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Utility maximization with ratchet and drawdown constraints on consumption in incomplete semimartingale markets

In this paper, we study expected utility maximization under ratchet and drawdown constraints on consumption in a general incomplete semimartingale market using duality methods. The optimization is considered with respect to two parameters: the initial wealth and the essential lower bound on consumption process. In order to state the problem and define the primal domains, we introduce a natural extension of the notion of running maximum to arbitrary non-negative optional processes and study its properties. The dual domains for optimization are characterized in terms of solidity with respect to an ordering that is introduced on the set of non-negative optional processes. The abstract duality result we obtain for the optimization problem is used in order to derive a more detailed characterization of solutions in the complete market case.


Introduction
The first mathematical study of optimal investment and consumption of an agent with intolerance for any decline in the standard of living appears in Dybvig [1995]. The constraint that the consumption process is non-decreasing, also called ratcheting of consumption, can be seen as an extreme form of habit formation.
"The model in this paper is close to modern models of habit formation such as those of Constantinides (1990), Detemple and Zapareto (1991), Ingersoll (1992), Shrikhande (1992), or Sunderesan (1989. The main difference is in the rapidity (immediate) of the habit formation and the severity (lexicographic) of the agent's preferences for maintaining a new standard of living." Dybvig [1995, p. 289] Dybvig suggests that this model might be suitable for situations where consumption involves long-term commitments.
"For example, it may be a good model of a university or foundation that at least some of the expenditures cannot be decreased quickly, due to implicit and explicit long-term commitments to faculty and due to the commitments to donors about the use of buildings or equipment." Dybvig [1995, p. 288] By several changes of variables, a constraint that consumption can fall no faster than at some given rate (i.e., the constraint of the type c t ≥ e − t s αudu c s for t ≥ s ≥ 0, where α · is given) can be transformed into ratcheting constraint (see Section 2 in Dybvig [1995] for the case when α · is constant), thus generalizing Dybvig's model to a less rigid form of habit formation. Dybvig [1995] finds the optimal investment and consumption strategies for an infinitelylived ratchet investor with CRRA utility function in a market with underlying risky asset following the Geometric Brownian Motion (GBM) by considering the corresponding Hamilton-Jacobi-Bellman (HJB) equation. Riedel [2009] shows that, in a complete market with pricing kernel driven by a Lévy process, an optimal consumption plan of an infinitely-lived ratchet investor with an arbitrary utility function is in fact equal to the running maximum of the optimal consumption plan of an unconstrained investor with the same utility function. Riedel's proof is elementary in the sense that, essentially, it relies on a concavity argument and integrations by parts. Koo et al. [2012] generalize Dybvig's portfolio selection result under the same assumptions on the market to an arbitrary utility function by using duality and the Feynman-Kac formula. Watson and Scott [2014] extend Riedel's result to finite time horizon by introducing a deterministic function of time, called a coupling curve, that reflects the effects of finiteness of horizon in optimization and is described as a free boundary of a certain free-boundary problem. Jeon et al. [2018] study the finite horizon GMB market case with general utility function by transforming it into an infinite family of optimal stopping problems.
The working paper of Arun [2012] is the first to study optimal investment and consumption problem under a drawdown constraint on consumption. Under this condition, the consumption is not allowed to fall below a fixed proportion λ ∈ [0, 1] of the running maximum of past consumption. In particular, λ = 0 corresponds to the unconstrained problem and λ = 1 corresponds to consumption ratcheting. Arun [2012] finds the optimal portfolio and consumption in a GBM market over infinite time horizon for an agent with CRRA utility function by writing down the HJB equation and using duality and a modification of Dybvig's verification argument. Jeon and Park [2021] extend Arun's result to a general class of utility functions by deriving a dual problem consisting of only the choice of an optimal (non-decreasing) maximum consumption process, converting it into an infinite two-dimensional family of optimal stopping problems, and characterizing the solutions to the latter by a family of free boundaries depending on the state variable of the maximum process.
The current paper studies both the ratchet constraint and the drawdown constraint on consumption in general incomplete semimartingale markets by duality methods. Clearly, the ratchet constraint is a special case of the drawdown constraint, but in this paper the case λ = 1 is thought of as the basic one and the solution for λ ∈ (0, 1) is derived by, loosely speaking, an interpolation between the ratchet investor problem with λ = 1 and the unconstrained problem with λ = 0.
Our market model is taken from Mostovyi [2015] with an additional assumption that the stochastic clock is equivalent to the Lebesgue measure on [0,T ), whereT is either finite or infinite time horizon. Mostovyi [2015] provides a simple necessary and sufficient condition, namely, finiteness of the primal and dual value functions, for the key assertions of utility maximization theory to hold in an incomplete semimartingale market model with intermediate consumption, stochastic clock, and utility stochastic field. This approach parallels the work of Kramkov and Schachermayer [2003] on maximization of the expected value of deterministic utility from terminal wealth. A different but common approach to establish duality for expected utility maximization with time-and/or scenario-dependent utility is to require the utility to satisfy additional assumptions, including uniform reasonable asymptotic elasticity of Karatzas andŽitković [2003] (see alsoŽitković [2005], Bank and Kauppila [2017], Yu [2015]), a generalization of the reasonable asymptotic elasticity condition of Kramkov and Schachermayer [1999] for deterministic utility applied to terminal wealth.
The main difficulty in applying Mostovyi's result for the ratchet/drawdown constraint is identifying the appropriate primal and dual domains. We handle this issue by introducing a natural extension of the notion of the running maximum to arbitrary non-negative optional processes, which we call a running essential supremum, and defining the primal domain as the solid hull of all consumption plans satisfying the ratchet/drawdown constraint (formulated in terms running essential supremum) together with the budget constraint. The dual domains, defined as the polar sets of the primal domains in the sense of Brannath and Schachermayer [1999], are characterized via a family of orderings we introduce on the set of non-negative optional processes. The corresponding ordering for the ratchet constraint, which we call chronological ordering, implicitly appears in convex duality method for optimization over the set of increasing processes described in Bank and Kauppila [2017], however, the approach of the current paper (i) seems to be more direct, (ii) allows for generalization to the drawdown constraint with λ ∈ (0, 1), (iii) allows to add another parameter to optimization, an essential lower bound on the consumption process. This parameter is omnipresent in the literature on optimization under the ratchet/drawdown constraint, since, if the market is Markovian, adding this second parameter turns the problem into Markovian: all the information about the past consumption that is necessary for the future optimization is contained in the current running maximum which serves (up to multiplication by λ) as a lower bound for the future consumption. Based on Mostovyi [2015], we derive a duality result for the two-parameter optimization problem, where the parameters are the initial wealth and the essential lower bound on consumption. In this regard, the formulation and derivation of our main optimization re-sult (Theorem 35) is similar to the derivation of Hugonnier and Kramkov [2004] from the results of Schachermayer [1999, 2003], and to the work of Yu [2015].
