An elementary approach to the Merton problem

In this article we consider the infinite-horizon Merton investment-consumption problem in a constant-parameter Black - Scholes - Merton market for an agent with constant relative risk aversion R. The classical primal approach is to write down a candidate value function and to use a verification argument to prove that this is the solution to the problem. However, features of the problem take it outside the standard settings of stochastic control, and the existing primal verification proofs rely on parameter restrictions (especially, but not only, R<1), restrictions on the space of admissible strategies, or intricate approximation arguments. The purpose of this paper is to show that these complications can be overcome using a simple and elegant argument involving a stochastic perturbation of the utility function.


Introduction and overview
In the Merton investment-consumption problem (Merton [10,11]) an agent seeks to maximise the expected integrated discounted utility of consumption over the infinite horizon in a model with a risky asset and a riskless bond. When parameters are constant and the utility function is given by a power law, it is straightforward to write down the candidate value function. However, it is more difficult to give a complete verification argument. For general strategies the wealth process may hit zero at which point the application of Itô's formula to the candidate value function breaks down; the local martingale which arises from the application of Itô's formula may fail to be a martingale; even for constant proportional strategies transversality may fail.
For all these reasons, it is difficult to give a concise, rigorous verification proof via analysis of the value function, and many textbooks either finesse the issues or restrict attention to a subclass of admissible strategies, and/or restrict attention to a subset of parameter combinations (especially R < 1, but even then there can be substantive points which are often overlooked). The need for such a verification argument has been obviated by the development of proofs using the dual method, which provides a powerful and intuitive alternative approach, see Biagini [1] for a survey (and also [5,7,8,13]). Nonetheless, it would be nice to provide a short proof based on the primal approach. 1 The goal of this paper is didactic -to give a simple, brief proof that the candidate value function is the value function via the primal approach, and moreover, * We thank Steve Shreve and Ioannis Karatzas for sharing their recollections of their motivations behind [9] and [6]. 1 Our original motivation for returning to the Merton problem arose from consideration of a problem involving stochastic differential utility. There, dual approaches are more involved and do not cover all parameter combinations, so that the primal method is not redundant, and indeed may provide a more direct approach.
to give a proof which is valid for all parameter combinations for which the Merton problem is well-posed.
The first full verification of the solution to the Merton problem of which we are aware (under an assumption of positive discounting and strictly positive interest rates) is Karatzas et al [6], which built on the previous work of Lehoczky et al [9]. There, the idea is to solve a perturbation of the original problem in which the agent may go bankrupt, at which point they receive a residual value P . (Part of their motivation was to better understand the results of Merton [11] on HARA utilities, see also Sethi and Taskar [14].) The solution to the perturbed problem is very clever, and is developed in the case of a general utility function, but it is also very intricate and takes many pages of calculation. Moreover, when specialised to the case of CRRA utilities, it places some assumptions on the parameter values beyond the necessary assumption of wellposedness of the Merton problem. The problem with bankruptcy is of independent interest, but more important for our purposes is the fact that, given the solution to the problem with bankruptcy for a CRRA utility, by letting P ↓ 0 (R < 1) or P ↓ −∞ (R > 1) Karatzas et al [6] recover the solution to the original Merton problem.
In their seminal paper on transaction costs, Davis and Norman [2, Section 2] briefly consider the Merton problem without transaction costs. They assume that the proportion of wealth invested in the risky asset is bounded, and for R < 1 they go on to prove a verification theorem for strategies restricted to this class. Further, in the case R > 1 they propose a different perturbation, this time of the candidate value function. The key point is that in the perturbed problem the candidate value function has a finite lower bound, and this allows Davis and Norman [2] to re-apply arguments from the R < 1 case, although the restriction to 'regular' investment strategies remains. The candidate value for the perturbed problem can be used to give an upper bound on the true value function, which converges to the candidate solution to the Merton problem as the perturbation disappears. Unlike the argument in Karatzas et al [6], the proof is quite short, but again it only works for certain parameter combinations, and more importantly it restricts attention to a subclass of admissible strategies.
Our goal is to give a complete, simple verification argument via primal methods. At its heart, our idea is a modification of the approach in [2]. We perturb the utility function, which leads to a perturbed value function. Moreover, rather than perturbing by the addition of a deterministic constant, we perturb by adding a multiple of the optimal wealth process. The great benefit is that the optimal consumption and the optimal investment are unchanged under the perturbation, which means that mathematical calculations remain strikingly simple. Moreover, these arguments are valid whenever the Merton problem is well-posed. This paper is structured as follows. In the next section we introduce the problem, and in Section 3 we give the candidate value function. In Section 4 we give a proof of the main result under a set of clearly-stated assumptions which are precisely designed to make the proof work. Often, proofs in the stochastic control literature (see, for example, Davis and Norman [2], Fleming and Soner [3, Example 5.2] and Pham [12]) artificially impose restrictions on the set of admissible strategies or on the parameter values to ensure that these assumptions are satisfied by default. In Section 5 we give a small amount of detail on the Karatzas et al [6] and Davis and Norman [2] approaches to the verification problem before in Section 6 we give our argument. Finally, in Section 7 we explain the insights which arise from considering a numéraire change of the problem, in particular on what are the appropriate parameter restrictions, and what is the best formulation of the problem. We also show how a change of numéraire can lead to a simplified solution of a slightly modified version of the problem with bankruptcy considered in Karatzas et al [6], and hence to a simplified verification argument.
Our proof is an improvement on the existing results in at least three important ways. First, it places no restrictions on the class of admissible strategies: for example, unlike much of the stochastic control literature, it does not require the fraction of wealth invested in the risky asset to be bounded. (The argument in Karatzas et al [6] also applies to general investment strategies.) Second, the proof covers all parameter combinations for which the Merton problem is well-posed (and does not assume that interest rates and discounting are positive -as we shall argue these quantities depend on the choice of currency units, and therefore are not absolutes in themselves). Third, our proof is simple, elegant and concise and not counting the derivation of the candidate solution and candidate value function can be written up in just over one page (Theorem 6.1 and Corollary 6.4). One final contribution of this article is to argue that some formulations of the Merton problem are more natural than others, in the sense that they are robust to changes of currency unit, and in consequence have simpler dependencies on parameter combinations.

