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Consumption investment optimization with Epstein-Zin utility in incomplete markets

In a market with stochastic investment opportunities, we study an optimal consumption investment problem for an agent with recursive utility of Epstein-Zin type. Focusing on the empirically relevant specification where both risk aversion and elasticity of intertemporal substitution are in excess of one, we characterize optimal consumption and investment strategies via backward stochastic differential equations. The supperdifferential of indirect utility is also obtained, meeting demands from applications in which Epstein-Zin utilities were used to resolve several asset pricing puzzles. The empirically relevant utility specification introduces difficulties to the optimization problem due to the fact that the Epstein-Zin aggregator is neither Lipschitz nor jointly concave in all its variables.


Introduction
Risk aversion and elasticity of intertemporal substitution (EIS) are two parameters describing two different aspects of preferences: risk aversion measures agent's attitude toward risk, while EIS regulates agent's willingness to substitute consumption over time. However the commonly used time separable utilities force EIS to be the reciprocal of risk aversion, leading to a rich literature on asset pricing anomalies, such as the equity premium puzzle, the risk-free rate puzzle, the excess volatility puzzle, the credit spread puzzle, and etc.
Recursive utilities of Kreps-Porteus or Epstein-Zin type and their continuous-time analogue disentangle risk aversion and EIS, providing a framework to resolve aforementioned asset pricing puzzles, cf. [2] and [1] for the equity premium puzzle and the risk-free rate puzzle, [4] for the excess volatility puzzle, and [5] for the credit spread puzzle. All these studies require the EIS ψ to be larger than 1 in order to match empirical observations. Bansal and Yaron [2] also empirically estimated ψ to be around 1.5. On the other hand, empirical evidence suggests the risk aversion γ to be in excess of 1. It then follows from γ > 1 and ψ > 1 that γψ > 1. Hence such an agent prefers early resolution of uncertainty (cf. [25] and [35]), therefore asks a sizeable risk premium to compensate future uncertainty in the state of economy.
In aforementioned applications, investment opportunities are driven by some state variable, leading to market incompleteness. On the other hand, the first step in all these applications is to understand the state price density of the representative agent, because it is the source to read out equilibrium risk-free rate and risk premium, cf. [2,Appendix]. Therefore, it is important to rigorously study the consumption investment problem for an agent with Epstein-Zin utilities in incomplete markets and derive the associated state price density. However, such a study, in a continuous-time setting and under the empirically relevant specification γ > 1 and ψ > 1, was still missing from the literature. This paper fills this gap.
In the seminal paper Duffie and Epstein [11], stochastic differential utilities (the continuous-time analogue of recursive utilities) are assumed to have Lipschitz continuous aggregators. Hence the Epstein-Zin aggregator, which is non-Lipschitz, is excluded. Schroder and Skiadas [33] studied the case where θ = 1−γ 1−1/ψ is positive. 1 Hence they did not cover the γ > 1 and ψ > 1 case, which leads to θ < 0. Kraft, Seifried, and Steffensen [24] imposed a special relation between γ and ψ (cf. Equation (H) therein), which excludes the case γ > 1 and ψ > 1. This special relation was removed by Kraft,Seiferling,and Seifried [23] in models with bounded value of risk. Regarding market models, Schroder and Skiadas [33] assumed bounded market value of risk. Schroder and Skiadas [34,Section 5.6] studied an incomplete market whose investment opportunities are driven by a state variable, when the Epstein-Zin utility has unit EIS. Kraft, Seifried, and Steffensen [24] studied a market model whose investment opportunities are driven by a square root process, leading to unbounded market price of risk. Regarding the state price density, its form can be obtained by a heuristic calculation using the utility gradient approach, cf. [13]. However, rigorous verification needs the aggregator to satisfy a certain Lipschitz growth condition, cf. [11] and [13], or joint concavity in both consumption and utility variables, cf. [14], or an integrability condition, cf. [33, Lemma 2]. As we shall see later, when γ > 1 and ψ > 1, the Epstein-Zin aggregator is neither Lipschitz continuous nor joint concave. Moreover, the integrability assumption in [33] is hard to verify when the market value of risk is unbounded.
In this paper, we analyze the consumption investment problem for an agent with an Epstein-Zin utility with γ, ψ > 1 and a bequest utility at a finite time horizon. This agent invests in an incomplete market whose investment opportunities are driven by a state variable, and the market price of risk can be unbounded. In Theorem 2.12 below, optimal consumption and investment strategies are characterized. In contrast to the dynamic programming approach in [24] and [23], we employ a probabilistic approach via backward stochastic differential equations (BSDE), which have been applied to optimal consumption investment problems for time separable utilities, cf. [17] and [9]. These optimal strategies are verified using a combination of techniques in [17] and the Lyapunov function, borrowed from the risk sensitive control literature (cf. [15], [32] and references therein). This combination allows to treat models with unbounded market value of risk. In Theorem 2.14, the state price density is verified, confirming an assumption in [33] and meeting the demand coming from aforementioned applications on asset pricing puzzles.