In a complete market, where the set of equivalent martingale deflators is a singleton, the characterization of dual domains simplifies (Proposition 39) and allows for a more detailed description of the structure of optimizers. It turns out that in the drawdown constraint case the optimizers exhibit tree possible types of behavior: the agent consumes either at the minimal level currently allowed by the drawdown constraint, or at the current running essential supremum level, or as an unconstrained agent (i.e., not restricted by the drawdown constraint) with the same utility function but a different initial wealth. In the case of ratchet constraint, the description of optimizers is linked to the Bank-El Karoui Representation Theorem for stochastic processes and to the related notion of envelope process introduced in Bank and Kauppila [2017]. As a special case, we derive from Corollary 43 Riedel's formula for the optimal consumption plans in a complete Lévy market model with exponential time preferences and infinite time horizon.
The paper is organized as follows. In Section 2, we describe the market model, define the domains for consumption processes, and formulate the utility maximization problem. In Section 3, we define the running essential supremum process and study its properties. Section 4 describes the structure of the Brannath-Schachermayer polar sets of the consumption domains. Section 5 introduces the two-parameter families of primal and dual domains and proves the main optimization result, Theorem 35. In Section 6, we derive more specific results for the complete market case. In the Appendix, we give a modification of a lemma by Bank and Kauppila [2017] concerning existence and uniqueness of the envelope process allowing us to say more about the structure of optimizers for the ratchet constraint in a complete market and, in fact, give an alternative solution (Proposition 51), not relying on the main duality result, for this case.
2 Mathematical framework 2.1 The market model LetT ∈ (0, ∞] be a deterministic time horizon and denote I := [0,T ] ifT < ∞ and I := [0, ∞) ifT = ∞. We consider a market consisting of one numéraire asset (for example, a savings account) and d risky assets with discounted price process S = (S) t∈I given by an R d -valued semimartingale on a filtered probability space (Ω, F, (F t ) t∈I , P), where the filtration (F t ) t∈I satisfies the usual conditions. We denote the corresponding optional σ-algebra on Ω × [0,T ) by O.
A portfolio is defined as a triplet Π = (x, H, c) of a constant initial wealth x, a predictable R d -valued S-integrable process H = (H t ) t∈I of risky asset quantities, and a non-negative finite-valued optional process c : Ω × [0,T ) → [0, ∞) representing the consumption rate relative to κ. The discounted value process V = (V t ) t∈I of portfolio Π is (1) Remark 2. (i) Our market model is essentially the model of Mostovyi [2015] but with additional assumptions on the stochastic clock κ: it is absolutely continuous with respect to the Lebesgue measure and has support of the form I = [0,T ] ∩ [0, ∞). Clearly, ifT < ∞ then we can embed our model into the infinite horizon Mostovyi's model by assuming that S t = ST , κ t = κT , F t = FT for all t ∈ (T , ∞). This allows us to freely apply Mostovyi's duality results in what follows. The reason why we separate the finite and infinite time horizon cases is because an x-admissible consumption plan c satisfying the drawdown constraint on [0,T ] forT < ∞ generally cannot be extended to an x-admissible consumption plan on [0, ∞) still satisfying the drawdown constraint on [0, ∞): if we extend c by taking c t = 0 for t ≥T , the drawdown constraint fails after timeT . Hence, the drawdown constraints on consumption throughout [0, ∞) and on [0,T ] forT < ∞ are two different (though similar) conditions, and we have to separate their treatment. The results of this paper remain valid if we assumeT to be a [0, ∞]-valued stopping time with respect to a filtration (F t ) t∈[0,∞) satisfying the usual conditions (with the exception of supplementary results in the Appendix, which rely on the Representation Theorem of Bank and El Karoui [2004] requiring the stopping timeT to be predictable, according to their Remark 2.1). In this case, I becomes a stochastic interval and we assume all processes after timeT to be equal to their value at timeT whenT is finite.
(ii) Since we work in discounted units, V t in (1) is given in the number of numéraire assets, and so is the cumulative consumption process C t := t 0 c s dκ s , t ∈ I. The initial model of Mostovyi [2015] allows the flexibility of changingκ t toκ t /n t and c t to c t n t , where n t is a strictly positive optional process, so that C t remains the same and the uniform boundedness in Assumption 1 is satisfied for the new stochastic clock with densityκ t n t . In the utility maximization problem, (6) below, optimizing over c t can be reformulated as optimizing over c t n t by a simple change of utility function U , as described in [Mostovyi, 2015, Remark 2.2]. However, in our case, this flexibility is no more useful since c t is the consumption rate on which we want to impose the ratchet/drawdown constraint and hence is uniquely determined: multiplication by time-dependent process n t does not preserve this type of constraint. Thus, if we are imposing a ratchet/drawdown constraint on consumption rate measured in currency per unit time (with respect to the usual Lebesgue clock dt),κ t is uniquely determined from (1) through the numéraire asset: if the numéraire asset is N t := N 0 exp t 0 r s ds , where N 0 > 0 and an optional process r t represents an interest rate, then the appropriate stochastic clock density is given byκ t = 1/N t and Assumption 1 must be satisfied for this clock. On the other hand, if we are imposing a ratchet/drawdown constraint on discounted consumption rate (i.e., measured in number of numéraire assets per unit time with respect to dt) then the appropriate stochastic clock density isκ t ≡ 1. In particular, this stochastic clock satisfies Assumption 1 only ifT is finite (or uniformly bounded in case of stopping timeT ). If we want to impose a ratchet/drawdown constraint on consumption rate measured in a different asset, this can be done in a similar way as long as Assumption 1 for the corresponding clock κ is satisfied.
Next, we define the set of all non-negative value processes associated with portfolios of the form Π = (1, H, 0), X := X ≥ 0 : X t = 1 + t 0 H s dS s for t ∈ I , and the set of equivalent martingale deflators, Z := { (Z t ) t∈I > 0 : Z is a càdlàg martingale such that Z 0 = 1 and XZ = (X t Z t ) t∈I is a local martingale for every X ∈ X } .
We make the following assumption, which is related to the absence of arbitrage in the market.
IfT is finite and S is locally bounded then S is a local martingale under any probability measure Q ∼ P given by dQ dP = ZT with Z ∈ Z. And vice versa, if S is a local martingale under a probability measure Q ∼ P then Z t = dQ dP Ft , t ∈ I, belongs to Z. Hence in this case, by [Delbaen and Schachermayer, 1994, Corollary 1.2], Assumption 3 is equivalent to no free lunch with vanishing risk (NFLVR) condition on the market, a version of no arbitrage condition.
According to Mostovyi [2015], Lemma 4.2, a consumption plan c is x-admissible if and only if (2) We adopt the following simplifying notation for non-negative optional processes c and δ, so, in particular, the x-admissibility condition (2) can be written as sup Z∈Z c, Z ≤ x.
Remark 4. The results of this paper hold as well if the set Z of equivalent martingale deflators is replaced by a set Z is a càdlàg local martingale such that Z 0 = 1 and XZ = (X t Z t ) t∈I is a local martingale for every X ∈ X } of equivalent local martingale deflators. Clearly, Z ⊆ Z ′ and the assumption Z ′ = ∅ is weaker than Assumption 3. By Proposition 1 in Chau et al. [2017], assumption Z ′ = ∅ is equivalent to another version of no arbitrage condition -no unbounded profit with bounded risk (NUPBR). Under this assumption, a consumption plan c is x-admissible if and only if sup Z∈Z ′ c, Z ≤ x, analogously to (2) (see Lemma 1 in Chau et al. [2017]), which makes it possible to replace Z with Z ′ in the arguments that follow.

