Epstein-Zin Utility Maximization on Random Horizons
EEpstein-Zin Utility Maximization on Random Horizons
Joshua Aurand ∗ Yu-Jui Huang † March 9, 2020
Abstract
This paper solves the consumption-investment problem under Epstein-Zin preferences on arandom horizon. In an incomplete market, we take the random horizon to be a stopping timeadapted to the market filtration, generated by all observable, but not necessarily tradable, stateprocesses. Contrary to prior studies, we do not impose any fixed upper bound for the randomhorizon, allowing for truly unbounded ones. Focusing on the empirically relevant case where therisk aversion and the elasticity of intertemporal substitution are both larger than one, we charac-terize optimal consumption and investment strategies through backward stochastic differentialequations (BSDEs). Compared with classical results on a fixed horizon, our characterizationinvolves an additional stochastic process to account for the uncertainty of the horizon. Asdemonstrated in a Markovian setting, this added uncertainty drastically alters optimal strate-gies from the fixed-horizon case. The main results are obtained through the development of newtechniques for BSDEs with superlinear growth on unbounded random horizons.
MSC (2010):
JEL:
G11, C61.
Keywords:
Consumption-investment problem, Epstein-Zin utilities, Random horizons, Back-ward stochastic differential equations.
Classical time-separable utilities unintentionally impose an artificial relation between agents’ riskaversion (denoted by γ ) and elasticity of intertemporal substitution (EIS, denoted by ψ ): the latterhas to be the reciprocal of the former. Such a relation is widely rejected empirically. Bansal andYaron [3], Bansal [2], Bhamra [6], and Benzoni [5] all point to the fact ψ > γ >
1. To disentangle EIS from risk aversion, Epstein and Zin [21] specifies arecursive utility in discrete time, whose continuous-time counterpart is formulated in Duffie andEpstein [18]. These Epstein-Zin type utilities have proved instrumental to resolve observed marketanomalies; see [3], [2], [6], and [5], among others.Since its inception in [21] and [18], the consumption-investment problem under Epstein-Zinpreferences has been extensively studied, yet predominantly on a fixed time horizon
T >
0; see e.g.[19], [42], [31], [43], [30], and [46]. In practice, an agent need not have a fixed planning horizon in ∗ University of Colorado, Department of Applied Mathematics, Boulder, CO 80309-0526, USA, email: [email protected] . † University of Colorado, Department of Applied Mathematics, Boulder, CO 80309-0526, USA, email: [email protected] . Partially supported by National Science Foundation (DMS-1715439) and the Universityof Colorado (11003573). a r X i v : . [ q -f i n . M F ] M a r ind, upon entering the market. The time to exit is usually random, depending on various factorswithin and/or beyond the market.In this paper, we investigate optimal consumption and investment under Epstein-Zin preferenceson a random horizon. Specifically, we focus on the empirically relevant specification γ , ψ >
1, andconsider an incomplete market in which an agent observes all state processes, but cannot trade allof them. The random horizon τ is taken to be a stopping time adapted to the market filtration,generated by all observable (but not necessarily tradable) state processes. That is, whether time τ has arrived depends on market situations, but the involved uncertainty can only be partially hedgedagainst by trading in the market.Prior studies on a random horizon τ , all with time-separable utilities, include [47], [36], [23](where τ is independent of the market), [27] (where τ depends completely on the market), [7],[26] (where τ depends on both the market and other external factors), among others. In all theseworks, τ is required a priori to be bounded (i.e. τ ≤ T a.s. for a known T >
T > τ drastically alters optimal consumption andinvestment strategies. Compared with the fixed-horizon optimal strategies in Xing [46], our optimalstrategies, given in (3.9) below, involve an additional process ˆ Z , indispensable to account for therandomness of τ . In a Markovian setting, our optimal strategies can be expressed through thesolution to a Dirichlet problem, with a boundary prescribed by τ . Such a result is comparable toKraft, Seifried, and Steffensen [30, Theorem 5.1] on a fixed horizon T >