The remaining of this paper is organized as follows. After the Epstein-Zin utility is introduced in Section 2.1, the consumption investment problem is introduced and main results are presented in Section 2.2. Then main results are specialized in two examples in Section 3, where statements of main results hold under explicit parameters restrictions. All proofs are postponed to the appendices.

Main results
2.1. Epstein-Zin preferences. We work on a filtered probability space (Ω, (F t ) 0≤t≤T , F, P). Here (F t ) 0≤t≤T is the smallest filtration generated by a k +n−dimensional Wiener process B = (W, W ⊥ ), where W and W ⊥ are the first k and the last n components, respectively, and satisfy the usual hypotheses of right-continuity and completeness.
Let C be the class of nonnegative progressively measurable processes on [0, T ]. For c ∈ C and t < T , c t stands for the consumption rate at t and c T represents a lump sum consumption at T . We consider an agent whose preference over C−valued consumption streams is described by a continuous time stochastic differential utility of Kreps-Porteus or Epstein-Zin type. To describe this preference, let δ > 0 represent the rate of time preference, 0 < γ = 1 be the relative risk aversion, and 0 < ψ = 1 be the (EIS). Define the Epstein-Zin aggregator f : This is a standard parametrization used, for instance, in [10]. Given a bequest utility function U (c) = c 1−γ /(1 − γ), the Epstein-Zin utility over the consumption stream c ∈ C on a finite time horizon T is Remark 2.1. The Epstein-Zin utility generalizes the standard time separable utility with constant relative risk aversion. Indeed, when γ = 1/ψ, the aggregator reduces to f (c, v) = δ c 1−γ 1−γ − δv. Then (2.2) can be represented explicitly as the standard time separable utility: As discussed in introduction, we are interested in the empirical relevant case where γ > 1 and ψ > 1. In this case, γ = 1/ψ is violated, hence (2.2) is not time separable.
When c follows a diffusion, the existence of V c was established by Duffie and Lions [12] via partial differential equation techniques. We work in a non-Markovian setting and construct V c via the following BSDE: Then f has super-linear growth in v when θ < 0. Therefore the BSDE (2.3) does not have a Lipschitz driver. Nevertheless, consider (Y t , Z t ) := e −δθt (1 − γ)(V c t , Z c t ) and the following transformed BSDE: When θ < 0, the driver F in (2.4) satisfies the monotonicity condition, i.e., y → F (t, c t , y) is decreasing. This allows us to establish the wellposedness of (2.3), hence define (2.2).
Let us introduce the set of admissible consumption streams as Proposition 2.2. Suppose γ, ψ > 1 and c ∈ C a . Then (2.4) admits a unique solution (Y, Z) in which Y is nonnegative and of class D. Moreover, Remark 2.3. When a BSDE satisfies the monotonicity condition, it is customary to assume its terminal condition to be square integrable, cf. [29,Theorem 1.3]. However this imposes unnecessary restrictions for later described utility maximization problem, in the sense that the bequest utility needs to be square integrable to define the associated Epstein-Zin utility. Therefore, Proposition 2.2 only asks for the terminal condition to be an integrable random variable.
Having defined V c 0 , we expect that, as a utility functional, C a ∋ c → V c 0 is concave. This would follow the standard argument when f (c, v) is jointly concave in c and v, cf. [11,Proposition 5]. However, calculation shows that f in (2.1) is not jointly concave when γ > 1 and ψ > 1. 3 Nevertheless, utilizing an orderly equivalent transformation of V c 0 , the following proposition confirms the concavity of Observe the driver of (2.5) is now jointly concave in all its variables when θ < 1.
Having established the Epstein-Zin utility in the previous section, we consider an optimal consumption investment problem for an agent with such a utility. Consider a model of a financial market with a risk free asset S 0 and risky assets S = (S 1 , . . . , S n ) with dynamics where diag(S) is a diagonal matrix with elements of S on the diagonal, 1 n is a n−dimensional vector with every entry 1. Given a correlation function ρ : R k → R n×k and ρ ⊥ : s defines a n−dimensional Brownian motion. In (2.6), X is a E−valued state variable satisfying Here E ⊆ R k is an open domain, r : E → R, µ : E → R n , σ : E → R n×n , b : E → R k , and a : E → R k×k . These model coefficients satisfy following assumptions.
Assumption 2.5. r, µ, σ, b, a, and ρ are all locally Lipschitz in E; r ≥ 0 and A := aa ′ is positive definite in any compact subdomain of E; moreover, (2.7) admits a unique strong solution.