Domains for consumption processes
In order to be able to formulate the drawdown constraint on consumption rate in this generalized framework, it is necessary to have a meaningful extension of the notion of running maximum from continuous processes to processes that are just measurable. Such an extension, which we call a running essential supremum process, is introduced in Definition 8 and studied in Section 3. The running essential supremumc of a non-negative optional process c belongs (by Proposition 12) to the following class of stochastic processes: Moreover,c is the smallest element of C inc that is greater or equal than c, P × dκ−almost surely on Ω × [0,T ) (Propositions 13, 15). Now we can formulate the constraint and define the corresponding domains. Let q ∈ R and λ ∈ [0, 1]. We introduce the following drawdown constraint on consumption: where "∨" denotes the pointwise maximum of two processes (one of which is the constant process q in this case). Similarly, we reserve the notation "∧" for pointwise minimum of two processes. Clearly, when q ≤ 0, the constraint reduces to simply For x > 0, λ ∈ [0, 1], and q ∈ R, we define the primal domains as follows: and denote C λ (x) := C λ (x, 0), the domain for optimization without parameter q, and C λ := C λ (1) so that C λ (x) = x · C λ . The set C λ (x, q) is non-empty as long as the constant process The sets C λ (x, q) are solid: if c ′ ≥ c ≥ 0, P×dκ−a.e., and c ′ ∈ C λ (x, q) then c ∈ C λ (x, q). As the following proposition states, C λ (x, q) is in fact the solid hull of all x-admissible processes satisfying (DC λ,q ). We take definition (3) as our working definition of the domains, rather than the solid hull definition (4), because it provides a concrete way of checking whether a given process c belongs to C λ (x, q): by checking whether c ∨ λ(c ∨ q) is x-admissible.
For every x-admissible process c satisfying (DC λ,q ), c ∨ λ(c ∨ q) = c, P × dκ−a.e., hence c ∨ λ(c ∨ q) is x-admissible and c belongs to C λ (x, q) by definition. It is easy to check directly from Definition 8 that for every non-negative optional c the running essential supremum of the process c ∨ λ(c ∨ q) is equal toc ∨ λq½ (0,T ) and, therefore, c ∨ λ(c ∨ q) automatically satisfies (DC λ,q ). Therefore, if c ∈ C λ (x, q) then c ∨ λ(c ∨ q) is x-admissible and satisfies (DC λ,q ), hence c ∨ λ(c ∨ q) belongs to C λ (x, q) by the first assertion. To show (4), we take c ′ := c ∨ λ(c ∨ q) and use the first two assertions.
Proposition 6 implies that if a functional U on C λ (x, q) satisfies the monotonicity property and has a maximizer c on C λ (x, q), thenĉ := c ∨ λ(c ∨ q) ≥ c is also a maximizer and, in particular, it is a maximizer over all x-admissible processes satisfying (DC λ,q ).
If λ = 0 then the domains C λ (x, q) = C 0 (x) consist of all x-admissible consumption plans and this case is handled in Mostovyi [2015], even with more general assumptions on the stochastic clock κ. Therefore, we will only consider the optimization problem for λ ∈ (0, 1] in this paper. If λ = 1 then we can say slightly more: the constraint (DC λ,q ) turns into c t ≥c t ∨q, P×dκ−a.e., which, by Proposition 13, holds if and only if c =c ≥ q, P×dκ−a.e. The definition (3) for λ = 1 turns into and for a functional U on C 1 (x, q) satisfying (5) we obtain: if c ∈ C 1 (x, q) is an optimizer thenc ∨ q½ (0,T ) is also an optimizer and, in particular, it is an optimizer over all xadmissible processes c ′ ∈ C inc with c ′ 0+ ≥ q. This argument allows us to completely embed an optimization problem over the set of x-admissible processes c ′ ∈ C inc with c ′ 0+ ≥ q into an optimization problem over the subset C 1 (x, q) of non-negative optional processes. Finally, notice that the set C 1 (x, q) contains all x-admissible non-negative optional c such that for almost every ω ∈ Ω the path t → c t (ω) is non-decreasing and c 0+ (ω) ≥ q. This holds because c =c ∨ q, P × dκ−a.e., for such c. Hence, it is completely legitimate to look at optimization of U satisfying (5) over all x-admissible non-decreasing optional processes that are (essentially) bounded from below by q as optimization over C 1 (x, q).

Formulation of the optimization problem
be a utility stochastic field satisfying the following conditions (same conditions as in Mostovyi [2015]): Assumption 7. For every (ω, t) ∈ Ω×[0,T ) the function x → U (ω, t, x) is strictly concave, increasing, continuously differentiable on (0, ∞) and satisfies the Inada conditions: where U ′ denotes the partial derivative with respect to x. At x = 0 we define, by continuity, U (ω, t, 0) := lim x↓0 U (ω, t, x), this value may be −∞. For every x ≥ 0 the stochastic process U (·, ·, x) is optional.
We consider the problem where an agent maximizes his expected utility of intertemporal consumption c under the x-admissibility constraint (2) and the drawdown constraint (DC λ,q ). This problem therefore can be seen as the optimization over the set C λ (x, q) defined in (3). The associated value function is where an appropriate domain K ⊆ R 2 for (x, q) is to be specified later. Here we use the convention where W − denotes the negative part of a stochastic field W .
Our goal is to develop duality arguments, based on Mostovyi [2015], describing the solutions of the two-parameter optimization problem (6) in similar fashion as it is done in Hugonnier and Kramkov [2004] for expected utility maximization problem with random endowments at maturity.
3 Running essential supremum of a non-negative optional process In this section, we introduce a notion of running essential supremum. This is the appropriate generalization to measurable processes of what running maximum is for continuous processes. To the best of our knowledge, this process has not been previously considered in the literature.
Let c be a non-negative optional process on (Ω, F, measurable, every path of c is Borel measurable. Therefore, the following pathwise definition of running essential supremum of c makes sense.
Definition 8. For every ω ∈ Ω, we definē where the essential supremum is taken with respect to the Lebesgue measure. We callc the running essential supremum of c.
In the sequel, we often denote by the bar over the process its running essential supremum without stating it explicitly.
Remark 9. Letc t (ω) := sup s∈[0,t] c s (ω) for (ω, t) ∈ Ω × [0,T ). This is the smallest nondecreasing process satisfying c t (ω) ≤c t (ω) for all (ω, t) ∈ Ω × [0,T ). Clearly,c ≥c and if c ≥ λ(c∨q) then c satisfies (DC λ,q ). However, there is a reason why we considerc and notc in the formulation of the drawdown constraint (DC λ,q ). An optimizer for (6) is determined up to P × dκ−nullsets, therefore, our definition of the drawdown constraint should be indifferent to changes of c on P × dκ−nullsets. Hence, it would be natural to adopt a definition of running maximum which gives out the same process, up to indistinguishability, for any two processes that are equal P × dκ−a.e. This holds forc (Proposition 14) but not forc. It also seems reasonable from an economic perspective to have a definition of running maximum that is unaffected by a jump of consumption rate c at a single point, since this jump does not even affect the expected utility functional eventually.
We will give an alternative definition of the running essential supremum, which is equivalent and makes it easier to prove certain properties ofc. First, we introduce the notion of the essential debut of level l ≥ 0 by process c.

