The Merton problem, also known as continuous time optimal portfolio and consumption via utility maximization 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 PDE theories and stochastic analysis. The celebrated dynamic programming principle enables one to solve 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 research as they enjoy the merits of homogeneity and scaling property. In abundant work on terminal wealth optimization, the value function can be conjectured in some convenient separation forms or the change of variables can be exercised, the dimension reduction can consequently be applied to simplify the HJB equation. When intermediate consumption is taken into account, the study of exponential utility becomes relatively rare in the literature due to its unnatural allowance of negative consumption behavior. To be precise, as the exponential utility is defined on the whole real 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 work simply ignore the constraint or interpret the negative consumption as the infusion of funds, i.e., the negative consumption control is assigned by different financial meanings so that the non-negativity constraint 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 form involving the state price density process. As illustrated in [5] , the structure of the value function and 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 control process. In the present paper, we aim to revisit this problem under the exponential utility binding strictly with the constraint that the consumption rate must be non-negative. Moreover, unlike the time separable utility studied in [5] , our paper further attempts to go beyond the conventional preference and investigate the consumption behavior when a reference point is combined in the utility as well. In particular, our new preference essentially concerns how far the investor is away from the past consumption maximum level, and this intermediate gap is chosen as the metric to generate the utility of the investor in a dynamic way. Due to the consumption running maximum process inside the utility function, the martingale method developed in [5] can no longer handle our path-dependent optimization problem because it turns to be difficult to conjecture the valid dual processes and the associated dual problem.
Our problem formulation is mainly motivated by the psychological viewpoint that the consumer's satisfaction level and risk tolerance sometimes depend on recent changes instead of the absolute rate. Some large amount of expenditures, such as purchasing a car, a house or some luxury goods, not only spur some long term continuing spending for maintenance and repair, but also lift up the investor's standard of living gradually. A striking decline in future consumption plan may result in intolerable disappointment and discomfort. To depict the quantitative influence of the relative change towards the investor's preference, it makes good sense to introduce the utility to measure the distance between the consumption control and a proportion of the past consumption peak. On the other hand, during some economic recession periods such as recent global economy battered by Covid-19, it is unrealistic to mandate that the investor needs to catch up with the past spending maximum all the time. To capture the possibility that the investor may strategically decrease the consumption budget to fall below the benchmark so that more wealth can be accumulated from the financial market to meet future higher consumption plan, we choose to work with the exponential utility instead of Isoelastic utility that is defined on the positive real line. As a direct consequence, the investor can bear a negative gap between the consumption control and the reference level. The flexibility to compromise the consumption plan below the reference point from time to time makes the model suitable to accommodate more versatile market environments and mathematically unique and interesting.
Utility maximization with a reference point has become an important topic in the research of prospect theory and behavioral finance, see [21] , [15] , [14] and [13] on portfolio management with either a fixed or an adaptive reference level. Our paper differs from the previous work as we do not distinguish the utility on gain and loss separately and our reference level has path-dependent nature and is dynamically updated by the control itself. The impact of the reference to the past consumption maximum becomes highly implicit in our setting, which makes the problem appealing and challenging. On the other hand, our formulation is closely related to the so-called consumption habit formation preference, which measures the deviation of the consumption from the standard of living conventionally defined as the average of the accumulative consumption. See some previous work on addictive consumption habit formation in [4] , [8] , [20] , [19] , [11] , [24] , [25] , [23] and nonaddictive consumption habit formation in [7] . Recently, there are also some emerging research on the 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 different utility functions are equipped when the consumption is above the habit and when the consumption is below the habit. It will be an interesting future work for us to also consider this S-shaped utility defined on the difference between the consumption and the consumption peak reference level and investigate the structure of the optimal consumption. Among the aforementioned work, it is worth noting that [7] considers the utility defined on the whole real line and also permits the admissible consumption to fall below the habit level from time to time, namely the consumption habit is not addictive. [7] extends the martingale method in [5] by using the adjusted state price density process, which produces a nice construction of the optimal consumption in the complete market model. 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 control problem with consumption constraint can be solved under the umbrella of dynamic programming and PDE approach. The optimal consumption and portfolio can be obtained in piecewise feedback forms across different regions. Furthermore, all free boundary curves to separate these regions, albeit complicated, can be fully explicitly characterized. Comparing with Merton problem with exponential utility, our value function and feedback optimal controls have distinctive and more interesting features. On the other hand, in terms of the control problem and the associated HJB equation, 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 of the consumption rate and the consumption running maximum, and [2] and [1] considers an optimal consumption and dividend control problem respectively with a standard power utility and the drawdown constraint is only mandated on the control and does not appear in the utility. As opposed to [12] and [1] , our utility measures the difference between the control and its running maximum and the non-negativity constraint on consumption is actively imposed under the exponential utility function. 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 function and its associated nonlinear PDE problem. Despite of its complex structure and the non-negativity constraint on optimal control, it is revealed in the present paper that the existence of the classical solution to the associated HJB equation can be obtained in the analytic form with the aid of the dual transform, the smooth-fit principle and other novel arguments.
In summary, by noting that the consumption control is restricted between 0 and the peak level, we first heuristically derive the HJB equation in different forms based on the decomposition of the domain {(x, h) ∈ R + × R + } into disjoint regions of (x, h) such that the feedback optimal consumption satisfies (i)
To overcome the obstacle from nonlinearity, we apply the dual transformation only with respect to the state variable x and treat h as the parameter that is involved in some free boundary conditions. The linearized dual PDE in different regions can be handled as ODE problem with the parameter h. By using 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 us to express the value function, the feedback optimal investment and consumption in terms of the primal variables after the inverse transform. Unlike [2] , [12] and [1] , taking the weight parameter 0 < λ < 1 for instance, we can explicitly characterize the thresholds x 1 (h),x(h), x 2 (h) and x 3 (h) for the wealth variable x as sophisticated nonlinear functions of the variable h such that we can provide the feedback optimal consumption in the way that:
and the instant c * (x, h) creates a new historical maximum level when x = x 3 (h). Two extreme cases λ = 0 and λ = 1 are also discussed separately. In particular, λ = 0 corresponds to Merton problem with 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 need to apply the dual transform in a restricted domain. Moreover, we reveal an interesting observation that there is no need to consider the singular consumption that increases its running maximum process, which differs from the case 0 < λ < 1. At last, the complete proof of the verification theorem is rigorously established.
Building upon the explicit value function and the feedback optimal controls, some quantitative properties and numerical examples are presented. The impacts of the variable h and the reference weight parameter on the boundary curves x 1 (h),x(h), x 2 (h) and x 3 (h) can be numerically illustrated and the financial insights are observed. We also perform some sensitivity analysis on the value function, the optimal consumption and portfolio with respect to the reference weight parameter, the drift of the risky asset, the volatility of the risky asset and the risk aversion parameter respectively and conclude some interesting financial implications.
The remainder of the paper is organized as follows. Section 2 introduces the market model and formulates 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 fully explicit 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 the rigorous 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≥0 satisfies the usual conditions. We consider a financial market consisting of one riskless asset and one risky asset. The riskless asset price satisfies dB t = rB t dt where r ≥ 0 represents the constant interest rate. The risky asset price follows the dynamics
where W is an F-adapted Brownian motion and both the mean return µ and volatility σ > 0 are given constants. The sharp ratio parameter is denoted by κ := µ−r σ . It is worth noting that our mathematical arguments and all conclusions can be readily generalized to the model with multiple risky assets as long as the market is complete. For the sake of simple presentation, we shall only focus on the model with a single risky asset. It is assumed that κ > 0 from this point onwards, i.e. µ > r that the return of the risky asset is higher than the interest rate.