An agent, whose preference is described by the Epstein-Zin utility, invests in this financial market. Given an initial wealth w, an investment strategy π, and a consumption rate c, the wealth of the agent follows (2.8) dW t = W t (r t + π ′ t µ t )dt + π ′ t σ t dW ρ t − c t dt, W 0 = w. Throughout the paper, r t , µ t , ρ t , σ t stand for r(X t ), µ(X t ), ρ(X t ), and σ(X t ), respectively. A pair of investment strategy and consumption stream (π, c) is admissible if its associated wealth process is nonnegative and c ∈ C a . The agent aims to maximize her utility V c 0 . We will further restrict admissible strategies to a permissible set. But let us first characterize the optimal value process via a heuristic argument. By the homothetic property of the Epstein-Zin utility, we speculate that the utility evaluated at optimal strategies has the following decomposition: where Y satisfies the following BSDE Let us determine the driver H in what follows. Parameterizing c byc = c/W, the wealth process satisfies We expect from the standard dynamic programming principle that V * + · 0 f (c * s , V * s )ds is a martingale. Using the previous three equations, calculation yields (2.11) Here, suppressing the subscript t, The supremum in (2.11) is attained at where W π * is the wealth process associated to the strategy π * . Coming back to (2.10), even though the driver H has an exponential term in y and a quadratic term in z, the parameter specification γ > 1 and ψ > 1 allows us to derive a priori bounds on Y . In particular, Y is bounded from above by a constant. Meanwhile, since the quadratic term of z in H will be shown to be nonnegative, the lower bound of Y can be obtained by studying a BSDE whose driver does not contain this quadratic term. As a result, a solution to (2.10) can be constructed under mild integrability conditions. Assumption 2.6.
Here E( α s dW s ) T := exp − 1 2 T 0 |α s | 2 ds + T 0 α s dW s denotes the stochastic exponential for T 0 α s dW s . Remark 2.7. The driver H has a nonnegative quadratic term in z and H(·, ·, 0) is bounded from above by −δθ, since γ > 1 and θ < 0. These properties are the reason why the standard exponential moment condition in [6] is avoid in Assumption 2.6.
In particular, since h ≤ 0, Y is bounded from above by −δθT .
Having constructed (Y, Z), the strategies (π * , c * ) in (2.12) are well defined. To verify their optimality, we need to further restrict the admissible strategies to a permissible set: (π, c) is permissible if c ∈ C a and (W π ) 1−γ e Y is of class D on [0, T ]. 4 To verify the optimality for (π * , c * ), let us introduce an operator F. For φ ∈ C 2 (E), where the dependence on x is suppressed on both sides. The function φ in the following assumption is called a Lyapunov function. Its existence facilities proving certain exponential local martingale is in fact martingale, hence verifying optimality of the candidate strategies. This strategy has been applied to portfolio optimization problems for time separable utilities, cf. [31].
The final assumption before the main results imposes an integrability assumption on the market price of risk λ. This ensures E T 0 e −δs (c * s ) 1−1/ψ ds < ∞, hence the admissibility for the candidate optimal consumption stream c * . Assumption 2.10. There exists λ : E → R n which satisfies µ = σλ and defines the minimal (local) martingale measure Q 0 for the discounted asset price via where W 0 := W ρ + · 0 λ s ds is a Q 0 −Brownian motion. 4 When the interest rate r and the square of sharp ratio µ ′ Σ −1 µ are bounded, (2.13) implies that Y is bounded from below as well. In this case, the definition of permissible strategies coincides with its analogue in [9] for time separable utilities with γ > 1.
Remark 2.11. When r and λ are bounded, Assumption 2.10 holds automatically and Assumption 2.9 is not needed. Indeed, Assumption 2.9 is used to prove the stochastic exponential in Lemma B.2 below is a martingale. When r and λ are bounded, h is bounded, hence H(·, 0, 0) is bounded as well. Therefore, (2.13) implies that Y is bounded, hence · 0 Z s dW s is a BMO-martingale, cf. eg. [28,Lemma 3.1]. Then the stochastic exponential in Lemma B.2 can be proved as a martingale directly. However many models do not have bounded market value of risk. Therefore we retain Assumptions 2.9 and 2.10 in their general forms. These conditions impose some market conditions which will be specified as explicit parameter restrictions in two examples in Section 3 below. Now we are ready to state our first main result.
Theorem 2.12. When γ, ψ > 1, let Assumptions 2.5, 2.6, 2.9, and 2.10 hold. Then π * and c * in (2.12) maximize the Epstein-Zin utility among all permissible strategies. Moreover, the optimal Epstein-Zin utility is given by The second main result below focuses on the state price density. Let us first define the optimal value process where π * is the optimal strategy in (2.12) and Y comes from Proposition 2.8. Schroder and Skiadas [33] conjectured in Assumption C3 therein that the state price density is This conjecture is verfied in [23, Lemma 2] under an integrability assumption. The constant w γ e −Y 0 in (2.17) normalizes D * 0 to be 1. Indeed, combining (2.1), (2.12) and (2.16), calculation shows Therefore the previous identity implies that D * 0 = 1 and D * is nonnegative. The following result confirms that D * is a state price density.
where Z comes from Proposition 2.8. Therefore, for any admissible strategy (π, c), W π D * + · 0 D * s c s ds is a nonnegative local martingale, hence a supermartingale.
The following result confirms that the local martingale in the previous result is in fact a martingale, when (π, c) is optimal. This result has been proved for recursive utilities with Lipszhitz continuous aggregator which is also jointly concave in all its variables, cf. [14,Theorems 4.2 and 4.3]. However, the Epstein-Zin aggregator here is neither Lipschitz, nor jointly concave, when γ > 1 and ψ > 1.