Essential debut
Let l ∈ [0, ∞). Define a stochastic process which is non-negative, non-decreasing, continuous, and adapted. Define the hitting time of (0, ∞) by the process X l , where by convention the infimum of an empty set isT . By construction, τ l can also be characterized as follows: where | · | denotes the Lebesgue measure of a set. That is, τ l is the essential debut (see Dellacherie and Meyer [1978], Chapter IV, p.108) of the set {(ω, t) : c t (ω) ≥ l}: given ω ∈ Ω, the set {s ∈ [0, t] : c s (ω) ≥ l} has a positive Lebesgue measure for every t ∈ (τ l (ω),T ), and τ l (ω) is the smallest time satisfying this property. Clearly, the essential debut τ l is no smaller than the hitting time inf{t ∈ [0,T ) : c t (ω) ≥ l}, but it could be strictly larger if the process c t does not spend a strictly positive amount of time (in the sense of the Lebesgue measure) at or above level l right after hitting it.
Proposition 10. For every l ≥ 0, τ l is a stopping time.
Proof. Since τ l is the hitting time of the open set (0, ∞) by the continuous adapted process X l and since the filtration (F t ) t∈[0,T ) is right-continuous, τ l is a stopping time (see, for example, Karatzas and Shreve [1991], p.7, 2.6 Problem).

Alternative characterization ofc t
In the sequel, we often omit writing ω in pathwise statements.
Proposition 11. The definition of the running essential supremumc t in Definition 8 is equivalent to the following: Note that the function l → τ l (ω) is non-decreasing for every ω ∈ Ω, hence Proposition 11 says that t →c t (ω) is a generalized inverse of the non-decreasing function l → τ l (ω).

Properties ofc t
Clearly, t →c t (ω) is non-decreasing. In the following two propositions we show that the processc is left-continuous, predictable, and c t (ω) ≤c t (ω) for every ω ∈ Ω, Lebesgue-a.e. in t.
Proposition 12. The running essential supremumc of a non-negative optional process c is left-continuous and predictable.
In particular, this proposition shows that t →c t (ω) is the left-continuous generalized inverse of the non-decreasing function l → τ l (ω) (cf. definition of the right-continuous generalized inverse, for example, in Karatzas and Shreve [1991], p.174, 4.5 Problem).
Proof. To prove left-continuity, let us fix a path ω ∈ Ω and t ∈ (0,T ). Since s →c s is nondecreasing, we need to show that for every ε > 0 there exists δ > 0 such thatc t−δ ≥c t − ε. Indeed, by the definition ofc t , To prove predictability, we write for every ν > 0 and a sequence of numbers ε n ↓ 0, The inclusion "⊆" in (7) holds because ifc t (ω) < ν then for n large enough (i.e., for ε n small enough) τ ν−εn (ω) ≥ t thanks to the characterization ofc t as the generalized inverse Note that for every ε > 0 the set {(ω, t) : τ ν−ε (ω) ≥ t} can be rewritten as Since τ ν−ε is a stopping time, the indicator function ½ (τ ν−ε ,T ) is measurable with respect to the predictable sigma-algebra. Hence, the right-hand side of (7) is a countable union of predictable sets, which implies that the set {c t (ω) < ν} is predictable for every ν > 0 and thereforec t is a predictable stochastic process.
Proposition 13. The running essential supremumc of a non-negative optional process c satisfies for every ω ∈ Ω the inequality Proof. Fix a path ω ∈ Ω. The statement would follow if we show that for every ε > 0 the set , where s k ∈ A ε and s k ↑ sup A ε as k → ∞, has measure zero, a contradiction. The fact that t 1 ∈ A ε and the definition ofc t 1 as the essential supremum imply that the set In the same way, we obtain that . Continuining this procedure analogously, after n steps we obtain n Lebesgue-negligible sets B 1 , ..., B n and a number t n ∈ (0,T ) such that . This cannot be repeated infinitely many times since the process c is non-negative and c t 1 < ∞, and the contradiction implies that |A ε | = 0.
By Proposition 12, the running essential supremumc of every non-negative optional process c belongs to C inc (see Definition 5). Note that since two left-continuous stochastic processes on (Ω, [0,T )) are equal P × Leb([0,T ))−a.e. if and only if they are indistinguishable, the same is true for any two processes in C inc . Moreover, the following property holds for running essential suprema: In particular, if c 1 = c 2 , P × Leb([0,T ))−a.e., thenc 1 andc 2 are indistinguishable. Proof.
proving the first assertion. The second assertion easily follows from the first.
The following proposition states thatc is the smallest element of C inc dominating c.
Proposition 15. If c is a non-negative optional process and Proof. Assume thatc t (ω) > c ′ t (ω) for some ω ∈ Ω and t ∈ (0,T ). Then, since c ′ is non-decreasing and by the definition of running essential supremum, Since c ≤ c ′ , P × Leb([0,T ))−a.e., by Fubini's theorem, the set of ω's satisfying (8) Note that if c(ω) has a jump at t, then dc t (ω)({t}) = c t+ − c t . This is the same random measure that comes from the non-decreasing càdlàg adapted process Integration with respect to non-decreasing càdlàg adapted processes, being a generalization of the Lebesgue-Stieltjes integration to stochastic processes, is a classical topic in the general theory of stochastic processes (see, for example, Dellacherie and Meyer [1982], Chapter VI, no. 51-58) and in the next proposition we list several properties of integration with respect to dc t = dc ′ t that will be extensively used in this paper. Throughout the paper, we adopt the following conventions: By the Optional Section Theorem, for optional processes D 1,2 this condition is equivalent to D 1 where o D denotes the optional projection of D.

For every
3. If two optional processes D 1,2 : P−a.e., for every stopping time T ∈ [0,T ], then Proof. Item 1 is proved in Dellacherie and Meyer [1982], Chapter VI, no. 57. Item 2 follows from and Fubini's theorem. Item 3 follows since the Optional Section Theorem implies that D 1 ≤ D 2 up to indistinguishability.
We will also need the following definition and lemma on several occasions.
Definition 17. For c ∈ C inc and ω ∈ Ω, we write dc t (ω) > 0 (or simply dc t > 0) and call Lemma 18. For a non-negative optional right-continuous process D and for c ∈ C inc , the following are equivalent: , which is only possible on a P−nullset by 1.
"2. ⇒ 1." For l ≥ 0, define a stopping time T l := inf{t ∈ [0,T ) : c t > l}, where the infimum of an empty set is taken to beT . This is indeed a stopping time by the right-continuity of the filtration. Since c is left-continuous and non-decreasing, c T l ≤ l < c s for any s ∈ (T l ,T ), hence T l is eitherT , or a point of increase of c. Therefore, by 2., D T l ½ {T l <T } = 0 for every l ≥ 0, P−almost surely. The time-change formula (55.1) in Dellacherie and Meyer [1982], Chapter VI, when applied to the càdlàg version c ′ t of c t , states that Since the right-hand side is zero, it follows that D t = 0, P × dc t −almost everywhere.
To prove that C λ is closed, let (c n ) n≥1 ⊆ C λ converge to c almost everywhere. After replacing c n byc n := inf k≥n c k ∈ C λ and using Proposition 14, we can assume without loss of generality that the sequence c n increases to c pointwise on Ω × [0,T ). Then it is easy to check directly from Definition 8 thatc n ↑c pointwise, therefore c n ∨ λc n ↑ c ∨ λc pointwise as well, and, by monotone convergence applied to the characterization (2) of x-admissiblity, c ∨ λc is 1-admissible, implying c ∈ C λ .