Let (π t ) t≥0 represent the dynamic amount that the investor allocates in the risky asset and (c t ) t≥0 denote the dynamic consumption rate of the investor. The resulting self-financing wealth process (X t ) t≥0 satisfies dX t = rX t dt + π t (µ − r)dt + π t σdW t − c t dt, t ≥ 0, with the initial wealth X 0 = x ≥ 0.
The consumption-portfolio pair (c, π) is said to be admissible, denoted by (c, π) ∈ A(x), if the consumption rate maintains non-negative, i.e. c t ≥ 0 a.s. for all t ≥ 0 and c is F-predictable and π is F-progressively measurable and both satisfy the integrability condition ∞ 0 (c t + π 2 t )dt < ∞ a.s. Moreover, no bankruptcy of the investor is allowed in the sense that X t ≥ 0 a.s. for t ≥ 0.
Let us focus on the exponential utility preference U (x) = − 1 β e −βx in the present paper with β > 0, x ∈ R. We are interested in the following infinite time utility maximization defined on the difference between the current consumption rate and its historical running maximum that
and the proportional constant 0 ≤ λ ≤ 1 depicts the intensity towards the reference level that the investor adheres to the past spending pattern. Moreover, as we are working in an infinite horizon framework with constant interest rate, it is assumed in this paper that the discount rate ρ = r to simplify some future computations. Here, H 0 = h ≥ 0 describes the reference level of the consumption that the individual aims to surpass at the initial time.
It is worth noting that the exponential utility function that we choose is defined on the whole real line, which implies that the current consumption rate is actually allowed to fall below the reference process H. To achieve the value function, it is not necessary for the optimal consumption control to exceed the reference level at any time. The lifetime average of the outperformance between the consumption and the reference level plays the key role. Meanwhile, as c t − λH t can be negative sometimes, the non-negativity constraint c t ≥ 0 a.s. is actively enforced for all time t ≥ 0 so that c t represents the consumption rate in the conventional sense. This control constraint spurs some new mathematical challenges when we handle the associated HJB equation using dynamic programming arguments in subsequent sections.
Remark 2.1. The problem (2.1) stems from some psychological consumption behavior that the investor sometimes can be very sensitive to the deviation from the past consumption pattern. In particular, the investor may have a strong memory of the past large amount of expenditures such as to purchase a large house or a fancy car, which may overturn the investor's living environment and future budget plan. In response to the psychological consistency on consumption stream, the investor can be prone to aggressively catch up with the past consumption peak to some extent, which motivates us to consider the preference that measures the distance between the current consumption choice 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 on consumption 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 is non-negative, [5] applied the martingale method and formulated the control problem with constraint into a relaxed form by introducing the Lagrange multipliers. By using the dual representation, the optimal non-negative consumption can be expressed in a technical integral form involving the unique state price density process. In the same complete market framework, this martingale method has been further refined by [7] to study the optimal consumption under non-addictive habit formation when the utility is generated by the difference between the non-negative consumption rate and the accumulative integral of past consumption control. By introducing the adjusted state price density process and the stochastic Lagrange multiplier process, the duality gap can be closed and the optimal non-negative consumption can be constructed and verified using the dual representation. Contrary to [5] and [7] , the presence of consumption running maximum inside the utility invalidates the martingale method because it becomes very complicated to construct the adjusted dual process and dual problem.
On the other hand, the problem (2.1) in the extreme case λ = 1 is related to the so-called ratcheting consumption behavior studied in the seminal paper [9] and several subsequent work. In [9] with power 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 function is 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 consumption in the sense that c t ≥ λ sup s≤t c s for some λ ∈ [0, 1]. Our present formulation differs substantially from [9] and [2] . The outperformed difference between the current consumption rate and a fraction of 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 ratcheting or drawdown constraint is no longer strictly enforced. The investor can strategically suppress the current consumption to some subsistence level that is below the reference level sometimes, which in turn may benefit the investor to attain a larger future consumption rate that is beyond the reference level with a higher probability and longer time periods.
To embed the control problem into the Markovian framework and derive the associated HJB equation using dynamic programming arguments, we treat both X t and H t as the controlled state processes given the control policy (c, π). The value function u(x, h) becomes two dimensional depending on variables x ≥ 0 and h ≥ 0, namely the initial wealth and the initial reference level for consumption. Let us consider the process Γ t := e −rt u(X t , H t ) + t 0 e −rs U (c s − λH s )ds.
(3.1)
The martingale optimality principle implies that (Γ t ) t≥0 is a local supermartingale under all admissible controls and (Γ t ) t≥0 is a local martingale given the optimal control (if it exists). If the function u(x, h) is smooth enough, by applying Itô's formula to the process (Γ t ) t≥0 , we can derive that
which heuristically leads to the associated HJB variational inequality
To guarantee the local martingale property of u(X * t , H * t ) under the optimal portfolio π * t and consumption control c *
In the present paper, we aim to find some deterministic functions π * (x, h) and c * (x, h) to provide the feedback form of the optimal portfolio and consumption strategy. To this end, if u(x, ·) is C 2 w.r.t the variable x, the first order condition gives the optimal portfolio in a feedback form by π * (x, h) = − µ−r σ 2 ux uxx . The previous HJB variational inequality (3.2) can first be written as
together with the free boundary condition u h = 0 on some set of (x, h) ∈ R + × R + that will be characterized later. To handle the control constraint 0 ≤ c ≤ h, we consider two extreme cases that λ = 0 and λ = 1 and the more interesting case 0 < λ < 1 respectively in the subsequent subsections.
To tackle the HJB equation (3.3), let us first consider the extreme case without the reference to its historical maximum, i.e. λ = 0. Recall that this case corresponds to the standard optimal consumption 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 manner based 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 be used later in the problem when λ > 0.
As λ = 0, the HJB variational inequality (3.3) can be simplified into a standard ODE problem without worrying about u h (x, h) ≤ 0. The first order condition without the non-negativity constraint gives the auxiliary feedback controlĉ(x) := − 1 β ln u x , and we need to distinguish two cases based on the value ofĉ(x) as below.
Region I: on the set {x ∈ R + : u x (x) ≥ 1}, we haveĉ(x) ≤ 0. The optimal consumption is therefore c * (x) = 0 and the ODE (3.3) is simplified to
Region II: on the set {x ∈ R + : u x (x) < 1}, we haveĉ(x) > 0. The optimal consumption is then c * (x) = − 1 β ln u x > 0 and the ODE (3.3) is written as
To guarantee the global regularity of the solution, we need to impose the smooth-fit condition along the free boundary {x ∈ R + : u x (x) = 1}. Moreover, with the aid of some boundary conditions at x = 0, we can actually determine its solution explicitly. To be precise, we observe that as the wealth level x declines to zero, the consumption rate c will first turn to zero at some point x * (to be determined later), then when x continues to tend to 0, the optimal investment π should also go to 0. Otherwise, we will confront the risk of bankruptcy by keeping trading with the nearly 0 wealth. Using the optimal portfolio π * (x) = − µ−r σ 2 ux uxx , the boundary condition becomes
In addition, note that if we start with 0 initial wealth, the wealth level will never change as there is no trading according to the previous condition, and the consumption rate should be 0 all the time consequently. Therefore, we can conclude that
On the other hand, as the wealth tends to infinite, one can consume as much as possible that leads to infinitely large admissible consumption rate and also a small variation in the wealth has the negligible effect on the change of the value function. It thus follows that lim x→+∞ u(x) = 0 and lim
To handle the nonlinear terms in the HJB equation (3.4) and (3.5), we employ the dual transform of the function u(x) that is defined by v(y) := sup x≥0 (u(x)−xy), y > 0. For the given x, we consider the variable y := u x (x) and it holds that u(x) = v(y) + xy. We can further deduce that
The nonlinear ODE (3.4) and (3.5) can be linearized as The next result gives the explicit solution to the dual ODE problem (3.9) and its proof is given in 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 explicitly by
where constants C 2 and C 3 are given by (3.12) in which the constants r 1 > 1 and r 2 < 0 are two roots of the algebraic equation
which are given by
By using the dual value function v(y) in Proposition 3.1, the optimal consumption and investment c * and π * can be expressed in terms of the dual value function and dual variable in the feedback form for y > 0 in the next main result.