Theorem 2.14. When γ, ψ > 1, let Assumptions 2.5, 2.6, 2.9, and 2.10 hold. Then, for optimal strategy (π * , c * ) given in (2.12), W π * D * + · 0 D * s c * s ds is a martingale. Therefore, for any admissible strategy (π, c), The previous theorem confirms that D * in (2.17) is indeed the state price density associated to optimal strategies. In an equilibrium setting where the representative agent has an Epstein-Zin utility, given the consumption stream, equilibrium risk-free rate and risk premium can be read out from D * , providing a framework to study various asset pricing puzzles as discussed in introduction.

Examples
This section specifies general results in the previous section to two extensively studied models, where explicit parameter restrictions are given so that all assumptions in the previous section are satisfied, hence statements of Theorems 2.12 and 2.14 hold. These parameter restrictions covers many empirically relevant specifications.
3.1. Linear diffusion. Both the interest rate and the excess return of risky assets in the following model are linear functions of a state variable, which follows a 1−dimensional Ornstein-Uhlenbeck process. This model has been studied in [21] and [36] for the time separable utility setting. The model dynamics is given by where r(x) = r 0 + r 1 x, µ(x) = σ(λ 0 + λ 1 x), with r 0 , r 1 ∈ R, λ 0 , λ 1 ∈ R n , σ ∈ R n×n , b, a ∈ R, and ρ ∈ R n . These coefficients satisfy Clearly this model satisfies Assumption 2.5 with E = R. Under following parameter restrictions, statements of Theorems 2.12 and 2.14 hold.
Proposition 3.2. When γ, ψ > 1, let Assumption 3.1 and the following parameter restrictions hold: Then statements of Theorems 2.12 and 2.14 hold.
In the above item i), observe that Therefore item i) assumes that either X is mean-reverting under P or the excess rate of return has a linear growth component of the state variable. In item ii), the inequality asks either b, the mean-reverting speed of the state variable under P, is large, or the volatility a is small, or EIS is close to 1. In particular, when ψ > 1, the inequality in item ii) is satisfied when Hence the left hand side of the inequality in Proposition 3.2 ii) is negative.
This condition already covers many empirically relevant specifications. For example, in [3] and [36], a single risky asset was considered and parameter values (in monthly units) are: Taken ψ = 1.5 from [2], (3.2) is verified by calculation. Figure 1 demonstrates the optimal consumption wealth ratio c * /W π * and optimal investment fraction π * with respect to the state variable X. The first two plots are results for an Epstein-Zin utility agent with ψ = 1.5. The last two plots are for a time separable utility agent with ψ = 1/γ. Both of them use parameters in (3.3), r = 0.0042, δ = 0.0067 (in monthly units), and γ = 5. Comparing the first and third plots, we can see the big impact of ψ on the optimal consumption wealth ratio. 3.2. Stochastic volatility. The following model has a 1−dimensional state variable, following a square-root process as suggested by Heston, which simultaneously affects the interest rate, the excess return of risky assets and their volatility. This model has been studied by [26] in the time separable utility setting. This model is specified as follows: where r(x) = r 0 + r 1 x, µ(x) = σλx, with r 0 , r 1 ∈ R, σ ∈ R n×n , λ, ρ ∈ R n , and b, ℓ, a ∈ R. These parameters satisfy The previous assumption ensures that X is strictly positive, hence Assumption 2.5 is satisfied with E = (0, ∞). The following result provides parameter restrictions such that statements of Theorems 2.12 and 2.14 hold. Proposition 3.4. When γ, ψ > 1, let Assumption 3.3 and the following parameter restrictions hold: Then statements of Theorems 2.12 and 2.14 hold.
In the above item i), either the interest rate or the excess rate of return has a linear growth component of the state variable; the condition in item ii) is interpreted similarly as in Proposition 3.2 ii). In particular, when r 1 = 0 (i.e., the interest rate is a constant) and ψ > 1, the condition in item ii) is satisfied when This condition covers the empirically relevant specifications in [27], where the parameter values are  Figure 2 demonstrates the optimal consumption wealth ratio c * /W * and optimal investment fraction π * with respect to volatility √ X. The first two plots are results for an Epstein-Zin utility agent with ψ = 1.5. The last two plots are for a time separable utility agent with ψ = 1/γ. Both of them use parameters in (3.5), r = 0.05, δ = 0.08, and γ = 5. We can see from the first and third plots that the impact of ψ on the consumption-wealth ratio is large, whereas little impact on the optimal investment fraction from the second and fourth plots. Let us first introduce several notation which will used throughout the appendices.
• Let S 2 denote the space of all 1−dimensional continuous adapted processes (Y t ) 0≤t≤T such that the norm E sup 0≤s≤T |Y s | 2 < ∞.
• Let S ∞ be the subspace of S 2 such that the norm sup 0≤s≤T |Y s | ∞ < ∞.