The Merton problem
Throughout this paper we will work on a filtered probability space (Ω, F, P, F = (F t ) t>0 ) satisfying the usual conditions and supporting a Brownian motion W = (W t ) t≥0 . We will assume a Black-Scholes-Merton financial market consisting of a risk-free asset with interest rate r ∈ R whose price process S 0 = (S 0 t ) t≥0 is given by S 0 t = exp(rt) and a risky asset whose price process S = (S t ) t≥0 follows a geometric Brownian motion with drift µ ∈ R and volatility σ > 0: An agent operating with this investment opportunity set and initial wealth x > 0 chooses an admissible investment-consumption strategy (ϑ 0 , ϑ, C) = (ϑ 0 t , ϑ t , C t ) t≥0 , where ϑ 0 t ∈ R denotes the number of riskless assets held at time t, ϑ t ∈ R denotes the number of risky assets held at time t, and C t ∈ R + represents the rate of consumption at time t. We require that ϑ 0 , ϑ, C are progressively measurable processes, ϑ 0 is integrable with respect to S 0 , ϑ 1 is integrable with respect to S, C is integrable with respect to the identity process, the wealth process X = (X t ) t≥0 defined by is P-a.s. nonnegative and the self-financing condition, is satisfied. We then denote by Π 0 t := Xt and Π t := ϑtSt Xt the fraction of wealth invested in the riskless and risky asset at time t, respectively. 2 Noting that Π 0 t + Π t = 1 by (1), it follows that X satisfies the SDE This means that we can describe an admissible investment-consumption strategy for initial wealth x > 0 more succinctly by a pair (Π, C) = (Π t , C t ) t≥0 of progressively measurable processes, where Π is real-valued and C is nonnegative, such that the SDE (2) has a unique strong solution X x,Π,C that is P-a.s. nonnegative. We denote the set of admissible investmentconsumption strategies for x > 0 by A (x). A consumption stream C is called attainable for initial wealth x > 0 if there exists an investment process Π such that (Π, C) ∈ A (x), and we denote the set of attainable consumption streams for x > 0 by C (x). The objective of the agent is to maximise the expected discounted utility of consumption over an infinite time horizon for a given initial wealth x > 0. To any attainable consumption stream C ∈ C (x), they associate a value Here, δ ∈ R can be seen as a discount or impatience parameter; see Section 7 for a discussion on the economic interpretation of δ. We assume that the agent has constant relative risk aversion (CRRA) or equivalently that U : In summary, the problem facing the agent is to determine