Theorem 3.1. Let x ≥ 0 be the initial wealth. We consider the process Y t := y * e rt M t , where
where we used the fact that r 1 (r 1 − 1) = r 2 (r 2 − 1) = 2r κ 2 . Note that as it is assumed that µ > r, we always have π * (y) > 0 be the definition of C 2 and C 3 .
Actually, we can further rewrite the optimal controls in terms of the primal variables using the inverse 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 1 (x) and k 2 (x) depending the value of x. Using the dual relationship, we can obtain that
Therefore, the free boundary point x * dividing the two regions is given by
with C 2 given in (3.12) . For x > −C 2 r 2 , the function k 1 (x) is uniquely determined by the im-
is decreasing and lim y→0 G(y) = +∞ and lim y→1 G(y) = −C 3 r 1 − κ 2
The next result follows directly from Theorem 3.1 and the arguments above.
Corollary 3.1. For initial wealth x ≥ 0, we can express the value function and feedback optimal consumption and portfolio by:
(3.14)
The optimal strategy c * and π * can therefore be written in the feedback form using x ≥ 0 by
Here, we choose market parameters r = 0.05, µ = 0.1, σ = 0.25 and β = 1 and graph the value function u(x) in Figure- 3.3 The case 0 < λ < 1
We next consider the original control problem binding with the reference to the historical consumption peak when 0 < λ < 1 and the non-negativity constraint c t ≥ 0 is enforced. In view of the constraints that 0 ≤ c t ≤ H t , we first need to decompose the domain (x, h) ∈ R + × R + into three different regions such that the feedback optimal consumption strategy satisfies:
Let us denote the auxiliary controlĉ(x, h) := − 1 β ln u x + λh, which is simply derived by the first order condition in the HJB equation (3.3) . We need to separate the following regions:
Region I : on the set R 1 := (x, h) ∈ R + × R + : u x (x, h) ≥ e λβh , we haveĉ(x, h) ≤ 0, and therefore the optimal consumption rate is c * (x, h) = 0 and the HJB variational inequality becomes
Region II : on the set
we have that 0 < c(x, h) < h, and therefore the optimal consumption rate is c * = − 1 β ln u x +λh. The HJB variational inequality can be written as
Remark 3.1. As pointed out in Remark 2.1, the main reason for us to consider the exponential utility 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 consumption during the economic recession periods. Based on the feedback form of the optimal consumption c * = − 1 β ln u x + λh in Region II, we can characterize the domain of (x, h) such that the investor lowers the consumption rate below the reference, i.e.
This subset will be further characterized explicitly in Remark 3.4 as a threshold (depending on h) of the wealth level x.
Region III : on the set
t coincides with the running maximum process H * t . However, two subtle cases may occur that motivate us to split this region further.
(i) In a certain region (to be determined), the historical maximum level is already attained at some previous time s before time t and the current optimal consumption rate is either to revisit this maximum level from below or to sit on the same maximum level. This is the case that the running maximum process H t keeps flat from time s to time t, and the feedback form of c * t = H * t = c * s for some time s < t. In this case, it is very natural to treat H * t as the state 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 the maximum level that is strictly larger than its past consumption, and the running maximum process H t is strictly increasing at the instant t. This corresponds to the case that c * t = H * t is a 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
we have a contradiction thatĉ(x) < h, and therefore c * t is not a singular control. We still need to follow the previous feedback form c * (x, h) = h, in which h is a previously attained maximum level. The corresponding running maximum process remains flat at the instant time. In this region of (x, h), we only know that u h (x, h) ≤ 0 as we have dH t = 0. The HJB variational inequality is written as
This corresponds to the singular control c * t that creates a new peak for the whole path and
is strictly increasing at the instant time so that H * t > H * s for s < t and we must require the following free boundary condition that
(3.20)
In this region, it is noted that c
. Therefore, the HJB equation follows the same PDE (3.19) but together with the new free boundary condition (3.20) .
This indicates that the initial reference level h is below the feedback controlĉ(x), and the optimal consumption is again a singular control c * (x, h) = ) in the direction of h and X * t remains the same, in which u(x, h) is the solution of the HJB equation (3.19) on the set D 2 . This suggests that for any given initial value (x, h) in the set D 3 , the feedback control c * (x, h) pushes the value function jumping immediately to the point (x,ĥ) on the boundary set D 2 whereĥ = 1 β(λ−1) ln( ux(x,ĥ) 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 original stochastic control problem that
The only possibility for (x, h) ∈ D 3 = C c occurs at the initial time t = 0, and the value function is just equivalent to the value function of (x,ĥ) on the boundary D 2 with the same x. In other words, if the controlled process (X * 0 , H * 0 ) starts from (x, h) in the region C, then (X * t , H * t ) will always stay inside the region C and will either reflect at the boundary or move along the boundary D 2 whenever it hits the boundary D 2 (but will never go across the boundary). On the other hand, if the process (X * 0 , H * 0 ) starts from the value (x, h) inside the region D 3 , the optimal control enforces an instant jump (and the only jump) of the process H from H 0− = h to H 0 =ĥ on the boundary D 2 and both processes X t and H t become continuous processes diffusing inside the effective domain C afterwards for t > 0.
In addition, to ensure the desired global regularity of the solution, we also need to impose the smooth-fit conditions along two free boundaries of (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.
Similar to the case when λ = 0, we can again employ the dual transform of the value function to linearize the HJB equation. In particular, we choose the dual transform only with respect to the variable x and treat the variable h as a parameter. Let h ≥ 0 be fixed, we consider x ≥ 0 such that (x, h) ∈ C and define the dual function on the domain y
For the given (
In view of the free boundary condition (3.20) , we obtain the boundary condition
To align with nonlinear HJB variational inequality (3.17), (3.18), (3.19) in three different regions, the transformed dual variational inequality can be written as (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
By using the previous conditions, we can solve the dual ODE (3.23) fully explicitly and its proof is provided in Section 5.2. 
, if e (λ−1)βh < y < e λβh ,
where functions C 2 (h), C 3 (h), C 4 (h), C 5 (h) and C 6 (h) are given explicitly in (3.27), (3.28), (3.29), (3.30) and (3.31) respectively that
(3.28)
(3.29)
(3.31)
Here, constants r 1,2 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 asymptotic results of the coefficients that
which will be used in later proofs.
We can now present the main result of this paper, which provides the optimal investment and consumption in the feedback form explicitly using the dual variables for 0 < λ < 1. The complete proof is deferred to Section 5.1.
Let (x, h) ∈ C and 0 < λ < 1, where x is the initial wealth and h ≥ 0 is the initial reference level and C stands for the effective domain (3.21). We consider the process Y t := y * e rt M t , where M t := e −(r+ κ 2 2 )t−κWt is the discounted state price density process and
is the reference process under the optimal control, and the constant
2r
2r
In particular, the running maximum process H * t is strictly increasing such that
βh at the initial time, the optimal consumption creates a new peak and brings H * 0− = h jumping immediately to a higher level
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−1)βHt , 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−1)βHt , the optimal 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 given explicitly in terms of the dual value function and the dual variables. We can also conduct the inverse dual transformation and express the primal value function u(x, h) and the feedback controls in terms of x and h, albeit in more complicated forms. In the main body of the proof of Theorem 3.2, we will take full advantage of the simplicity in the dual feedback formulas and verify the optimality of the feedback controls using the duality relationship and some estimations based on the 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 dual transform becomes necessary, which will be established as follows.