• Denote by T the set of all F−stopping time τ such that 0 ≤ τ ≤ T . The process Y is of class D if the family {Y τ ; τ ∈ T } is uniformly integrable. • Let M 2 denote the class of (multidimensional) predictable processes (Z t ) 0≤t≤T such that Proof of Proposition 2.2. The proof is split into several steps. First when the terminal condition is bounded, the solution is constructed by slightly modifying the proof of [29,Theorem 1.3]. For general terminal conditions, the solution is obtained by the monotone convergence theorem in [22] and the localization technique in [6]. Finally, uniqueness is proved and (2.2) is verified. For simplicity of notation, we denote ξ = e −δθT c 1−γ T throughout this proof.
Step 1: Bounded terminal condition. When ξ 2 ≤ C for some constant C, consider the following truncated BSDE: it is differentiable at y = 0 due to θ < 0. Therefore (A.1) admits a unique solution (Y n , Z n ) ∈ S 2 × M 2 . The first component of such solution is also nonnegative. Indeed, consider (A.1) with zero as the terminal condition. Such BSDE admits a unique solution (Ỹ n ,Z n ) ≡ (0, 0) in S 2 × M 2 . Since ξ ≥ 0, it follows from the comparison theorem for BSDEs with Lipschitz drivers that Y n ≥Ỹ n = 0. On the other hand, since θ < 0, F n is decreasing in n, the comparison theorem then implies that (Y n ) n is decreasing. Hence Y :=↓ lim n→∞ Y n is well defined and nonnegative.
To take the limit of (Y n , Z n ) n , let us derive the following uniform estimate. Applying Itô's formula to (Y n ) 2 yields where the first inequality follows from Y n ≥ 0 and F n ≤ 0. It then follows that Therefore there exists Z ∈ M 2 such that (Z n ) n converges to Z weakly. Note that lim n→∞ F n (t, c t , y) = F (t, c t , y), lim n→∞ Y n = Y , and for any n, where the second inequality follows from θ < 0 and the third inequality holds due to the first estimate in (A.2). The dominated convergence theorem then implies that lim n→∞ T t |F n (s, c s , Y n s ) − F (s, c s , Y s )| ds = 0, for any t.
Now we prove the convergence of (Z n ) n in M 2 . Applying Itô's formula to |Y n − Y m | 2 yields where the first inequality holds due to the fact that y → F n (t, c t , y) is decreasing and the second inequality follows from the first estimate in (A.2). Since c ∈ C a , the dominated convergence theorem implies the right hand side of (A.3) converges to zero as n, m → ∞. Combining the previous convergence with the weak convergence of (Z n ) n , we obtain where P−lim stands for convergence in probability. Passing to a subsequence, we can obtain that the last convergence is P−a.s.. Therefore, sending n → ∞ in (A.1), we obtain that (Y, Z) ∈ S ∞ × M 2 solves (2.4) and Y is nonnegative. On the other hand, and taking limits on m and supremum over t, we obtain Therefore Y n converges to Y uniformly in t, implying that Y is a continuous process.
Step 2: General terminal condition. When ξ is not bounded, set ξ n = ξ ∧ n and consider which admits a solution (Y n , Z n ) ∈ S ∞ × M 2 with Y n ≥ 0 from results of the previous step. Moreover, since y → F (·, ·, y) satisfies the monotonicity condition, the comparison result in [29, for all n and t ∈ [0, T ]. Therefore Y :=↑ lim n→∞ Y n is well defined and satisfies This a priori bound allows us to construct solution to (2.4) via the localization technique in [6]. To connect to the proof of [6, Theorem 2], we observe that Then c ∈ C a implies E τ k 0 F (s, c s , Y s )ds < ∞. Therefore, even though F does not have at most linear growth in y due to the fact that c may not be bounded, [6,Lemma 3] can be replaced by [22,Proposition 2.4]. Following the same argument as in [6] after its Lemma 3, we obtain a solution (Y, Z). It also satisfies the terminal condition Y T = ξ. Indeed, on {ξ ≤ k}, τ k = T and Y n τ k = ξ for any n ≥ k. Then Y τ k = lim n→∞ Y n τ k = ξ on {ξ ≤ k}, which implies Y T = ξ since ↑ lim k→∞ {ξ ≤ k} = Ω. The construction of the solution also implies (A.4) is satisfied, hence Y is of class D.
Step 3: Uniqueness and (2.2). For future reference, we prove a comparison result for (2.4). Let (Y, Z) (resp. (Ỹ ,Z)) be a super-solution (resp. sub-solution) to (2.4), i.e., Moreover, both Y andỸ are of class D. Then Y ≥Ỹ . Indeed, define which is negative since y → F (·, ·, y) is decreasing. We then obtain implying that e · 0 αsds (Y −Ỹ ) is a local supermartingale, hence a supermartingale, since the exponential factor is bounded and both Y andỸ are of class D. Therefore, Y T ≥Ỹ T implies Y ≥Ỹ . The uniqueness follows from the comparison result directly.
