y r − [ C ( h ) y r − r + C ( h )] + 1 βr > . Using the expression of C ( h ) and C ( h ), at the point e λβh this boils down to prove that e λβh ( r − r − r ) β ( e λβh (1 − r ) r − r + e ( λ − βh (1 − r ) − r r + ( r − − λ ) r − r r × (cid:20) − r − r e ( λ − − r ) βh − λ (1 − r ) λ (1 − r ) − ( r − r ) e [ λ (1 − r ) − ( r − r )] βh (cid:21) ) + 1 rβ > . Using the fact that e λβh (1 − r ) > e ( λ − βh (1 − r ) , the above is larger than e λβh ( r − r − r ) β n e λβh (1 − r ) r − r + e λβh (1 − r ) − r r o + 1 rβ − e λβh ( r − × (1 − r )(1 − λ ) r − r ( r − r ) βr (cid:20) − r − r e ( λ − − r ) βh − λ (1 − r ) λ (1 − r ) − ( r − r ) e [ λ (1 − r ) − ( r − r )] βh (cid:21) = − e λβh ( r − (1 − r )(1 − λ ) r − r ( r − r ) βr " − r − r e ( λ − − r ) βh − λ (1 − r ) λ (1 − r ) − ( r − r ) e [ λ (1 − r ) − ( r − r )] βh , which is strictly positive. Hence yv yy ( y, h ) > e λβh .For the point e ( λ − βh , similar as before, it is enough to show that e ( λ − βh ( r − r − r ) β ( e [ λ (1 − r ) − ( r − r )] βh r − r + e ( λ − βh (1 − r ) − r r + ( r − − λ ) r − r r × (cid:20) − r − r e ( λ − − r ) βh − λ (1 − r ) λ (1 − r ) − ( r − r ) e [ λ (1 − r ) − ( r − r )] βh (cid:21) ) + 1 rβ > . Using e [ λ (1 − r ) − ( r − r )] βh < e ( λ − βh (1 − r ) , similar calculation as at the point e λβh shows that theabove term is also strictly larger than − e λβh ( r − (1 − r )(1 − λ ) r − r ( r − r ) βr (cid:20) − r − r e ( λ − − r ) βh − λ (1 − r ) λ (1 − r ) − ( r − r ) e [ λ (1 − r ) − ( r − r )] βh (cid:21) > , and hence is strictly positive. Step 2 : In this step, we show that the function γ ( y ) := yv yy ( y, h ) = 2 rκ C ( h ) y r − + 2 rκ C ( h ) y r − + 1 rβ is either monotone or first increasing then decreasing. Combining with Step 1, this guaranteesthe statement of the lemma. Indeed, the extreme point y ∗ of γ ( y ) should satisfy the first ordercondition γ ′ ( y ∗ ) = 0, i.e. C ( h )( r − y ∗ ) r − r + C ( h )( r − 1) = 0 . 39e remark that C ( h ) < 0, while C ( h ) can be negative or positive. If C ( h ) ≤ 0, there is nosolution for y ∗ , hence γ ( y ) is monotone. If C ( h ) > 0, there exists a unique real solution to theabove equation y ∗ = (cid:18) C ( h )(1 − r ) C ( h )( r − (cid:19) r − r , which might fall into the interval [ e ( λ − βh , e λβh ]. Noticing that C ( h ) < 0, and γ ′ ( y ) = 2 rκ y r − (cid:0) C ( h )( r − y ) r − r + C ( h )( r − (cid:1) , it follows that when y ≤ y ∗ , γ ′ ( y ) ≥ 0; when y ≥ y ∗ , γ ′ ( y ) ≤ 0. Hence γ ( y ) increases beforereaching y ∗ , then decreases after passing y ∗ . Proof of Corollary 3.4. We first verify the conclusion for λ = 0. In view of c ∗ ( x ) in (3.16), itis sufficient to check lim x → + ∞ ln g ( x ) x . As g ( x ) satisfies the equation x = − C r ( g ( x )) r − − rβ (cid:16) ln g ( x ) + κ r (cid:17) and lim x → + ∞ g ( x ) = 0, it is easy to see that lim x → + ∞ ln g ( x ) x = − rβ and hencelim x → + ∞ c ∗ ( x ) x = r follows. By lim x → + ∞ g ( x ) = 0 again, we also get lim x → + ∞ π ∗ ( x ) = µ − rrβσ usingthe feedback form in (3.15).For the case λ > 0, as we consider the asymptotic behavior along the boundary x ( h ), we firsthave lim x → + ∞ , ( x,h ) ∈ x ( h ) c ∗ ( x, h ) x = lim h → + ∞ hx ( h ) . Taking into account the explicit form of x ( h ) in (3.43), we need to compute two limitslim h → + ∞ − C ( h ) r (1 − λ ) r − e ( λ − r − βh h = lim h → + ∞ − r (1 − λ ) r − (1 − r ) κ r − r ) βr [1 − e (1 − r ) βh ] h = 0 , and lim h → + ∞ − C ( h ) r (1 − λ ) r − e ( λ − r − βh h = lim h → + ∞ − r (1 − λ ) r − ( r − κ r − r ) βr h − r − r − λ (1 − r ) λ (1 − r ) − ( r − r ) e (1 − r ) βh i h = 0 . Therefore, we obtain that lim x → + ∞ , ( x,h ) ∈ x ( h ) c ∗ ( x, h ) x = r. Similarly, thanks to the explicit form of π ∗ ( x, h ) in (3.59), we need to compute two limits along x ( h ) thatlim h → + ∞ rκ C ( h )(1 − λ ) r − e ( λ − βh ( r − = lim h → + ∞ (1 − λ ) r − (1 − r )( r − r ) βr [1 − e (1 − r ) βh ]= (1 − λ ) r − (1 − r )( r − r ) βr , h → + ∞ rκ C ( h )(1 − λ ) r − e ( λ − r − βh = lim h → + ∞ (1 − λ ) r − ( r − r − r ) βr (cid:20) − r − r − λ (1 − r ) λ (1 − r ) − ( r − r ) e (1 − r ) βh (cid:21) = (1 − λ ) r − ( r − r − r ) βr . Therefore, we conclude thatlim x → + ∞ , ( x,h ) ∈ x ( h ) π ∗ ( x, h ) = µ − rσ (cid:18) (1 − λ ) r − (1 − r )( r − r ) βr + (1 − λ ) r − ( r − r − r ) βr (cid:19) = ( µ − r )(1 − λ ) r − rβσ . Acknowledgements : H. Pham and X. Yu appreciate the financial support by the PROCORE-France/Hong Kong Joint Research Scheme under no. F-PolyU501/17. X. Yu is partially supportedby the Hong Kong Early Career Scheme under grant no. 25302116. X. 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