By using the dual relationship between u and v, we have that the optimal
Defining f (·, h) as the inverse of g(·, h), we have that
Note 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 1 , f 2 and f 3 . By the definition of g in (3.35), the invertibility of the map x → g(x, h) is guaranteed by the following 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) > 0, ∀h > 0 and the inverse Legendre transform u(x, h) = inf y≥(1−λ)e −(1−λ)βh [v(y, h) + xy] is well defined. Moreover, this implies that the feedback optimal portfolio π * (y, h) > 0 always holds.
Using (3.35) and Proposition 3.2, the function f is implicitly determined in different regions by the following equations:
(3.37) (ii) If e (λ−1)βh < f 2 (x, h) < e λβh , Lemma 3.1 implies that v y (y, h) is strictly increasing in y and f 2 (x, h) can be uniquely determined by
is strictly increasing in y and f 3 (x, h) can be uniquely determined by
.
In region R 1 , we can obtain the explicit form of f 1 (x, h) = −x C 2 (h)r 2 1 r 2 −1 . The condition f 1 (x, h) ≥ e λβh gives us that this is valid when x ≤ x 1 (h), where we define the free boundary by
(3.40)
In region R 2 , the function f 2 is uniquely determined implicitly by (3.38) when
In view of the definition of f 2 (x, h) in (3.38), we can obtain the free boundary point explicitly as
Remark 3.4. In addition, as in Remark 3.1, we know that the optimal consumption falls below the reference level if and only if 1 < f 2 (x, h) < e λβh . Using (3.38) again, we can determine the critical pointx(h) byx
It then follows that if and only if the wealth level x is sufficiently small that satisfies x 1 (h) < x < x(h), the optimal consumption rate meets the compromised plan 0 < c * t (x, h) < λH * t (x, h).
In region D 1 ∪ D 2 , the expression of f 3 is uniquely defined implicitly by the equation (3.39) . This expression of f 3 holds when
It follows from (3.39) that the free boundary point x 3 (h) is explicitly given by Figure  4 , 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 investor would expect larger wealth thresholds to trigger the change of consumption from 0 to c * > 0 and from c * < H * to the historical maximum c * = H * . We also recall that we only consider the effective domain that is the region below the boundary curve x 3 (h) (including the boundary curve x 3 (h)). It is interesting to observe from Figure 5 that x 1 (h; λ) and x 2 (h; λ) are both decreasing in 0.01 ≤ λ ≤ 0.98, whilex(h; λ) and x 3 (h) are both increasing in 0.01 ≤ λ ≤ 0.98. That is, if the investor clings to a larger proportion of the past spending maximum, it is more likely that the investor 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 other hand, with a higher proportion λ towards the consumption peak, the investor needs to accumulate larger wealth to consume at the reference level c * = λH * t or consume at the peak to create a new historical maximum record that H * t = c * t > H * s for s < t. In particular, Figure 5 illustrates thatx(h; λ) is increasing in terms of λ, which indicates that if the investor adheres more to the past consumption peak with a larger proportion λ, it is more likely that the investor will suppress the optimal consumption rate c * t below λH * t due to the larger thresholdx(h; λ) for the wealth level. That is, the more the investor cares about the past consumption peak H * t , the more conservative the investor will become by comparing c * t and λH * t . This observation can partially explain the real life situations that the constantly aggressive consumption behavior may not lead to a long term happiness. A high consumption plan also creates a high level of psychological competition with the past pattern such that this aggressive consumption behavior may not be sustainable for the whole lifetime. A wise investor who takes into account the past reference will strategically lower the consumption rate from time to time (triggered by a wealth threshold) below the target reference such that the reference process can be maintained at a 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 following result, in which the primal value function u and optimal feedback controls are all given in terms of the primal variables x and h.
For (x, h) ∈ C and 0 < λ < 1, the value function u(x, h) of the control problem in (2.1) can be explicitly expressed in a piecewise manner by
where the free boundaries x 1 (h), x 2 (h) and x 3 (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 terms of primal variables (x, h) accordingly:
whereh(x) is given in (3.44), and
2r given by (3.42) .
Remark 3.5. In all referred work [2] , [12] and [1] , the domain of (x, h) is split into several regions by linear free boundaries such as ν 1 ≤ x h ≤ ν 2 for some constants ν 1,2 , in which different optimal consumption policies (or dividends) need to be followed. In contrary, our free boundary curves 
At last, we present some main results for the extreme case λ = 1. We separate this subsection from the previous case 0 < λ < 1 because there are some new and distinct features in the optimal feedback controls when λ = 1. Solving the HJB equation essentially follows the same arguments in the case 0 < λ < 1. However, the effective domain C defined in (3.21) needs to be modified to
(3.48)
Equivalently, C = R 1 ∪ R 2 ∪ R 3 = R 2 + , where R 1 , R 2 are defined the same as in the previous subsection for 0 < λ < 1.
In particular, we recall that R 1 = (x, h) ∈ R + × R + : u x (x, h) ≥ e βh , in which the optimal consumption c * t = 0. We also have R 2 = (x, h) ∈ R + × R + : 1 < u x (x, h) < e βh , in which the feedback optimal consumption c * t = − 1 β ln u x + h. As opposed to the case 0 < λ < 1, we now consider R 3 = {(x, h) ∈ R + × R + : 0 ≤ u x (x, h) ≤ 1} and note that the previous auxiliary singular controlĉ(x) = 1 β(λ−1) ln( ux 1−λ ) to further split the region R 3 for the case 0 < λ < 1 is no longer well defined when we have λ = 1.
In fact, in the extreme case λ = 1, there is no need to consider the singular optimal consumption that excesses the previous maximum level h. In the whole region R 3 , the optimal consumption is no longer unique, but one feedback optimal consumption c * (x, h) = h is to follow the previously attained maximum level, which is the initial level H * 0 = h and c * (x, h) ≤ h for any x ≥ 0. Comparing with other work [2] , [12] and [1] , this unique and interesting phenomenon that we can only focus on the optimal control such that H * t will never increase for λ = 1 results from the nature of the formulation U (c t − H t ) where the utility is defined on the difference. For the case 0 < λ < 1, the utility U (c t − λH t ) allows the investor to gain positive outperformance c t − λH t > 0 if he chooses a large c t to increase H t . On the other hand, for the case λ = 1, the investor can only obtain 0 = c t − H t by choosing to consume more than the past maximum. However, the investor can also easily achieve the same goal of zero difference c t − H t by following the previously attained maximum level without creating any new record. Therefore, to achieve the largest gap c t − H t = 0, one equivalent optimal way is to sit on the previous consumption peak and the investor has no incentives to switch to a singular control to increase the reference process H t at any time. Even if the initial wealth is sufficiently large, the investor will choose to consume the constant initial level h = H 0 such that c t = H t as long as it is sustainable but never to excess this reference level during the life time. Consequently, in this subsection, we shall only adopt the feedback control c * (x, h) = h in the region R 3 .
Based on the observations above, if the wealth x is larger than or equal to the subsistence level x ≥ x * := h r , the investor can always choose to invest zero amount π * t ≡ 0 in the risky asset and save the initial wealth x * in the bank account such that the interest rate can support the constant consumption at the initial reference level c * t = H 0 = h, t ≥ 0. That is, we have c * t − H * t = 0 for t ≥ 0. Note that x * = h r can also be obtained by x 3 (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) = − 1 rβ for x ≥ h r . That is, the primal 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 ≥ h r , which differs substantially from the case 0 < λ < 1. Therefore, we have the asymptotic conditions that which are completely different from the boundary condition (3.24) for 0 < λ < 1. Based on the same analysis in the case 0 < λ < 1, we can write down the linear dual ODE for the case λ = 1 as
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 → 0 in the third region, we can establish the next result. We can similarly present the result of the verification theorem when λ = 1 as below. 