Finally, we verify that V c satisfies (2.2). To this end, since (Y, Z) solves (2.4), (V c t , Z c t ) = e δθt (Y t , Z t )/(1 − γ) satisfies (2.3), implying that V c + · 0 f (c s , V c s )ds is a local martingale. Taking a localizing sequence (σ n ) n≥1 for V c + · 0 f (c s , V c s )ds, we obtain Sending n → ∞ on both sides, note that V c ≤ 0 and ψ > 1, therefore the integrand on the left side is negative and the integrand on the right side is positive. The monotone convergence theorem and the class D property of V c then yield where the second inequality holds since γ > 1 and θ < 0, the third inequality follows from (A.4) and γ > 1. Subtracting δθE T 0 V c s ds on both sides of (A.5), we confirm (2.2).
The concavity of c → V c is proved in the following. This proof utilizes simultaneously the joint concavity of the driver for (2.5) and the class D property of the solution to (2.4).

Proof of Proposition 2.4.
Denote the driver of (2.5) as F(t, c t , y, z) = δe −δt c Its joint concavity in all its variables implies, for any α ∈ [0, 1], c,c ∈ C a , ∆Y t ≤ e −δT ∆c where ∆X = αX + (1 − α)X, for X = c, Y, Z andX =c,Ỹ,Z, respectively. Define It then follows from θ < 0 that On the other hand, since θ < 0, Jensen's inequality yields where Y (resp.Ỹ ) is the first component of the solution to (2.4) with c (resp.c). The previous two inequalities and θ < 0 combined yield Therefore (αY + (1 − α)Ỹ , ∆Z) is a super-solution to (2.4) where c is replaced by ∆c. Moreover αY + (1 − α)Ỹ is of class D, because both Y andỸ are. Therefore, it follows from the comparison result in Step 2 of the previous proof that where Y ∆c is the first component of the solution to (2.4) where c is replaced by ∆c. Dividing the previous inequality by (1 − γ) on both sides, we confirm αV c Appendix B. Proofs in Section 2.2 Even though the driver H in (2.10) has an exponential term in y, the parameter specification γ > 1 and ψ > 1 allow us to derive a priori bounds for Y . Then a solution to (2.10) is constructed via the localization technique in [6].
Proof of Proposition 2.8. Due to Assumption 2.6 i), W := W − · 0 1−γ γ ρ ′ σ ′ Σ −1 µ(X s )ds is a P−Brownian motion. Therefore, we can rewrite (2.10) under P, and all expectations are taken with respect to P throughout this proof. On the other hand, in many widely used models, µ ′ Σ −1 µ is an unbounded function of the state variable, hence H(t, 0, 0) = h t − δθ + θ δ ψ ψ is not bounded. Therefore we introduce Since γ > 1 and ψ > 1, θ < 0. Due to r ≥ 0 and µ ′ Σ −1 µ ≥ 0, γ > 1 implies h ≤ 0. Consider a truncated version of (B.1): Here ξ n = T 0 h s ∨ (−n) − δθ ds is bounded and This truncated driver H n is Lipschitz in y and quadratic in z. Indeed, since eigenvalues of It then follows from [  On the other hand, when y − t 0 h s ∨ (−n) − δθ ds ≤ −δθ(T − t), the first inequality in (B.3) and θ < 0 imply H n (t, y, z) ≥ θ δ ψ ψ e δψT . Therefore consider where Y admits a representation Y t = E t [ξ] + θ δ ψ ψ e δψT (T − t). Now since H n is sandwiched between two drivers with simpler form, Y n satisfies for any n > 0. These uniform bounds on Y n allow us to construct a solution (Y, Z) to (B.1) using the localization technique in [6], and Y satisfies The previous inequalities imply that Y satisfies the terminal condition of (B.1). The desired estimates on Y follows after subtracting t 0 h s − δθ ds on both sides of the previous inequalities.
The following several results prepare the proofs of Theorems 2.12 and 2.14. First we show w 1−γ 1−γ e Y 0 is an upper bound for the optimal value among permissible strategies.
Lemma B.1. Let Assumption 2.6 hold. For any permissible (π, c), where V c is defined in Proposition 2.2, Y is constructed in Proposition 2.8, and c is financed by π via (2.8).
Proof. This proof extends the technique in [17] to recursive utilities. For a permissible (π, c), define Using the second identity in (2.11), calculation shows that R has a negative and decreasing in time finite variation part. Hence, we obtain, using Y T = 0, for some Z R . Here due to the assumption that (W π ) 1−γ e Y is of class D and In what follows we will show that (π * , c * ) is a permissible strategy and it attains the upper bound w 1−γ 1−γ e Y 0 . First, we establish an important result that certain exponential local martingale associated to π * is a martingale. Proof. It follows from (2.12), the definition of W ρ and M that Here we suppress time subscripts to simplify notation.
Here F W = σ(W s ; 0 ≤ s ≤ T ), the third identity follows from [18,Lemma 4.8] since L (2) and W ⊥ are independent, and the fourth identity holds due to the martingale assumption on Q (1) . In the remaining of the proof, we will prove the martingale property of Q (1) .