The candidate value function
From the homogeneous structure of the problem we expect (see for example, Rogers [13, Propo- For this reason, we may guess that it is optimal to invest a constant fraction of wealth in the risky asset, and to consume a constant fraction of wealth. (Of course, this will be verified later.) So, consider an investment-consumption strategy that at each time t, invests a constant proportion of wealth Π t = π into the risky asset and consumes a constant fraction ξ > 0 of wealth per unit time, i.e., C t = ξX t . Then the agent's wealth process X = X x,π,ξX is given by Denoting the market price of risk or Sharpe ratio by λ := µ−r σ , we obtain Multiplying this by e −δt and taking expectations gives where Provided that F (π, ξ) > 0, we find that In order to maximise this over π, we want to minimise (1 − R)F (π, ξ), which is equivalent to maximising λσπ − π 2 σ 2 2 R. This is achieved atπ = λ σR . In this case λσπ −π 2 σ 2 2 R = λ 2 2R and the problem then becomes to maximise provided that η > 0. (If η ≤ 0, no maximum exists.) Therefore, when η > 0, the agent's optimal behaviour (at least over constant proportional strategies) and corresponding value function are given bŷ When η ≤ 0, the problem is ill-posed. Indeed, if R < 1, then F (π, ξ) ↓ 0 as ξ ↓ − ηR (1−R) and hence J(ξX) ↑ ∞ by (4). If R > 1, then F (π, ξ) ≤ F (π, ξ) = Rη + (1 − R)ξ ≤ Rη ≤ 0 for every π ∈ R and ξ ≥ 0. Hence, at least for constant proportional strategies J(ξX) = −∞. We will see in Corollary 6.5 that J(C) = −∞ for every admissible consumption stream C ∈ C (x).

The verification argument under fiat conditions
In this section, we prove that our candidate optimal strategy (π,ξX) from (5) is optimal in a subset of the class of all admissible strategies. Since the conditions defining that class are chosen precisely in such a way that the proof works, we call them fiat conditions.
is called fiat admissible if the following three conditions are satisfied: (P) The wealth process X x,Π,C is P-a.s. positive.
We denote the set of all fiat admissible investment-consumption strategies for We denote the set of fiat attainable consumption streams by C * (x). Remark 4.2. As far as we are aware, the above notion of fiat admissible strategies has not been explicitly used in the literature before. However, the conditions (P), (M) and (T) or stronger versions thereof have been used explicitly or implicitly throughout the stochastic control literature on the Merton problem: 1. Condition (P) is (implicitly) assumed throughout most of the stochastic control literature dealing with the Merton problem; a notable exception is [6]. However, for R > 1, (P) can be assumed without loss of generality because any admissible strategy (Π, C) ∈ A (x) violating (P) has J(C) = −∞.