Chronological ordering
The following ordering will be crucial in our definition and characterization of the dual domains. This ordering appears implicitly in Bank and Kauppila [2017] through their definition (11). Borrowing this idea, we solve the singular control problem of Bank and Kauppila [2017] type, corresponding to λ = 1 and q = 0 in our framework, in what seems to be a more direct way (i.e., with a more straightforward definition of the dual domain and of the dual optimization problem; see Remark 36(iii)) and extend the method to constraints given by parameters λ ∈ (0, 1) and q > 0.
Remark 21. (i) By the Optional Section Theorem and by the definition of optional projection, condition (12) is equivalent to saying that for every stopping time T ∈ [0,T ], Intuitively, all the processes obtained from δ by "moving some of the mass of δdκ to earlier times", possibly removing some of the mass, and taking the optional projection afterwards, are -dominated by δ, hence the name "chronological". This is best seen in the deterministic case, i.e., on a trivial filtered probability space, when the definition of turns into comparison of mass ofδdκ and δdκ on each of the intervals (t,T ) for t ∈ [0,T ). The most subtle application of this intuition of moving mass to earlier in time is in Step 1 of the proof of Lemma 27 below.
The following is the remarkable property of the chronological ordering: Proposition 22. For two non-negative optional processesδ and δ, In the informal language from above, the most important message of this proposition reduces to the obvious statement that "moving some of the mass of δdκ to earlier times can only make the integral of an increasing process with respect to δdκ smaller".
Proof. To show "⇐", take c := ½ (T,T ) for every stopping time T and apply the equivalent characterization (14) of . For "⇒", when c is finite-valued, we have An arbitrary c ∈ C inc can be approximated by finite-valued c ∧ N ∈ C inc as N → ∞ and the right-hand side of (15) follows by monotone convergence.
Now we introduce an ordering λ that will be used to characterize dual domains for λ < 1.
Informally,δ λ δ if, up to taking optional projections,δ is obtained from δ by leaving some mass of δdκ where it is, "moving some of its mass to earlier time" while also multiplying it by λ (i.e., the (1 − λ) fraction of mass gets lost), and removing the rest.
An alternative characterization of λ is given in the following proposition.
Hence,δ λ δ if and only if the conditions of Definition 23 hold with δ 1 =δ 1 =δ ∧ δ, i.e., in the case when we choose not to move all the mass that is possible not to move if we want to obtainδ from δ.
The following proposition is an extension of Proposition 22 to λ .

An important lemma
Lemma 27. For all λ ∈ [0, 1] and c ≥ 0 optional, the following holds: Informally, in order to achieve the value of c,δ close to the value on the left-hand side of (18), we obtainδdκ by splitting the mass of δdκ into two parts: (i) if c t ≥ λc t then we leave the mass of δdκ at time t where it is (i.e., the corresponding part of δ goes to δ 1 of Definition 23), (ii) if c t < λc t then we move the mass of δdκ at time t (i.e., the corresponding part of δ goes to δ 2 of Definition 23) to an earlier time where c is close to its current running essential supremumc t , with the penalty of (1 − λ) fraction of the mass moved.
Step 1. λ = 1 and c = ½ A , where A ∈ O. Since c ∨c =c, P × dκ−a.e., by Proposition 13, we are proving that c, δ ≤ supδ δ c,δ for every optional process δ ≥ 0. Let τ A be the essential debut of A. Then τ A is a stopping time by Proposition 10 applied with l = 1, andc t = ½ (τ A ,T ) (t) by the definition ofc.
Comparing the first and the last terms in the above sequence of equalities and inequalities shows that "≤" in (18) holds for c 1 + c 2 .
Step 4. Monotone convergence. Let (c n ) n≥1 be an increasing sequence of non-negative optional processes such that c n ↑ c and c n ∨ λc n , δ ≤ sup δ λ δ c n ,δ for every n.
Combining the results of Steps 2 and 3 shows that the conclusion of the lemma holds for all λ ∈ [0, 1] in case when c is a non-negative simple O-measurable function on Ω × [0,T ).
Step 4 allows to conclude the proof for an arbitrary non-negative optional process c.
Remark 29 (Monotonicity in λ). By the definition (3) of primal domains, where C 0 is the set of all x-admissible consumption plans. This reflects the fact that the drawdown constraint gets more restrictive as λ ∈ [0, 1] increases. The dual domains D λ , being the Brannath-Schachermayer duals of C λ , have the reverse monotonicity property: where, according to Proposition 28, D 0 is the smallest solid, convex, and closed subset of L 0 + (Ω × [0,T ), O, P × dκ) containing Z. Proposition 30. For every λ ∈ [0, 1], the following holds: Proof. For a fixed λ, let us define Clearly, Z ⊆ D ′ . On the other hand, the set D ′ inherits the properties of D λ : it is convex, it is closed by Fatou's lemma, and it is λ -solid, since it is easy to check that E T 0δ dκ ≤ E T 0 δdκ forδ λ δ. By minimality of D λ established in Proposition 28, we must have D ′ = D λ and therefore sup δ∈D λ E T 0 δdκ = α.
Using Ito's formula for semimartingales and localization, one can show that E T 0 Zdκ = E[Z T κ T ] ≤ A for every finite T ≤T , Z ∈ Z, and for constant A from Assumption 1. Sending T → ∞ ifT = ∞ and taking T =T otherwise, we conclude that α ≤ A < ∞.

The optimization problem
In this section, we describe the solution of the optimization problem (6) using convex duality methods. We fix λ ∈ (0, 1] and for brevity omit mentioning the dependence on λ explicitly, wherever it is possible. That is, we omit λ from notations C λ , D λ , C λ (x), D λ (y), C λ (x, q). The domain K for parameters (x, q), the domain L for dual variables (y, r), dual domains D(y, r), as well as the value function u from (6), the dual value function v defined in (29), and the primal and dual optimizers, all implicitly depend on λ ∈ (0, 1]. Note that the definition (3) of the primal domains C(x, q) (i.e., C λ (x, q)) can be reformulated in terms of C(x) (i.e., C λ (x)) as follows: without any reference to the drawdown constraint, only to the essential lower bound on consumption. In fact, all of the statements in this section (except Propositions 32 and 33) hold for arbitrary subsets C, D, and C(x, q) defined as in (22), of L 0 + (Ω × [0,T ), O, P × dκ) satisfying the following three conditions: 1. The sets C and D are polar to each other in the sense of Brannath and Schachermayer [1999].
2. The set D contains an element that is strictly positive P × dκ−almost everywhere.
All of these conditions hold for C λ and D λ , λ ∈ (0, 1], the second one being satisfied since Z ⊆ D λ . Additionally, this means that the arguments of this section can be adopted in order to introduce, with the definition analogous to (22), essential lower bound on consumption in the usual unconstrained model of Mostovyi [2015]. Some trivial but useful properties of the domains (C(x, q)) (x,q)∈K and (D(y, r)) (y,r)∈L * are summarized in the following remark.