For the sequence of subdomains (E n ) n in Assumption 2.9 i), define τ n := inf{t ≥ 0 | X t / ∈ E n }∧T . we first prove that Y ·∧τn is bounded. Since we have seen in Proposition 2.8 that Y is bounded from above, it suffices to show E P ·∧τn T ·∧τn h s ds is bounded from below. Then (2.13) implies that Y ·∧τn is bounded as well. Due to the Markovian structure, define The Feynman-Kac formula (see [16] when the equation is not uniformly parabolic) implies that, under Assumption 2.5, y ∈ C 1,2 ([0, T ] × E) and it is the unique solution to where L is the infinitesimal generator of X under P. Now since E n is compact, the continuity of y implies that y(· ∧ τ n , X ·∧τn ) is bounded.
As a solution to (2.10), (Y, Z) satisfies Since both X ·∧τn and Y ·∧τn are bounded, H(·, y, z) is Lipschitz in y and quadratic in z, and H(·, 0, 0) is bounded. It then follows from the BMO-estimate for quadratic BSDEs (cf. eg. s dW s ) τn defines Q n on F τn which is equivalent to P.
Assuming that lim n→∞ Q n (τ n < T ) = 0, by the monotone convergence theorem, proving the martingale property of E( L s dW s ) on [0, T ]. It remains to prove lim n→∞ Q n (τ n < T ) = 0. To this end, (2.10) yields On the other hand, by Itô's formula, Taking difference of the previous two identities, when t ≤ τ n , s ds is a Q n −Brownian motion on [0, τ n ]. On the right hand side, the quadratic term is nonnegative, −θ δ ψ ψ e − ψ θ Yt is nonnegative since θ < 0, and δθ − F[φ] is also bounded from below due to Assumption 2.9 ii). Therefore, The stochastic integral on the right hand side has zero expectation under Q n . Indeed, since (Z s − ∇φ ′ a(X s ))dW n s is a BMO(Q n )−martingale. Therefore its expectation is zero. Therefore Since Y is bounded from above and φ is bounded from below due to Assumption 2.9 i), there exists a constant C, such that Now sending n → ∞ in (B.9), Assumption 2.9 i) and the previous inequality confirms lim n→∞ Q n (τ n < T ) = 0.
The martingale property in the previous result helps to verify the permissibility of (π * , c * ). Proof. Denote W π * by W * . The calculation leading to (2.11) yields where the second identity follows from the form of c * in (2.12). Therefore, Since θ < 0 and Y is bounded from above, the second exponential term on the right is bounded, uniformly in t. Meanwhile, due to Lemma B.2, the stochastic exponential on the right is of class D on [0, T ]. Then the statement is confirmed.
Lemma B.4. Let Assumptions 2.5, 2.6, 2.9, and 2.10 hold. Let c * be in (2.12) and On the other hand, the expression ofc * in (2.12) implies Since ψ > 1, θ < 0, and Y is bounded from above, the first three terms on the right hand side are bounded. Therefore it suffices to prove To this end, it follows from Assumption 2.10 that Here the first inequality follows from r ≥ 0 and ψ > 1; the second inequality holds due to Hölder's inequality; the third inequality is obtained using the fact that e − · 0 rsds W π * is a nonnegative Q 0 −local martingale, hence a Q 0 −supermrtingale; and the fourth inequality holds thanks to (2.15). Now we are ready to prove the first main result.
Proof of Theorem 2.12. Corollary B.3 and Lemma B.4 have already shown that (π * ,c * ) is permissible. Choosing (π * , c * ), the inequality in (B.7) is an identity. Then the class D property of W π * 1−γ e Y and Proposition 2.8 combined imply Therefore the upper bound in Lemma B.1 is attained by (π * , c * ).
Finally, we prove Lemma 2.13 and Theorem 2.14.
Proof of Theorem 2.14. It follows from (2.12) and (2.18) that Here Since θ < 0, C is bounded from above by a constant. We have already seen in Lemma 2.13 that W π * D * + · 0 D * s c * s ds is a nonnegative local martingale. It suffices to prove that it is of class D. To this end, it follows from (B.11) that Here since W π * 1−γ e Y is of class D, E W π * s 1−γ e Ys is bounded uniformly in s. Therefore the previous inequality holds. On the other hand, using the expression of c * in (2.12), Then ψ > 1 and the previous two equations combined yield that the second term on the right hand side of (B.12) is bounded from above by an integrable random variable, hence is of class D.
Meanwhile, using the class D property of W π * 1−γ e Y again, the first term on the right of (B.12) is also of class D. This confirms the class D property of W π * D * + · 0 D * s c * s ds.

Appendix C. Proofs in Section 3
To prove Proposition 3.4, let us recall the following result on the Laplace transform of the integrated square root process; see [30,Equation (2.k)] or [7,Equation (3.2)].
Lemma C.1. Consider X with dynamics where W is a 1−dimensional Brownian motion. When

Now Propositions 3.2 and 3.4 are proved.
Proof of Proposition 3.2. Assumptions 2.6, 2.9, and 2.10 are verified one by one. Then statements of Theorems 2.12 and 2.14 follow. We denote Θ = σ ′ Σ −1 σ throughout the proof to simplify notation.