Condition (M) is implied by the stronger condition
It is not difficult to check that for R < 1, (M1) is implied by the even stronger condition (B) Π is uniformly bounded.
A common approach in the stochastic control literature is to assume (B), see e.g. Davis
It is clear that C * (x) ⊂ C (x). The following result shows that the candidate optimal strategy (π,ξX) from (5) is optimal in the class of fiat admissible strategies.
where the corresponding optimal investment-consumption strategy is given by Proof. First, we show that V * (x) ≥V (x) = J(Ĉ). By the arguments in Section 3, it only remains to show thatĈ is fiat attainable. It follows from the construction ofĈ, that the wealth process X x,Π,Ĉ is P-a.s. positive. Next, a similar calculation as in (3) shows that for each T > 0, This implies that the local martingale and hence a supermartingale. Finally, (3) together with the fact that F (π, η) = η > 0, implies that (Π,Ĉ) satisfies the transversality condition (T1).
Next, we show that we may in addition assume without loss of generality that C 1−R is integrable with respect to the identity process; for otherwise J(C) = −∞. It suffices to argue that J(C) ≤V (x).
Set X := X x,Π,C for brevity and define the process We want to apply Itô's formula to M . This is indeed possible asV is in C 2 (0, ∞) and X is positive by fiat admissibility of (Π, C). Note thatV x (X t ) is positive andV xx (X t ) is negative. Then, noting that the argument ofV and its derivatives is X t throughout, and adding and subtracting R Vxx for the second equality, Here, N given by Finally, a simple calculation using the definition ofV and η shows that A 3 = 0. It follows that M t ≤V (x) + N t , t ≥ 0.
Taking expectations and using fiat admissibility of (Π, C) to ensure that N is a supermartingale, we find for each t ≥ 0, Taking the limit as t goes to infinity, and using the monotone convergence theorem as well as the transversality condition, we obtain This establishes the claim.  Proof. It is sufficient to show that (P), (M) and (T) are satisfied for general strategies, or to find a way of bypassing the relevant part of the argument. First, (T) is automatically satisfied by the fact that X 1−R /(1 − R) is nonnegative. Next, M is nonnegative and hence N is bounded below by −V (x) by (7). Therefore, N is always a supermartingale and (M) is automatically satisfied.
Finally, to avoid imposing (P), one has to refine the argument in Theorem 4.3 by a stopping argument. To wit, fix an admissible strategy (Π, C) ∈ A (x). Then for n ∈ N , set τ n := inf{t ≥ 0 : X x,Π,C ≤ 1 n } and let τ ∞ := lim n→∞ τ n . Then it is not difficult to check that Now first taking the limit t → ∞, we obtain Next, taking the limit n → ∞, the result follows from the monotone convergence theorem and the fact that ∞ τ∞ C s ds = 0 P-a.s. Remark 4.6. The above approach of avoiding (P) is taken in [6,Theorem 4.1]. Note, however, that there the stopping argument is slightly more involved as it also requires stopping when the wealth process X x,Π,C or the quadratic variation of · 0 σΠ dW gets too large. But this additional stopping rather obfuscates the argument. 7 More precisely, we have ∞ τ∞ Cs ds = 0 P-a.s. Remark 4.7. If R > 1, extending Theorem 4.3 to general admissible strategies is far more involved. While condition (P) can be assumed without loss of generality (recall Part 1 of Remark 4.2), condition (M) is in general not satisfied as there are investment strategies Π and consumption strategies C such that N fails to be a supermartingale, see Example 4.8 below. Note that these strategies are suboptimal because A 1 and A 2 are (very) negative. Finally, we have no reason to expect that the transversality condition (T) is satisfied. Indeed, (T) even fails for constant proportional strategies: If ξ > ηR R−1 , then F (π, ξ) < 0, and it follows from (3) that lim t→∞ E e −δt 1−R X x,π,ξX t = −∞.
Example 4.8. For R > 1, the process N in the proof Theorem 4.3 can fail to be supermartingale. We first give an abstract version of an example and then two concrete specifications. Let (Π, C) ∈ A (x) be such that X = X x,Π,C has P-a.s. positive paths. Define the stopping time If τ is bounded, then N fails to be a supermartingale because The above abstract situation can be achieved either by "wild" investment or by "too fast" consumption, or a combination of the two.
For an example of a "wild" investment strategy Π, assume that µ ≥ r > 0 and define the stopping timeτ Then the corresponding wealth process X is a stopped and time changed CEV process: Since R > 1, X remains positive. Since τ =τ P-a.s. we have τ < 1 P-a.s. and N fails to be a supermartingale.
For an example of a "too fast" consumption strategy C (with bounded investment strategy Π), assume that µ ≥ r > 0 and define the stopping timē Note thatτ < 1 P-a.s. since Then the corresponding wealth process satisfies the SDE

It is not difficult to check that this has the solution
which is well-defined and positive by the fact thatτ < 1 P-a.s.. Since τ =τ P-a.s., we have τ < 1 P-a.s. and N fails to be a supermartingale.
5 Verification approaches for R > 1 As we have explained in Remark 4.7, a verification argument for general admissible strategies requires some new ideas for the case R > 1. In this section, we discuss the two most general approaches in the extant stochastic control literature. Both approaches first consider a perturbation of the problem (or the candidate solution) and then let the perturbation disappear.