The value function u(x, h) can be attained by employing 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 ≥ 0, which are given by:
2r
2r
Following the same inverse dual transform arguments in the previous subsection that u(x, h) = inf y>0 [v(y, h) + xy] for 0 ≤ x < h r and u(x, h) = − 1 rβ for x ≥ h r , we can also obtain the functions f i (x) = u x (x, h) in different regions that:
(3.55)
(ii)f 2 (x, h) that is uniquely determined by
We can conclude the corollary below on the value function and feedback optimal controls in the whole domain. 
where the free boundariesx 1 (h) andx 2 (h) are given explicitly in (3.55) and (3.57) respectively and f 2 (x, h) is given implicitly by (3.56). The feedback optimal consumption and portfolio are given by:
and At last, based on Corollaries 3.1, 3.2 and 3.3, we present the result of the asymptotic behavior of the optimal consumption-wealth ratio c * t X * t and the investment amount π * t when the wealth is sufficiently large and its proof is given in Section 5.2. x = r and lim x→+∞ π * (x) = µ−r rβσ 2 . For 0 < λ < 1, as x ≤ x 3 (h), we consider the asymptotic behavior along the boundary curve x 3 (h) as x, h → +∞, and we have
As the wealth level gets sufficiently large, the optimal consumption is asymptotically proportional to the wealth level that c * t ≈ rX * t and the optimal investment converges to a constant level that
That is, the investor will only allocate constant amount of wealth into the risky asset and save most of the wealth into the bank account. For λ = 1, the investor will stop the investment in the risky asset when the wealth exceeds the constant level h r and one optimal consumption is to constantly spend the initial reference amount c * t = h for t ≥ 0.
This section reports some numerical examples of sensitivity analysis using the previous explicit value 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 market parameters that r = 0.05, µ = 0.1, σ = 0.25, β = 1, λ = 0.5 and we plot the graphs for the variable h ∈ [0.5, 1.5] and x ∈ [0, 25]. We first perform the sensitivity analysis by plotting graphs of the value function, the feedback optimal consumption and the feedback optimal portfolio for different values of the reference weight parameter λ = 0.1, 0.2, ..., 0.8, 0.9. Here, we choose the market parameters that r = 0.05, µ = 0.1, σ = 0.25, β = 1 and fix the variable h = 1 and plot all graphs as functions of x for 0 ≤ x ≤ x 3 (1). From Figure 10 , for x 1 (1) < x < x 2 (1), we can see that the feedback consumption c * (x, 1) is increasing in x. More importantly, for the fixed 0 < x < x 3 (1), the feedback optimal consumption c * (x, 1; λ) is increasing in the parameter λ ∈ (0, 1], which is consistent with the intuition: the stronger that the investor adheres to the past consumption maximum level, the more likely that the investor will consume more during the life-cycle. We can observe from Figure 11 (see also Figure 14 , Figure 17 and Figure 20 ) that for x ≤ x 1 (1), the feedback π * (x, 1) is increasing and linear in x; and for x 1 (1) < x < x 2 (1), the feedback π * (x, 1) is increasing and concave in x; and for x 2 (1) ≤ x ≤ x 3 (1), the feedback π * (x, 1) is increasing and convex in x, Moreover, for the fixed 0 < x < x 3 (1), π * (x, 1; λ) is decreasing in the parameter λ ∈ (0, 1], which coincides with the observation in Figure 10 that the optimal consumption level is lifted up by a larger value of λ. The investor may strategically invest less in the market to save enough cash for higher consumption plan influenced by λ. From Figure 9 , for each 0 < x < x 3 (1), the graphs illustrate that the value function u(x, h; λ) is decreasing in λ ∈ (0, 1]. This suggests that even the optimal consumption rate increases because of the increase in λ, it does not necessarily imply that the value function also increases. In our preference formulation, the utility function is defined on the difference between the consumption rate c * t and the reference process λH * t . As both λ and c * t increase, the reference process λH * t increases as well. From Figure 9 , we can see that λH * t actually increases faster than the consumption rate c t when λ increases, which leads to a drop of the difference c * t − λH * t so that the resulting value function actually decreases.
We next present the impact of the drift parameter µ on the value function, the feedback optimal consumption and the feedback optimal portfolio by considering µ = 0.1, 0.12, 0.14, 0.16, 0.18. We again fix marker parameters r = 0.05, σ = 0.25, λ = 0.5, β = 1 and the maximum reference variable h = 1 and plot the graphs as functions of x for 0 ≤ x ≤ x 3 (1). From Figure 13 , we can see that the feedback function of the optimal consumption is increasing in terms of the drift parameter µ. This implies that if the market performance is getting better as the return of the risky asset increases, the investor can accumulate more wealth from the financial market to support a higher consumption plan. Likewise, Figure 14 illustrates that the investor's optimal portfolio in the financial market increases as the stock return increases. Figure 12 shows that the primal value function is increasing with respect to the drift parameter µ. It illustrates that when the return parameter µ increases, the increase in optimal consumption rate c * t dominates the increase in the running maximum process H * t so that the life time value function is lifted up. Similarly, for the market parameters r = 0.05, µ = 0.1, β = 1, λ = 0.5 and the fixed variable h = 1, we continue to present the sensitivity analysis of the value function u(x, 1), the feedback controls c * (x, 1) and π * (x, 1) with respect to different volatility parameters σ = 0.1, 0.2, 0.4, 0.6, 0.8. From Figure 16 , we observe that the monotonicity of the optimal consumption c * (x, 1) on the volatility σ is not guaranteed and the dependence becomes much subtle and complicated. We can also see that the consumption c * (x, 1) is not simply concave or convex in the variable x for the region x 1 (1) < x < x 2 (1), which depends on the volatility σ and other market parameters. In this example, we can observe that the threshold x 2 (1; σ) is increasing in σ but the dependence of the threshold x 1 (1; σ) on σ is unclear. Figure 15 and Figure 17 show that both the value function and the optimal portfolio are decreasing in the volatility σ. These graphs are consistent with the real life situation and are similar to some classical models such as the Merton problem that if the risky asset has a higher volatility, the investor will allocate less wealth in the risky asset and the value function on consumption also becomes lower. At last, for the market parameters r = 0.05, µ = 0.1, σ = 0.25, λ = 0.5 and the fixed h = 1, we plot the sensitivity analysis of the value function u(x, 1; β), the feedback controls c * (x, 1; β) and π * (x, 1; β) with respect to the risk aversion parameter β = 0.2, 0.5, 1, 2, 5 in the following figures for 0 ≤ x ≤ x 3 (1). We can see from Figure 19 that the threshold x 1 (1; β) is decreasing in the risk aversion parameter β and x 2 (h) is increasing in β. That is, for a more risk averse investor, it is more difficult to start a positive consumption c * > 0 but it becomes much easier to consume at the maximum level. However, the optimal consumption c * (x, 1; β) has a very complicated dependence on the parameter β. Within this numerical example, the optimal portfolio π * (x, 1; β) is still decreasing in β from Figure 20 , but the impact of β on the value function u(x, 1; β) is no longer monotone. Only when the wealth is sufficiently large, the value function behaves increasing with higher risk aversion β. 
In this subsection, we only provide the complete proof of Theorem 3.2 for the case 0 < λ < 1, which exhibits more structures in the feedback optimal controls. One can follow similar arguments to prove Theorem 3.1 and Theorem 3.3 and their proofs are omitted for the sake of length.