When −b + 1−γ γ aλ ′ 1 Θρ < 0, since γ > 1, 1−γ 2γ λ ′ 1 Θλ 1 ≤ 0, we can choose sufficiently small c such that F[φ] ↓ −∞ as |x| ↑ ∞. When λ ′ 1 Θλ 1 > 0, then 1−γ 2γ λ ′ 1 Θλ 1 < 0, we can also choose sufficiently small c such that F[φ] has the same asymptotic behavior. In both cases, F[φ] is bounded from above on R, hence Assumption 2.9 is verified. Assumption 2.10: Consider the martingale problem associated to L 0 : x on R. Since all coefficients have at most linear growth, this martingale problem is well-posed and its solution, denoted by Q ρ , satisfies dQ ρ Here, due to the independence between X and W ⊥ , proof similar to (B.8) implies that the first stochastic exponential on thr right is a P−martingale; hence Q 0 is well defined. Therefore λ in Assumption 2.10 can be chosen as λ 0 + λ 1 X and Q 0 is the minimal martingale measure for the discounted asset price.
To verify (2.15), note Combining the previous two change of measures, the dynamics of X can be rewritten as where W ψ := W + · 0 (1 − ψ)(λ 0 + λ 1 X s ) ′ ρds is a 1−dimensional Q ψ −Brownian motion. On the other hand, calculation shows, for any ǫ > 0, where C is a constant and C ǫ is a constant depending on ǫ.
In order to appeal Lemma C.1 to calculate the expectation on the right hand side of (C.2), let us introduce another measureQ ψ via dQ ψ dQ ψ = E −(ψ − 1)λ ′ 0 ρW ψ T . Under this measure, X has dynamics dX t = − b − (ψ − 1)aλ ′ 1 ρ X t dt + adW ψ t , whereW ψ := W ψ + · 0 (ψ − 1)λ ′ 0 ρ ds is aQ ψ −Brownian motion. Let Y := X 2 . It then has dynamics which is of the same type of X in Lemma C.1. Come back to (C.2), Hölder's inequality implies, for any δ > 0, Observe that the first expectation on the right hand side is finite, since dQ ψ dQ ψ = E (ψ − 1)λ ′ 0 ρW ψ T has all finite moments. For the second expectation, we can choose sufficiently small δ and ǫ such that according to Lemma C.1, when the second expectation is finite. Now combining the previous estimates and (C.2), we confirm (2.15). Finally, note that (C.3) is exactly the assumption in Proposition 3.2 ii).
Consider the martingale problem associated to L := bℓ − b − 1−γ γ aλ ′ Θρ x ∂ x + 1 2 a 2 x∂ 2 x on (0, ∞). Since bℓ > 1 2 a 2 , Feller's test of explosion implies that the previous martingale problem is well-posed. Then [8,Remark 2.6] implies that the stochastic exponential in Assumption 2.6 i) is a P−martingale, hence P is well defined. For Assumption 2.6 ii), h(x) = (1 − γ)r 0 + (1 − γ)r 1 + 1−γ 2γ λ ′ Θλ x. Since X has the following dynamics under P: where W is a P−Brownian motion. Then E P [ T 0 h(X s )ds] > −∞ follows from the fact that E P [X s ] is bounded uniformly for s ∈ [0, T ]. Assumption 2.9: The operator F in (2.14) reads λ ′ Θλx, whereM = 1 + 1−γ γ ρ ′ Θρ. Consider φ(x) = −c log x + cx, for two positive constants c and c determined later. It is clear that φ(x) ↑ ∞ when x ↓ 0 or x ↑ ∞. On the other hand, calculation shows where C is a constant. Since bℓ > 1 2 a 2 , the coefficient of 1/x is negative for sufficiently small c. When r 1 or λ ′ Θλ > 0, since γ > 1, the coefficient of x is negative for sufficiently small c. Therefore, these choices of c and c imply that F[φ](x) ↓ −∞ when x ↓ 0 or x ↑ ∞, hence F[φ] is bounded from above on R, verifying Assumption 2.9. Assumption 2.10: Consider the martingale problem associated to L 0 := [bℓ − bx − aρ ′ λx] ∂ x + + 1 2 a 2 x∂ 2 x on (0, ∞). Since bℓ > 1 2 a 2 , Feller's test of explosion implies that this martingale problem is well-posed and its solution, denoted by Q ρ , satisfies dQ ρ dP = E −λ ′ ρ √ X s dW s T . Define Q 0 via Argument similar to (B.8) implies that Q 0 is well defined and λ in Assumption 2.10 can be chosen as λ √ X. To verify (2.15), note where W 0 := W ρ + · 0 λ √ X s ds is a Q 0 −Brownian motion. Following the construction of Q 0 , one can similarly show E ψλ ′ √ X s dW 0 s is a Q 0 −martingale. Hence Q ψ can be defined via dQ ψ dQ 0 := E ψλ ′ X s dW 0 s T .
Then Lemma C.1 implies that the expectation on the right hand side is finite when This is exactly the assumption in Proposition 3.4 ii).