Perturbation with finite bankruptcy
The first perturbation approach is by Karatzas et al [6] who study an optimal investmentconsumption problem with bankruptcy for a general utility function which is of interest in its own right, building on earlier work [9] by a subset of the authors. In the following, we only describe their contribution towards the solution of the Merton problem for CRRA utilities. We assume R > 1, and we use our notation. Assume that δ > 0 and r > 0. For an admissible strategy (Π, C) ∈ A (x), denote the bankruptcy time τ 0 = τ x,Π,C 0 = inf{t : X x,Π,C t = 0}. Then choose a finite bankruptcy value P ∈ (−∞, 0) and consider the problem with bankruptcy: Note that the classical Merton problem corresponds to the limiting case P = −∞. Karatzas et al [6] show the following: (A) Suppose that a C 2 -functionV P : (0, ∞) → (P, 0) solves the HJB equation corresponding to the optimisation problem (9) given by subject to lim x↓0Ṽ (x) = P .
Here, the argument for (A) is relatively straightforward; see [6,Theorem 4.1] and not more difficult than the proof of our Theorem 4.3. Similarly, the argument for (C) is easy: the first inequality follows from the fact that V (x) ≤ V P (x) ≤V P (x) for each x > 0 and P ∈ R − by the definition of V P and (A); the second inequality is straightforward using the explicit form for V P . But the main difficulty -and great ingenuity -of the argument in [6] is (B). Indeed, a direct calculation for r > 0 case takes at least two pages and yields the answer: where the functionĈ P (x) describing the optimal consumption is the inverse of the function and ν is the negative root of the equation λ 2 2 ζ 2 + (r − δ − λ 2 2 )Rζ − rR 2 = 0. We return to this approach in Section 7.2.

Perturbation of the value function
The second perturbation approach is by Davis and Norman [2] who study the Merton problem with transaction costs; the perturbation argument for R > 1 in the frictionless case is a fortunate by-product, and not the main contribution of the paper. Again we will use our notation to describe their approach.
Assume that δ > 0, r > 0 and that condition (B) of Remark 4.2 is satisfied. Moreover, denote by A b (x) all admissible strategies (Π, C) for which Π is uniformly bounded, write C b (x) for the corresponding set of attainable consumption strategys and set V b (x) := sup C∈C b (x) J(C). For ζ > 0, consider the perturbed value functionV ζ (x) =V (x + ζ) and for (Π, C) ∈ A b (X) (such that C 1−R is integrable with respect to the identity process), consider the process M ζ defined by Then the same argument as in the proof of Theorem 4.3 but withV replaced byV ζ yields where the only difference is thatV and its derivatives are replaced byV ζ and its derivatives. Then as in the proof of Theorem 4.3 it follows that A 1,ζ , A 2,ζ ≤ 0. Moreover, using that x (X t )e −δt ≤ 0, which crucially uses that r ≥ 0. Finally, using that Π andV ζ x are bounded, it is not difficult to check that N ζ is a square integrable martingale. Now following the proof of Theorem 4.3, and using that |V ζ | is bounded and δ > 0 it follows that We may conclude that V b (x) ≤V ζ (x) and taking the limit as ζ ↓ 0, it follows that V b (x) =V (x).