Proof of Theorem 3.2. The proof of the verification theorem boils down to show that the solution of the PDE indeed coincides with the value function, i.e. there exists (π * , c * ) ∈ A(x) such that
Taking into account the definition of H * t = h ∨ sup s≤t c * s , let us further definê
where Y t (y) = ye rt M t is the discounted martingale measure density process. For any admissible strategy (π, c) ∈ A(x), similar to the standard proof of Lemma 1 in [2] , we have
Regarding (λ, h) as fixed parameters, let us consider the dual transform of U with respect to c in the constrained domain that
We remark that when λ = 0, V (y, h) is independent of h. Moreover, V (y, h) can be attained by the 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 , and for all y > 0, we see that
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 inequality becomes equality with the choice of c * t = c * t (Y t (y * ), H * t (y * )), in which y * is the unique solution to the equation E ∞ 0 c * (Y t (y * ), H * t (y * ))M t dt = x for the given x > 0 and h ≥ 0. In conclusion, we arrive at
which completes the proof of verification theorem.
We then proceed to prove some auxiliary results that have been used to support the previous proof of the main theorem.
Proof. Note that the martingale measure density process M t satisfies the equation
By (3.9) and (3.23) , v(y, h) satisfies the ODE
By Itô's formula, we have that
Let us define the stopping time
By integrating the above equation from 0 to T ∧ τ n and taking expectation on both sides, we have that v(y, h) = E T ∧τn 0 e −rt V (Y t (y)Ĥ t (y))dt + E e −r(T ∧τn) v(Y T ∧τn (y),Ĥ T ∧τn (y)) .
To wit, the integral term with respect to dĤ t (y) vanishes asĤ t (y) increases only if c * t (y) =Ĥ t (y) and we have v h (Y t (y),Ĥ t (y)) = 0 by the free boundary condition. The expectation of the integral of dW t also vanishes as the local martingale T ∧τn 0 κv y (Y t (y),Ĥ 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 2 .
By passing to the limit as n → +∞, the first term in (5.4) tends to E T 0 e −rt V (Y t (y)Ĥ t (y))dt by the monotone convergence theorem. Moreover, the second term in (5.4) can be written as
As n → +∞, the first term in (5.5) clearly converges to E e −rT v(Y T (y),Ĥ T (y)) . We will further
show that the transversality condition holds in the sense that E e −rT v(Y T (y),Ĥ T (y)) converges to 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, it follows by the definition of τ n that for all t ≤ τ n , we have inf s≤t Y s (y) ≥ 1 n and Y t (y) ≤ n. Using the fact that when y is sufficiently large, v(y, h) is of order C 2 (h)y r 2 and Remark 3.2 gives that
We can then compare the order of v(Y τn (y),Ĥ τn (y)) accordingly for the fixed τ n . First of all, we have Y t (y) r 2 ≤ ( 1 n ) r 2 = n −r 2 . Secondly, it is easy to see that O e . Note that all these three terms have an order smaller than O (1). Thirdly, similar to the proof of (A.25) in [12] , we have that
for any κ ≥ 1. Putting all pieces together, the desired claim holds that lim T →+∞ E e −rτn v(Y τn (y),Ĥ τn (y))1 {T >τn} = 0.
Proof. The proof is similar to [12] . For the sake of completeness, we present the argument in sketch. Suppose that H * t is strictly increasing at t, the fact that H * t = c * t implies that the optimal consumption is given by c * t = 1 (λ−1)β ln( 1 1−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−1)βH * s and the formula that c * s = − 1 β ln Y s (y) + λH * s or c * s = H * s , we have that c * s ≤ H * s . Thus, we derive that 
The inequality holds as an equality with the control c * t . In other words, for any admissible (c t ) 0≤t≤T , we have that for all t ∈ [0, T ],
Multiplying both sides by e −rt and integrating from 0 to T , we have that
To turn (5.3) into an equality, the equality in (5.2) needs to be attained with some y to be determined later, and
also needs to hold. Hence, we choose to employ
where we define
Note that the construction of c * t (y) guarantees the validity of the equality (5.6). In view of the definition ofĤ t (y) in (5.1), one can obtain that: (i) If y ↓ 0, thenĤ t (y) ↑ +∞ and F t (y, Y t (y)) > 0, which yields that E ∞ 0 M t c * t (y)dt ↑ +∞; (ii) If y ↑ +∞, thenĤ t (y) ↓ h and F t (y, Y t (y)) ↓ 0, which yields that E ∞ 0 M t c * t (y)dt ↓ 0. The existence of y * satisfying the budget constraint (5.2) can be verified from the previous asymptotic behavior ofĤ t (y) and F t (y, Y t (y)) by passing to the limit y → 0 and y → +∞ and the fact that E ∞ 0 M t c * t (y)dt is continuous in the variable y.
We then prove the transversality condition, which is a key step in the proof of Lemma 5.1: Lemma 5.4. For all y > 0, the following transversality condition holds:
Proof. Let us first recall that
From Proposition 3.2, in the interval e (λ−1)βh < y < e λβh , which corresponds to the case
In the interval y ≥ e λβh , which corresponds to the case c t = 0, we have
In the interval (1 − λ)e (λ−1)βh ≤ y ≤ e (λ−1)βh , which corresponds to the case
We first deal with the case 0 < c t < H t and check the asymptotic behavior of the following expectation
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
which converges to 0 as T → +∞. For the fourth term, we have that
Similar to (5.7), we can show that E[yM T (rT + ln y + κ 2 2r )] converges to 0 and moreover we have
which also converges to 0 as T → +∞. Furthermore, we can deduce that
The first term clearly vanishes as T → +∞ by repeating similar computations as above. As for the second term, we first know that
Let us define the equivalent measure Q under which W ( κ 2 ) t := W t + κ 2 t is a Brownian motion, with the Randon-Nikodym derivative
It follows by Girsanov's theorem that
which clearly vanishes when T → +∞.
(ii)
Step 2 : Let us continue to consider the term with C 3 (h). In view of the constraint Y T (y) < e λβĤ T , we have
and it follows that E e −rT C 3 (Ĥ T (y))(Y T (y)) r 1
It is thus verified that the third term converges to 0 as T → +∞. (iii)
Step 3 : Now let us work with the term C 4 (Ĥ T (y))e −rT (Y T (y)) r 2 . Remark 3.2 asserts that
Let us define the set
and two auxiliary processes Using the formula ofĤ t defined in (5.1), we have that
in which we define
and the third terms comes fromh in the definition ofĤ t (y) with A c being the complementary set of A.
We then proceed to show that all three terms Υ 1 (T ) and Υ 2 (T ) and E e −rT (Y T (y)) r 2 1 A c converge to 0 as T → +∞.