Dual relations between domains
Remark 31. (i) We have C(x, q) = C(x) for q ≤ 0. For all (x, q) ∈ K, the set C(x, q) is solid and contains the constant process c ≡ x/α > 0. For x > 0 and q 1 < q 2 < x/(αλ), we have C(x, q 2 ) ⊆ C(x, q 1 ), i.e., the sets C(x, q) decrease as q increases.
The following is a simple sufficient condition for an element of D(y) in order to be an element of D(y, r). It will be used in Section 6 to characterize the optimizers for (6) in a complete market.
Proposition 33. Letδ λ δ ∈ D(y) andδ is strictly positive on a set of positive measure (i.e., excluding the trivial caseδ = 0, P × dκ−a.e.). Thenδ ∈ D(y, r) for Proof. First, since (δ − δ) ∨ 0 λ(δ −δ) ∨ 0 by Proposition 25, r ≤ 0. Second, and the strict inequality holds between the first and the last terms becauseδ > 0 on a set of positive measure. Hence, (y, r) ∈ L * so that D(y, r) is well-defined. According to (23), it is enough to test the defining property of D(y, r) only for (x, q) ∈ K, c ∈ C(x, q) with where the last inequality follows from (c ∨ q) − q ∈ C inc and (δ − δ) ∨ 0 λ(δ −δ) ∨ 0. Rearranging the terms and using c ∈ C(x), δ ∈ D(y), we obtain c,δ ≤ c, δ +qr ≤ xy+qr, completing the proof. Now we prove duality relations between the families (C(x, q)) (x,q)∈K and (D(y, r)) (y,r)∈L * . The next proposition is an analogue of Proposition 1 in Hugonnier and Kramkov [2004] for our setting and is crucial for reducing, similarly to Hugonnier and Kramkov [2004], the two-dimensional optimization problem to one dimension in the proof of Theorem 35 below.
(ii) For every (y, r) ∈ L * , D(y, r) contains a strictly positive process. For a non-negative optional process δ, the following holds: Proof. We show (ii) first. The implication "⇒" in (25) holds by the definition of D(y, r). Assuming the right-hand side of (25) and substituting q = 0 implies that δ ∈ D(y), therefore the implication "⇐" also follows from the definition of D(y, r). To show that D(y, r) contains a stictly positive process, we will find an ε > 0 such that D(ε) ⊆ D(y, r).
Since Z ⊆ D, the claim then follows from strict positivity of equivalent martingale deflators Z ∈ Z. A sufficient condition for D(ε) ⊆ D(y, r) is εx ≤ xy + qr, ∀(x, q) ∈ K, since then for every δ ∈ D(ε) and c ∈ C(x, q) ⊆ C(x), c, δ ≤ εx ≤ xy + qr. Equivalently, ε ≤ y + (q/x)r for all (x, q) ∈ K. By the definition of the open cone K this means that ε ≤ y + r/(αλ). Finally, by the definition of L * , y + r/(αλ) > 0, therefore such an ε > 0 exists. Now we show (i). The constant consumption plan c ≡ x/α belongs to C(x, q) and is strictly positive. The implication "⇒" in (24) follows from the definition of D(y, r). To show "⇐" when q ≤ 0, we conclude that c ∈ C(x) = C(x, q) by testing the right-hand side of (24) with δ ∈ D = D(1, 0). It remains to prove "⇐" for the case q > 0. Let us fix δ ∈ D.
According to (22), we want to show that the right-hand side of (24) implies c ∨ λq, δ ≤ x or, equivalently, The idea of the proof of this inequality below is to show that δ½ {c≥λq} ∈ D(1, r) for an appropriate r and then apply the right-hand side of (24) to this process.

The main duality result for optimization
Following Mostovyi [2015] (and the standard duality theory), we introduce the conjugate stochastic field V to U : and denote by I(ω, t, y) := −V ′ (ω, t, y), the derivative of −V with respect to y. It is well-known that −V satisfies Assumption 7 and I(ω, t, ·) = (U ′ ) −1 (ω, t, ·).
We define the dual optimization problem for (6) through the value function v(y, r) := inf δ∈D(y,r) where the convention is used and W + denotes the positive part of a stochastic field W . We often omit writing down the dependence of U , U ′ , V , and I on ω and t in what follows.
The following theorem describes the duality between (6) and (29), and establishes existence and uniqueness of optimizers under Mostovyi's finiteness assumptions for value functions u and v.
Then we have 1. u(x, q) < ∞ for all (x, q) ∈ K, v(y, r) > −∞ for all (y, r) ∈ L * . The functions u and v are conjugate: 2. The subdifferential of u maps K into L * : ∂u(x, q) ⊆ L * for all (x, q) ∈ K.
Remark 36. (i) Since each of the sets C(x, q), (x, q) ∈ K, contains the strictly positive constant consumption plan c ≡ x/α, the condition that u(x, q) > −∞ for all (x, q) (ii) It follows from the proof of Proposition 34(ii) that for all (y, r) ∈ L * the set D(y, r) contains D(ε) for ε > 0 small enough. Hence, the condition that v(y, r) < ∞ for all (y, r) ∈ L * is equivalent to v(y, 0) < ∞ for all y > 0.
The infimum on the right-hand side is the dual value function of the unconstrained problem, when both the drawdown and the essential lower bound constraints are dropped (v(ε), in the notation of Mostovyi [2015]). Therefore, the condition that v(y, r) < ∞ for all (y, r) ∈ L * is satisfied as long as the dual value function of the unconstrained problem is finite (from above), in particular, it is always true if the unconstrained problem satisfies the assumptions of Mostovyi [2015], Theorem 2.3.
(iii) Under Assumptions 1, 3, 7 and assuming u(x, 0) > −∞ and v(y, 0) < ∞ for all x, y > 0 (in particular, under assumptions of Theorem 35) we can apply [Mostovyi, 2015, Theorem 3.2] to the sets C := C λ and D := D λ for λ ∈ (0, 1] and thus solve the initial problem (6) in case q = 0. We use this observation in the proof of Proposition 40 below. In particular, for the case λ = 1, Theorem 3.2 in Mostovyi [2015] applied to the sets C 1 and D 1 is an analogue of Theorem 3.2 in Bank and Kauppila [2017].
The following two lemmas are analogues of Lemmas 11 and 12 in Hugonnier and Kramkov [2004] for our setting and will help us prove Theorem 35.
then, for every x > 0, where cl(E) denotes the closure of E with respect to convergence in measure P × dκ.
The proof below follows very closely the proof of Lemma 11 in Hugonnier and Kramkov [2004], except a small adjustment due to the stochastic utility and working on the product space Ω × [0,T ).
Fix x > 0 and c ∈ xcl(E). Let (c n ) n≥1 be a sequence in xE that converges P × dκ−a.e. to c. For any ε > 0, we have where the first inequality holds true because U is increasing, the second one follows from Fatou's lemma, since U (t, c n t + c ε t ) → U (t, c t + c ε t ) almost surely and all the terms are bounded from below by the integrable process −U − (t, c ε t ), and the third one follows from the fact that E is convex and therefore c n + c ε ∈ (x + ε)E for n ≥ 1. Since φ is concave, it is continuous. It follows that Lemma 38. Let (y n , r n ) ∈ L * and δ n ∈ D(y n , r n ), n ≥ 1, converge to (y, r) and to an optional process δ ≥ 0, respectively. If δ > 0, P × dκ−a.e., then (y, r) ∈ L * and δ ∈ D(y, r).
For (y, r) ∈ L * , we define the sets A(y, r) = {(x, q) ∈ K : xy + qr ≤ 1}, and denote by C (y,r) the closure ofC (y,r) with respect to convergence in measure P × dκ. By Lemma 37, for z > 0, sup c∈zC (y,r) By Proposition 34, the sets D(y, r) and C (y,r) are polar sets of each other. Therefore, they satisfy the assumptions of Mostovyi [2015], Theorem 3.2. This theorem implies that there exists a unique solutionδ(y, r) to (29) and the second conjugacy relation in (30) holds: v(y, r) = sup z>0 sup c∈zC (y,r) The function u is clearly concave on K. The first equation in (30) follows from Rockafellar [1970], Section 12.