The general verification argument
In this section, we present our general verification argument. It is inspired by the perturbation argument of Davis and Norman. The key idea is to use the candidate optimal consumption strategy as a stochastic perturbation of the utility function. This yields a very elegant and simple argument that has the trio of advantages that it is no more difficult than the fiat verification argument in Theorem 4.3, it does not need to distinguish between the case R > 1 and R < 1 and it does not involve any stopping argument. 8 The following theorem contains the solution to the stochastically perturbed Merton problem. The subsequent corollary then lets this perturbation disappear. Recall the notations of Theorem 4.3: Theorem 6.1. Suppose η > 0. Denote by Y = (Y t ) t≥0 the candidate optimal wealth process started from unit initial wealth 1, i.e., Y t := X 1,Π,ηX t , and by G = (G t ) t≥0 , the corresponding optimal consumption stream, i.e., G t = ηY t . Fix ε > 0, define the function , and for an attainable consumption stream C consider Then for x > 0, Moreover, the supremum is attained when Π =Π and C =Ĉ whereĈ = ηX x,Π,Ĉ .
Proof. First, from the SDE for the wealth process (2) we have that X x,Π,ηX + εY = X x+ε,Π,ηX . It follows thatĈ + εG = ηX x+ε,Π,ηX ∈ C (x + ε), which together with Theorem 4.3 implies that J ε (Ĉ) = J(Ĉ + εG) =V (x + ε). It remains to show that V ε (x) ≤V (x + ε). The argument is very similar to the one in the proof of Theorem 4.3. Let (Π, C) ∈ A (x) be arbitrary and set X := X x,Π,C for brevity. The dynamics of X + εY are given by Define the process M ε = (M ε t ) t≥0 by We proceed to apply Itô's formula to M ε . Adding and subtracting R 1−R (V x ) 1−1/R + εηY tVx and λ 2 2 V 2 x Vxx + λ 2 R εY tVx and noting that the argument ofV and its derivatives is (X t + εY t ) throughout, we obtain By the same arguments as in the proof of Theorem 4.3, it follows that A 1,ε ≤ 0, A 3,ε ≤ 0 and Next, define the process Λ ε = (Λ ε t ) t≥0 by Then Λ ε ≤ M ε by monotonicity of U andV . Using that Λ ε is a (UI) martingale by Remark 4.4, it follows that N ε is bounded below by the (UI) martingale −V (x + ε) − Λ ε and hence a supermartingale.
Taking expectation in (19), we find for each t ≥ 0, Next, note that X + εY satisfies the transversality condition (T) since Taking the limit in (20) as t goes to infinity and using (21), we may conclude that for any C ∈ C (x), Remark 6.2. The perturbation of the problem by the additional consumption of εG elegantly and simply transforms the problem to one in which the fiat conditions (P), (M) and (T) are satisfied. Since Y is positive P-a.s., the same is trivially true for X + εY . Moreover, J(εG) = ε 1−R J(G) > −∞ and this allows us to easily find an integrable lower bound on N ε and hence conclude it is a supermartingale. Again Y satisfies a transversality condition (T) and so the same is trivially true for X + εY .
Remark 6.3. One interpretation of the theorem is that a financially-savvy benefactor gives the agent an additional consumption stream based on an initial wealth ε which is invested optimally by the benefactor. Then, if the agent behaves optimally with their own wealth, the two consumption streams and investment strategies remain perfectly aligned to each other, and the derivation and valuation of the candidate optimal strategy is simple and immediate. Proof. The equality J(Ĉ) =V (x) follows from Theorem 4.3. It remains to establish that V (x) ≤V (x). Using the notation of Theorem 6.1, for any C ∈ C (x), we get Proof. Fix C ∈ C (x). It suffices to show that J(C) = −∞. We use an approximation argument. For n ∈ N set δ n := δ + R( 1 n − η). Then δ n > δ and η n := 1 R [δ n − (1 − R)(r + λ 2 2R )] = 1 n > 0. Then using that U (c) < 0 for c ≥ 0, it follows from Theorem 6.1

We finish this section by showing that in the case
Taking the limit on the right hand side as n goes to ∞, it follows that J(C) = −∞.

Change of numéraire arguments
We close the paper with some remarks on change of numéraire ideas. As we have seen in Section 5, using the perturbation arguments of Karatzas et al [6] or Davis and Norman [2], we get verification arguments for the case R > 1 under the parameter restrictions δ > 0 and r > 0.
In this section we show by using a change of numéraire that this parameter restriction can be weakened, although not to the extent that it covers all the parameter combinations for which η > 0. We then apply these ideas to present another new verification argument that is based on Karatzas et al [6] but far simpler.