Thanks to Corollary A.7 of [12] with
we obtain some upper bounds of the limit by
We now show that the above bounds are either negative or not attainable. For the first bound a 1 (a 1 +2ζ) 2 − r, direct calculations lead to
However, this zero bound can never be reached, as the corresponding condition of attaining the bound is a 1 + ζ < 0, but we instead have that
We can show that the second upper bound (a 1 +b 1 )(a 1 +b 1 +2ζ) 2 − r is strictly negative. From Corollary A.7 of [12] , the conditions under which this bound can be attained are a 1 + b 1 + ζ > 0, and 2a 1 + b 1 + 2ζ > 0. Recall that κ > 0, we have that
Now direct computations yield that
We can also derive that the upper bound (a 1 +b 2 )(a 1 +b 2 +2ζ) 2 − r is strictly negative. Once again from Corollary A.7 of [12] , the conditions under which this bound can be attained are a 1 +b 2 +ζ > 0, and 2a 1 + b 2 + 2ζ > 0. As κ > 0, we equivalently need
where we define λ * := (r 1 −r 2 )−λ(1−r 2 ) 1−λ . Noting that λ * > 0, we can show by straightforward computations that
Putting all the pieces together, we conclude that Υ 1 (T ) → 0 and Υ 2 (T ) → 0 as T → +∞. For the last term, we have that lim T →+∞ E e −rT (Y T (y)) r 2 1 A c = 0 from Lemma 5.5 below with a = −κr 2 , b = 0, η = r κr 2 + 1 2 κ. Let us now deal with the case c t = 0. Noting that in this case we need to calculate the order of e −rT C 2 (Ĥ T (y))(Y T (y)) r 2 − e −rT 1 rβ e λβĤ T (y) when T → +∞, and C 2 takes the order O e (λ−1)(1−r 2 )βh + O e [λ(1−r 2 )−(r 1 −r 2 )]βh + O e λ(1−r 2 )βh . As the first two terms have been calculated in the C 4 term above, here we only need to focus on O e λβh(1−r 2 ) . By virtue of the condition e λβĤ T (y) < e rT yM T , it holds that e λβĤ T (y)(1−r 2 ) < (e rT yM T ) 1−r 2 .
It follows that e −rT e λβĤ T (y)(1−r 2 ) (e rT yM T ) r 2 < yM T .
The term e −rT 1 rβ e λβĤ T (y) is also bounded by 1 rβ yM T using condition e λβĤ T (y) < e rT yM T . It has been shown that the expectation of the last term converges to 0 as T → +∞, which verifies the claim in this case.
We now turn to the proof of the case c t = h. Similar as before, we need to calculate the order of e −rT C 5 (Ĥ T (y))(Y T (y)) r 1 + C 6 (Ĥ T (y))(Y T (y)) r 2 − 1 rĤ T (y)Y T (y) − 1 rβ e (λ−1)βĤ T (y) when T → +∞. Remark 3.2 states that we have C 5 (h) = O e λβh(1−r 1 ) + O e (λ−1)(1−r 1 )βh and C 6 (h) = O e (λ−1)(1−r 2 )βh + O e [λ(1−r 2 )−(r 1 −r 2 )]βh . Firstly, as C 6 (h) has the same asymptotic behavior as C 4 (h), the desired convergence result holds. For the term C 5 (h), noting that its first term can be bounded using the same argument as the term C 3 (h). For the second term, thanks to the condition e (λ−1)βĤ T (y) ≤ Y T (y), we hence have that
Following the same computations for the term C 3 (h), the desired result for C 5 (h) also holds. The term e −rT 1 rβ e (λ−1)βĤ T (y) is first bounded in view of the condition e (λ−1)βĤ T (y) < e rT yM T . Using similar arguments for the case c t = 0, we can obtain its convergence result. The last term 1 rĤ T (y)Y T (y) term has already been handled in the proof for the case 0 < c t < h, which eventually completes the whole proof.
The following result has been used in the previous proof, which is essentially similar to Corollary A.7 of [12] . We present it here for the completeness. 
In particular, we have that
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 the following results of the function f . Lemma 5. 6 . The function f is C 1 within each of the subsets of R 2 + :
x ≤ x 1 (h), x 1 (h) < x < x 2 (h) and x 2 (h) ≤ x ≤ x 3 (h), and it is continuous at the boundary of x = x 2 (h) and x = x 3 (h). Moreover, we have that:
and
Proof. The proof of the lemma is similar to lemma 6.1 of [10] . As the inverse of g, the function f satisfies that
By definition, the function g(·, h) and its inverse f (·, h) are C 1 and decreasing, for any h > 0. Direct computation leads to (5.8) . From the definition of g in (3.35), we can calculate the partial derivative g h explicitly. As g h is clearly a continuous function in each of the closed intervals, it is bounded, i.e. ∃ a constant α > 0, such that g h (x, h) ≤ α, ∀(x, h) ∈ R 2 + . Now in order to prove that f is C 1 within each of the intervals x ≤ x 1 (h), x 1 (h) < x < x 2 (h) and x 2 (h) ≤ x ≤ x 3 (h), we can verify that f is differentiable in each variable with continuous partial derivative.
First, let us prove that f ∈ C 0 in each of the closed intervals, which implies that f x ∈ C 0 in each of the closed intervals. Indeed, for a pair (x, h) belonging to one of the intervals and a l 2 small enough, we have that
0. Now using the continuity of f (·, h), we obtain
Finally, for sufficiently small l 1 , we have that
which will tend to 0 when l 1 , l 2 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 that
Since f x ∈ C 0 and g h (f (x, h), ·) is continuous, we obtain
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 the primal variables as in (3.58). Using Lemma 5.6 which implies the C 1 regularity of f and (3.58), together with the continuity of f at the boundary between the three regions, we can draw the conclusion that c * (x, h) is Lipschitz on C.
Recall that from Proposition 3.2, the coefficients C 2 (h), C 3 (h), C 4 (h), C 5 (h) and C 6 (h) are C 1 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 can conclude 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
, 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 given by
where C 1 , C 2 , C 3 , C 4 are some constants to be determined and r 1,2 are given in (3.13) . Note that we have r 1 > 1 by a straightforward calculation. In view of the general solution in (5.11), the boundary condition v y (y) → −∞ as y → 0 first implies that C 4 ≤ 0. Then v(y) − yv y (y) → 0 as y → 0 further gives that C 4 ≡ 0.
Suppose the convergence v y (y) → 0 holds as y → y 0 for some y 0 . It is obvious that y 0 > 1 as x = 0 belongs to the region c * (x) = 0. It thus follows that
If C 1 = 0 and C 2 = 0, the first equation and the third equation lead to a obvious contradiction. If C 1 = 0 and C 2 = 0, we get another contradiction from the zero-order and first-order smooth-fitting condition along the boundary y = 1. Noting r 1 > 1 and r 2 < 0, the only possible choice is y 0 = +∞ and C 1 ≡ 0. By taking the inverse transform, we note that y 0 = +∞ and C 1 = 0 actually imply the hidden boundary condition that u x → +∞ as x → 0.
Substituting C 1 = 0 and C 4 = 0 back into the general solution (5.11) and using the smoothfitting condition at y = 1, we get equations
which give constants C 2 and C 3 in (3.12).
Proof of Proposition 3.2. We can first obtain the special solution v(y, h) = − 1 rβ e λβh for the first equation, v(y, h) = − y rβ + y rβ (ln y − λβh + κ 2 2r ) for the second equation, and v(y, h) = − 1 r hy − 1 rβ e (λ−1)βh for the third equation in (3.23) . Therefore, we can summarize the general solution of the ODE (3.23) by v(y, h) =
, if e (λ−1)βh < y < e λβh , C 5 (h)y r 1 + C 6 (h)y r 2 − 1 r hy − 1 rβ e (λ−1)βh , if (1 − λ)e (λ−1)βh ≤ y ≤ e (λ−1)βh , Similar to the case when λ = 0, the free boundary condition v y (y, h) → 0 in (3.25) implies that 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 1 (h) ≡ 0. Moreover, it is easy to see that as h → +∞, we get y → 0 in the third region (1 − λ)e (λ−1)βh ≤ y ≤ e (λ−1)βh and therefore the boundary conditions in (3.24) also implies the asymptotic condition that C 6 (h) → 0 as 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−1)βh . After simple manipulations, we can deduce the system of equations:
1 rβ e λβh = C 3 (h)e λβhr 1 + C 4 (h)e λβhr 2 − 1 rβ e λβh + 1 2r 2 β e λβh κ 2 , C 2 (h)r 2 e λβhr 2 = C 3 (h)r 1 e λβhr 1 + C 4 (h)r 2 e λβhr 2 + 1 2r 2 β e λβh κ 2 , C 3 (h)e (λ−1)βhr 1 + C 4 (h)e (λ−1)βhr 2 + 1 2r 2 β e (λ−1)βh κ 2 = C 5 (h)e (λ−1)βhr 1 + C 6 (h)e (λ−1)βhr 2 − 1 rβ e (λ−1)βh , C 3 (h)r 1 e (λ−1)βhr 1 + C 4 (h)r 2 e (λ−1)βhr 2 + 1 2r 2 β e (λ−1)βh κ 2 = C 5 (h)r 1 e (λ−1)βhr 1 + C 6 (h)r 2 e (λ−1)βhr 2 .