Complete market
In this section, we examine the complete market case, when Z = {Z} is a singleton, in much greater detail. This becomes possible due to the fact that the description of the dual set D λ given in Proposition 28 simplifies significantly in a complete market: In particular, for any consumption plan c satisfying (DC λ ), sup δ∈D λ c, δ = c, Z .
Proof. Clearly, the set on the right-hand side of (35) is λ -solid, convex, and contains Z. By Proposition 28, it remains to verify that it is closed with respect to convergence in measure P × dκ. Let δ n λ Z, n ≥ 1, be a sequence converging almost surely to an optional process δ ≥ 0. To show that δ λ Z, we check the right-hand side of (16): for every c satisfying (DC λ ), by Fatou's lemma and by (16) applied to δ n λ Z, c, δ ≤ lim inf n→∞ c, δ n ≤ c, Z .
The last assertion follows from (16) as well.
We note that from (35) it is easy to show the strict inclusion D λ 2 D λ 1 for 1 ≥ λ 2 > λ 1 ≥ 0 in a complete market. Namely, it suffices to find a process δ such that δ λ 2 Z but δ λ 1 Z. One can take a stopping time T ∈ (0,T ] such that P(T <T ) > 0, define and check the required properties which, by Proposition 25, are equivalent to δ 2 λ 2 Z½ (T,T ) and δ 2 λ 1 Z½ (T,T ) .
Proposition 40. Suppose all the assumptions of Theorem 35 hold and the market is complete. Let x > 0, y = u ′ (x), and assume a consumption plan c satisfies (DC λ ) and c, Z = x. Then c ∈ C(x). The consumption plan c is the optimizer in C(x) if and only if forδ t := U ′ (t, c t ) the following holds: with equality dc t − almost everywhere.
In this case, the optimizer c is related with its running essential supremumc in the following way (up to P × dκ−nullsets): and, P-almost surely,c satisfies with equality dc t − almost everywhere.
Remark 41 (Interpretation). Let λ ∈ (0, 1).We see that only the following three types of behavior are possible for the optimal consumption plan c =ĉ(x) (up to P × dκ−nullsets): (i) c t = λc t . The agent consumes at the minimal level allowed by the drawdown constraint.
(ii) c t =c t . The agent consumes at the current running essential supremum level.
(iii) c t = I(t, yZ t ). The agent consumes as an unconstrained agent with a different initial wealth x 0 given by u ′ 0 (x 0 ) = y, where u 0 is the value function for the unconstrained problem.
How exactly the timeline separates into these three regions is encoded in (37), however, this is more difficult to interpret. For the simpler ratchet constraint λ = 1, see Corollary 43 below and the discussion after it.
Remark 42. The jointly measurable process defined by Dellacherie and Meyer [1982], Chapter VI, Theorem no. 47 and Remark no. 50(f), the optional projection o D t is right-continuous (in fact, càdlàg). Therefore, by Lemma 18, condition 3. of Proposition 40 is equivalent to the following: P-almost surely, with equality when dc t > 0.
The same is true for (37): "equality dc t −almost everywhere" can be replaced with "equality when dc t > 0".
Proof of Proposition 40. The consumption plan c belongs to C(x) because it satisfies (DC λ ) and is x-admissible by Proposition 39: sup δ∈D λ c, δ = c, Z = x.
Finally, we show that the expression (36) of c through its running essential supremum and the condition (37) for the latter follow from 1. For the ratchet constraint, the characterization of the optimizers given in Proposition 40 simplifies significantly.
Corollary 43 (λ = 1). Suppose all the assumptions of Theorem 35 hold and the market is complete. Let x > 0, y = u ′ (x), and c ∈ C inc such that c, Z = x. Then c ∈ C 1 (x). The consumption plan c is the optimizer in C 1 (x) if and only if, P−almost surely, yZdκ t for all t ∈ [0,T ), with equality when dc t > 0. (40) This characterization is closely related to the notion of the envelope process introduced in Lemma A.1 in Bank and Kauppila [2017] and to the Representation Theorem of Bank and El Karoui [2004] it is based on. In Lemma 49 of the Appendix, we slightly modify Lemma A.1 of Bank and Kauppila [2017] to show the following: if, in addition to Assumption 7 on utility U , we assume that E T 0 U ′ (ω, t, x)dκ t < ∞ for every x > 0 then for every y > 0 there exists a unique c = c y ∈ C inc for which (40) holds. By Corollary 43, this consumption plan has to be the optimizer, c y =ĉ(x) ∈ C 1 (x). In the Appendix, we also give a short proof, inspired by the arguments of Riedel [2009] and not involving duality, that, under an additional assumption on utility (Assumption 50), the optimal consumption plans have this structure.
In the example below, we derive, as a special case of Corollary 43, the formula of Riedel [2009] for the optimal consumption plans.
Example 44. LetT = ∞ and assume that stochastic clock is given byκ s = e −rs , where r > 0 is the interest rate (cf. Remark 2(ii)). We take the utility field given by U (t, x) = e −δt u(x) e −rt , where u is a strictly concave, increasing, continuously differentiable deterministic function on (0, ∞) satisfying the Inada conditions u(0) = +∞, u(+∞) = 0 and δ > 0 is the parameter of exponential time preferences of the agent. This utility satisfies Assumption 7 and the expected utility functional in (6) becomes E ∞ 0 e −δt u(c t )dt . Assume further that log(Z t ) is a Lévy process (starting at zero) for the unique equivalent martingale deflator Z. With these choices ofT , κ, U , and Z, we turn out in the framework of Riedel [2009]. The case of the GBM market considered in Dybvig [1995] corresponds to log(Z t ) = −θB t − 1 2 θ 2 t, where B t is the underlying Brownian motion and θ is the market price of risk.
Let i = (u ′ ) −1 . We will show that for a suitable constant K > 0 the consumption plan satisfies (40). By Lemma 49, this is then the unique process in C inc satisfying (40), hence by Corollary 43 it is the optimizerĉ(x) ∈ C 1 (x). The definition (41) corresponds to the formula (3) of Riedel [2009] for the optimal consumption in case q = 0; in order to obtain the result analogous to Ridel's for the case q > 0, we can simply apply Corollary 48 below.
With the martingale property of Z and withκ being deterministic and exponential, it is easy to check for the right-hand side of (40) that o ∞ . yZdκ t = yZ t ·dκ([t, ∞)) = y e −rt r Z t . With c defined as in (41) where "=" holds in place of "≤" on {dc t > 0} ∈ O. Due to the Lévy assumption on log(Z t ), the integral I t satisfies E[I t |F t ] = E ∞ 0 e −δs inf u∈[0,s) Z u e (δ−r)u ds := I ∈ (0, ∞), hence o I t ≡ I. Taking the optional projection in (42) This is precisely (40) if we take K := y/(Ir).