The Merton problem under a change of numéraire
We say that a pair (S 0 ,S) = (S 0 t ,S t ) t≥0 of semimartingales is economically equivalent to (S 0 , S) if there exists a positive continuous semimartingale D = (D t ) t≥0 such thatS 0 = DS 0 and S = DS. Here, the interpretation of D is an exchange rate process and (S 0 ,S) describes the financial market in a different currency unit; see [4, Section 2.1] for more details.
Next, recall that if (ϑ 0 , ϑ, C) is a admissible investment-consumption strategy for initial wealth x > 0, (where ϑ 0 and ϑ 1 denote the number of shares held in the riskless and risky asset, respectively), then the corresponding wealth process X = ϑ 0 Now if (S 0 ,S) is economically equivalent to (S 0 , S) with corresponding exchange rate process D, it is not difficult to check that the corresponding wealth processX := ϑ 0S0 + ϑS = DX satisfies the SDE whereC = DC. This means that if C describes an attainable consumption strategy in units corresponding to (S 0 , S), thenC = DC describes the same consumption strategy in units corresponding to (S 0 ,S) (which is also attainable for those units).
Nonetheless, the condition δ + r(R − 1) > 0 is stronger than the condition for a well-posed problem (namely η > 0) and there are parameter values which we would like to consider (and which are covered by Theorem 6.1) for which the verification arguments of [6] and [2] do not apply, even after the change of numéraire arguments of this section.
Remark 7.1. An alternative formulation of the Merton problem is to associate to an attainable consumption stream C the expected utility where φ := δ + r(R − 1). Then K(C; φ) = J(C, φ − r(R − 1)). In order to emphasise the dependence of the problem on the currency units which are being used we might expand the notation to write J(C; S 0 , S; δ) and K(C; S 0 , S; φ) and then (23) becomes whilst, for K(C, φ) = K(C; S 0 , S, φ) we find In particular, K defined via (24) has the advantage that (unlike J) it is numéraire-independent in the sense that a change of currency unit leaves the problem value unchanged. With this in mind it makes sense to call φ (rather than δ) the impatience rate.
so that the optimal consumption rate is a linear (convex if R > 1) combination of the impatience rate and (half of) the squared Sharpe ratio per unit of risk aversion, with the weights depending on the risk aversion.

The Merton problem with bankruptcy revisited
Using the ideas of this section we can revisit the argument of Karatzas et al [6] to give a much simpler proof for V (x) =V (x) in the case that δ + (1 − R)r > 0.
The idea is to consider the case r = 0, which is not studied in Karatzas et al [6]. For r = 0, it is not difficult to check that the HJB equation (10)  which is substantially simpler than the solution (11) for r > 0. Since the argument in (A) and (C) of Section 5.1 carry verbatim over to r = 0, we have a verification argument for the Merton Problem in the case that δ > 0 and r = 0. Now if δ + r(R − 1) > 0, we choose γ = −r, so that δ = δ + r(R − 1) > 0 andr = 0, and the above change of numéraire argument in Section 7.1 give a verification argument also in this case.
Remark 7.2. Motivated by the above change of numéraire ideas, one might want to consider a generalisation of (9) to allow for different discount rates on the utility of consumption and the bankruptcy payout: For δ, χ > 0, let J P (C; δ, χ) = E Now let D t = e γt for some γ ∈ R be an exchange rate process, (S 0 ,S) = (DS 0 , DS) the corresponding economically equivalent Black-Scholes-Merton model with interest rater = r + γ, driftμ = µ + γ and volatilityσ = σ, C an attainable consumption strategy in units corresponding to (S 0 , S) andC = DC the corresponding attainable consumption strategy in units corresponding to (S 0 ,S). Then, noting that τ 0 is numéraire independent and recalling that C/S 0 =C/S 0 , we have J P (C; δ, χ) = E It follows from the above calculation that the problem with two different discount rates is mathematically equivalent to the problem with the same discount rates; indeed, choose γ = χ−δ 1−R , then J P (C; δ, χ) = J P (C; χ, χ). However, the problem (25) with the same discount rates does not have a simpler solution than the one with different discount rates-unless r = 0.
This suggests that it might be useful to consider the numéraire invariant analogue to (25) and to set Then K P has the property that it is invariant under a change of currency units: K P (C; φ, χ) = K P (C; φ, χ). Note that by construction J P (C; δ, χ) = K P (C; δ + r(R − 1), χ). In particular, J P (C, δ, δ) = K P (C, δ + r(1 − R), δ) so that the numéraire-dependent problem (25) with the same discount rates corresponds to the numéraire-independent problem with different discount rates-unless r = 0. Unlike the numéraire-dependent problem (25), the numéraire-independent problem (26) has a much simpler solution for the same discount rates than for different discount rates. Indeed, where 2R . This gives a further justification for considering the numéraireindependent formulation of the Merton problem (cf. Remark 7.1) or its bankruptcy variant.