(5.14)
The system of equations above can be solved fully explicitly. To this end, the linear system can be regarded as linear equations in terms of variables C 3 (h), C 2 (h) − C 4 (h), C 4 (h) − C 6 (h) and C 3 (h) − C 5 (h). We can solve the first two equations and obtain C 3 (h) explicitly in (3.28) and C 2 (h) − C 4 (h). By solving the last two equations, we also get C 3 (h) − C 5 (h), which yields C 5 (h) in (3.30) by substituting the function C 3 (h).
Plugging the derivative C ′ 5 (h) back into the boundary condition (5.13), we obtain that C ′ 6 (h)(1 − λ) r 2 e (λ−1)βhr 2 =(1 − λ) r 1 e (λ−1)βhr 1 (r 2 − 1)κ 2 2(r 1 − r 2 )βr 2 × (λ − 1)(1 − r 1 )e (λ−1)(1−r 1 )βh − λ(1 − r 1 )e λ(1−r 1 )βh .
By using the asymptotic condition that C 6 (h) → 0 when h → +∞ and λ(1 − r 2 ) − (r 1 − r 2 ) < 0, we can integrate the equation above on both sides, and get C 6 (h) explicitly in (3.31). Substituting C 6 (h) back to (r 1 − r 2 )(C 6 (h) − C 4 (h))e (λ−1)βhr 2 = (r 1 − 1) e (λ−1)βh 2r 2 β κ 2 ,
we can get C 4 (h) in (3.29 ). Substituting C 4 (h) to the equation that (r 1 − r 2 )(C 2 (h) − C 4 (h))e λβhr 2 = (r 1 − 1) e λβh 2r 2 β κ 2 , we can at last obtain C 2 (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 2 (r 2 −1)C 2 (h)y r 2 −2 , as r 2 (r 2 −1) = 2r κ 2 > 0 and C 2 (h) > 0 from its expression (3.27), we draw the conclusion easily.
(ii) In the region (1 − λ)e (λ−1)βh ≤ y ≤ e (λ−1)βh , v yy (y, h) = r 1 (r 1 − 1)C 5 (h)y r 1 −2 + r 2 (r 2 − 1)C 6 (h)y r 2 −2 , the conclusion follows by the fact that C 5 (h) > 0, C 6 (h) > 0 and the identity that r 1 (r 1 − 1) = r 2 (r 2 − 1) = 2r κ 2 > 0.
(iii) In the region e (λ−1)βh < y < e λβh , we can proceed in the following two steps:
Step 1 : We can first equivalently check that yv yy (y, h) > 0 at the two endpoints e (λ−1)βh and e λβh , i.e. 2r κ 2 y r 2 −1 [C 3 (h)y r 1 −r 2 + C 4 (h)] + 1 βr > 0.
Using the expression of C 3 (h) and C 4 (h), at the point e λβh this boils down to prove that e λβh(r 2 −1) 1 (r 1 − r 2 )β e λβh(1−r 2 ) r 2 − 1 r + e (λ−1)βh(1−r 2 ) 1 − r 1 r + (r 2 − 1)(1 − λ) r 1 −r 2 r
Using the fact that e λβh(1−r 2 ) > e (λ−1)βh(1−r 2 ) , the above is larger than e λβh(r 2 −1) 1 (r 1 − r 2 )β e λβh(1−r 2 ) r 2 − 1 r + e λβh(1−r 2 ) 1 − r 1 r + 1 rβ − e λβh(r 2 −1)
which is strictly positive. Hence yv yy (y, h) > 0 at the point e λβh . For the point e (λ−1)βh , similar as before, it is enough to show that e (λ−1)βh(r 2 −1) 1 (r 1 − r 2 )β e [λ(1−r 2 )−(r 1 −r 2 )]βh r 2 − 1 r + e (λ−1)βh(1−r 2 ) 1 − r 1 r + (r 2 − 1)(1 − λ) r 1 −r 2 r × 1 − r 1 1 − r 2 e (λ−1)(1−r 2 )βh − λ(1 − r 1 ) λ(1 − r 2 ) − (r 1 − r 2 ) e [λ(1−r 2 )−(r 1 −r 2 )]βh + 1 rβ > 0.
Using e [λ(1−r 2 )−(r 1 −r 2 )]βh < e (λ−1)βh(1−r 2 ) , similar calculation as at the point e λβh shows that the above term is also strictly larger than −e λβh(r 2 −1) (1 − r 2 )(1 − λ) r 1 −r 2 (r 1 − r 2 )βr 1 − r 1 1 − r 2 e (λ−1)(1−r 2 )βh − λ(1 − r 1 ) λ(1 − r 2 ) − (r 1 − r 2 ) e [λ(1−r 2 )−(r 1 −r 2 )]βh > 0, and hence is strictly positive.
Step 2 : In this step, we show that the function γ(y) := yv yy (y, h) = 2r κ 2 C 3 (h)y r 1 −1 + 2r κ 2 C 4 (h)y r 2 −1 + 1 rβ is either monotone or first increasing then decreasing. Combining with Step 1, this guarantees the statement of the lemma. Indeed, the extreme point y * of γ(y) should satisfy the first order condition γ ′ (y * ) = 0, i.e. C 3 (h)(r 1 − 1)(y * ) r 1 −r 2 + C 4 (h)(r 2 − 1) = 0.
We remark that C 3 (h) < 0, while C 4 (h) can be negative or positive. If C 4 (h) ≤ 0, there is no solution for y * , hence γ(y) is monotone. If C 4 (h) > 0, there exists a unique real solution to the above equation
which might fall into the interval [e (λ−1)βh , e λβh ]. Noticing that C 3 (h) < 0, and γ ′ (y) = 2r κ 2 y r 2 −2 C 3 (h)(r 1 − 1)(y) r 1 −r 2 + C 4 (h)(r 2 − 1) , it follows that when y ≤ y * , γ ′ (y) ≥ 0; when y ≥ y * , γ ′ (y) ≤ 0. Hence γ(y) increases before reaching 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) , it is sufficient to check lim x→+∞ ln g 1 (x) x . As g 1 (x) satisfies the equation x = −C 3 r 1 (g 1 (x)) r 1 −1 − 1 rβ ln g 1 (x) + κ 2 2r and lim x→+∞ g 1 (x) = 0, it is easy to see that lim x→+∞ ln g 1 (x) x = −rβ and hence lim x→+∞ c * (x)
x = r follows. By lim x→+∞ g 1 (x) = 0 again, we also get lim x→+∞ π * (x) = µ−r rβσ 2 using the feedback form in (3.15) .
For the case λ > 0, as we consider the asymptotic behavior along the boundary x 3 (h), we first have Similarly, thanks to the explicit form of π * (x, h) in (3.59), we need to compute two limits along (1 − λ) r 1 −1 (r 2 − 1) (r 1 − r 2 )βr 1 − r 1 1 − r 2 − λ(1 − r 1 ) λ(1 − r 2 ) − (r 1 − r 2 ) e (1−r 1 )βh = (1 − λ) r 1 −1 (r 1 − 1) (r 1 − r 2 )βr .
Therefore, we conclude that lim x→+∞,