Monotonicity and continuity of optimizers
Next, we establish monotonicity and continuity of the optimizers with respect to the initial wealth. These results will be useful when dealing with the case q > 0 in the following subsection.
Proof. For fixed x 2 > x 1 > 0 we denote for brevity y i = u ′ (x i ), c i =ĉ(x i ) andc i its running essential supremum, i = 1, 2. By (36), c i = λc i ∨ I(y i Z) ∧c i for i = 1, 2. We split the product space Ω × [0,T ) into two regions: has a positive P × dκ measure, a contradiction. Therefore,c 1 t ≤c 2 t , P × dκ−a.e. on R c , completing the proof of monotonicity.
To prove continuity, we take an increasing sequence x n ↑ x (for a decreasing sequence the argument is analogous). By lim n→∞ĉ (x n ) ≤ĉ(x), P × dκ−a.e., and by monotone convergence so the equality should hold in place of inequality, i.e.,ĉ(x) = lim n→∞ĉ (x n ), P×dκ−a.e.

Case q > 0
Now we consider the presence of a lower bound q > 0 on initial consumption. It turns out that optimizers for q > 0 can be described in terms of appropriate optimizers for q = 0, as the following proposition states. Note that in the case of a complete market the constant α defined in Proposition 30 becomes simply α = E T 0 Zdκ .
Proposition 46. Suppose all the assumptions of Theorem 35 hold and the market is complete. Fix q > 0. For x > 0, y = u ′ (x), letc x be the running essential supremum of the optimizerĉ(x) ∈ C(x) and define for t ∈ [0,T ) and π(x) := c, Z , the price of the consumption plan c. Then π(x) ∈ [(αλ)q, ∞), and if π(x) > (αλ)q then c ∈ C(π(x), q) and c is the optimizer in C(π(x), q).
Remark 47 (Interpretation). In view of Proposition 40,ĉ(x) = λc x ∨ I(yZ) ∧c x , therefore, the optimizer in C(x ′ , q), (x ′ , q) ∈ K, can be described as follows: for some x > 0, the optimizersĉ(x) andĉ(x ′ , q) behave identically on {c x ≥ q}, i.e., starting from the time when the running essential supremum ofĉ(x) becomes at least q. On {c x < q},ĉ(x ′ , q) behaves as an unconstrained agent's consumption I(t, yZ t ) restricted to stay between λq and q:ĉ(x ′ , q) = λq ∨ I(t, yZ t ) ∧ q. This agrees with the observation that the drawdown constraint (DC λ,q ) with q > 0 becomes the simple drawdown constraint (DC λ ) on the set {c ≥ q} and becomes simply a lower bound c ≥ λq on {c < q}.
As in the case q = 0, we have three possible types of behaviour: either the agent consumes at the minimal level allowed by the drawdown constraint (DC λ,q ), c t = λ[c t ∨ q] = λ[c x t ∨ q], or the agent consumes at the current running essential supremum level c t =c t =c x t ∨ q, or the agent consumes as an unconstrained agent, c t = I(t, yZ t ). This separation into three regions agrees with the result of [Arun, 2012, Theorem 1] on the drawdown constraint in a GBM market with CRRA utility, but it does not seem obvious from Arun's result that the optimal consumption can be linked to the optimal consumption of an unconstrained agent.
Let c ′ :=c x ∨ q − q ∈ C inc . The equality c,δ = x ′ y + qr is equivalent to c,δ − yZ = qr, which can be checked with the following sequence of equalities: where the expectation on the last line vanishes by (37) due to the fact that the measure dc ′ is absolutely continuous with respect to dc x .
In Proposition 51 in the Appendix, we achieve a similar conclusion for the ratchet constraint without relying on Theorem 35, but under additional assumptions on U , by using the Bank-Kauppila envelope process.

Ratcheting consumption in complete market -envelope process and alternative solution
The following lemma is a modification of [Bank and Kauppila, 2017, Lemma A.1] for less restrictive assumptions on U and possibly finite time horizonT . The proof is based on the Representation Theorem of Bank and El Karoui [2004] and follows closely the proof of [Bank and Kauppila, 2017, Lemma A.1]. This lemma is used for showing existence and uniqueness of c ∈ C inc satisfying (40) (see the discussion following Corollary 43), as well as for an alternative solution of the ratchet constraint problem in a complete market given in Proposition 51 below.
In a complete market, Z = {Z} is a singleton and we denoteZ t := T t Zdκ for 0 ≤ t ≤T .
Lemma 49. Assume that stochastic clock satisfies Assumption 1, utility satisfies Assumption 7, and E T 0 U ′ (ω, t, x)dκ t < ∞ for every x > 0. Then for every y > 0 there exists a unique process c y ∈ C inc such that, P-almost surely, o T . U ′ (c y )dκ t ≤ y oZ t for all t ∈ [0,T ), with equality when dc y t > 0.
The process c y is finite-valued.
In the language of Bank and Kauppila [2017], o T . U ′ (c y )dκ is called the envelope process of y oZ , and Lemma 49 states its existence and uniqueness under given assumptions on U .
SinceZ is a continuous non-increasing jointly measurable process with ZT = 0 and E[Z 0 ] = α < ∞, the optional projection oZ is a class (D), continuous in expectation, non-negative supermartingale with lim t↑T oZ t = 0, P−a.s. By Theorem 3 and Remark 2.1 in Bank and El Karoui [2004] there exists an optional process L t taking values in R∪{−∞} such that for every stopping time S ≤T , The process L t is non-positive on [0,T ) up to indistinguishability. Assuming otherwise, by the optional section theorem there exists a stopping time S ≤T such that P(S <T ) > 0 and L S > 0 on {S <T }, which implies a contradiction. Furthermore, almost every path of the non-positive left-continuous process L t := sup s∈[0,t) L s is strictly negative on (0,T ): otherwise, for the stopping times S := inf{t ∈ (0,T ] :L t = 0} and S n := inf{t ∈ (0,T ] :L t > −1/n}, n ≥ 1, we have P(S <T ) > 0, S n ↑ S as n ↑ ∞, sup s∈[S n ,t) L s =L t for every n ≥ 1, t > S n , and, by monotone convergence, Hence, the process c y t := 0, t = 0, −1/L t , 0 < t <T , belongs to C inc , is finite-valued, and, by (49), satisfies (48).
For uniqueness, assume c ∈ C inc is such that o T . U ′ (c)dκ t ≤ y oZ t for all 0 ≤ t <T with "=" when dc t > 0. For l > 0, we show below that T l := inf{t ≥ 0 : c t > l} is the largest stopping time minimizing E Proposition 51. Let κ satisfy Assumption 1, U satisfy Assumptions 7 and 50, assume Z = {Z} is a singleton, and let y > 0, q ≥ 0. Define the consumption planĉ := c y ∨ q½ (0,T ) ∈ C inc , where c y is given by Lemma 49, and let x := ĉ, Z . If x < ∞ and U(ĉ) < ∞ thenĉ is the unique maximizer of U in C 1 (x, q).
Sinceĉ t ≥ c y t and U ′ is decreasing, T s U ′ (c y t )dκ t ≥ T s U ′ (ĉ t )dκ t for all s ≥ 0. Moreover, for s ≥ 0 such that dĉ ′ s > 0, we haveĉ t = c y t for all t > s and, since dκ has no atoms, T s U ′ (c y t )dκ t = T s U ′ (ĉ t )dκ t . As a consequence, Finally, combining (50), (51), using the defining property (48) of c y and the fact that dĉ ′ s > 0 implies dc y s > 0, we obtain The uniqueness of optimizer, up to P × dκ−nullsets of Ω × [0,T ), follows by strict concavity of U and finiteness of U(ĉ).