0. The key distinctionis that optimal strategies in [30] are specified via the solution to a Cauchy problem, where theboundary condition is imposed only at the fixed terminal time T .In the Heston model of stochastic volatility, we demonstrate explicitly the effect of a randomhorizon. Under a practical parameter specification, the fixed-horizon optimal investment strategydictates a constant proportion of wealth in the risky asset, regardless of market evolution. Bycontrast, on a random horizon that reflects an agent’s tolerance to extreme market situations,the optimal investment strategy becomes a function of market states; see Section 4.2 for detailedderivations and numerical illustrations.Our analysis is based on techniques of backward stochastic differential equations (BSDEs). On afixed horizon, the BSDE approach is fairly versatile for Epstain-Zin utility maximization, as shownin [46]. A random horizon, nonetheless, poses a series of nontrivial challenges.The first step of our investigation is to prove the existence of the Epstein-Zin utility process,given a consumption stream ( c t ) t ≥ . This translates into solving a random-horizon BSDE with non-uniform superlinear growth: its generator grows super-linearly in one variable, and the growthis not uniform in other variables. The BSDE literature on random horizons τ frequently imposes“ τ ≤ T a.s. for a fixed T > τ (see e.g. [16], [39], [29], [41], and [9]), none of them allows for non-uniform superlineargrowth of the generator. In response to this, we introduce a truncated BSDE on the interval [0 , n ],for all n ∈ N . With a fixed horizon n , the construction in [46] can be used to deal with the non-uniform superlinear growth, so that a unique solution to each truncated BSDE exists. Motivatedby Pardoux [39] and Briand and Carmona [9], we prove that this sequence of solutions is Cauchyin a complete space of stochastic processes. The limit, as n → ∞ , exists and solves the originalrandom-horizon BSDE. See Proposition 2.1 and Theorem 2.1 for details.Next, we look for consumption and investment strategies that maximize the Epstein-Zin util-ity. By dynamic programming, we derive a random-horizon BSDE, i.e. (3.6) below, from whichcandidate optimal strategies can be deduced. This BSDE is non-standard in that its generator has2uadratic and exponential growth in several different variables. To find a solution to it, our planis to contain the exponential growth of the generator, so that results for random-horizon quadraticBSDEs can be applied. This results in a new, delicate truncation technique, forcing the generatorto be of linear growth and strictly increasing in one specific variable. Note that the truncation usedin [46] does not work here, as it requires a bounded horizon; see Remark B.1. With the exponentialgrowth contained, a careful use of the existence result in Briand and Confortola [10, Theorem 3.3],followed by a comparison principle in Kobylanski [29, Theorem 2.3], yields a solution to the BSDE;see Proposition 3.1.Now, with the candidate optimal strategy ( π ∗ , c ∗ ) constructed, it remains to show its optimalityamong the set of permissible strategies, defined in (3.11) below. On a fixed horizon, BMO argumentsare useful in establishing the permissibility of strategies, as shown in [46]. The same technique,however, breaks down in our case: the BMO norms can easily blow up on an unbounded randomhorizon, even when the solution to the aforementioned BSDE is desirably bounded. To proceed, weimpose appropriate exponential moment conditions on the random horizon τ (i.e. Assumption 2below), from which the permissibility of ( π ∗ , c ∗ ) can be extracted; see Lemma 3.1. The optimalityof ( π ∗ , c ∗ ) then follows from standard arguments; see Theorem 3.1, the main result of this paper.In view of the literature, the exponential moment conditions imposed on τ are not very re-strictive. They cover all prior studies on random-horizon consumption-investment problems (where“ τ ≤ T a.s. for a fixed T >
0” is imposed), and allow for many unbounded τ ’s. Moreover, thistype of conditions are commonly-seen for random-horizon BSDEs; see Remark 3.4.Finally, we recast our general results in a Markovian setting. The BSDE used to find optimalstrategies reduces to a Hamilton-Jacobi-Bellman equation, which in turn simplifies to an ellipticboundary value problem. Under appropriate conditions, this boundary value problem has a uniqueclassical solution; see Theorem 4.1, the Markovian counterpart of Proposition 3.1. This analysisfacilitates comparing our paper with prior studies on Epstein-Zin utilities, many of which focus onthe Markovian case. It also facilitates numerical computation, from which we directly observe thesignificant impact of a random horizon on optimal strategies, as mentioned above.It is of great interest to consider more general random horizons, than the ones adapted solely to the market filtration as we focus on in this work. The companion paper, Aurand and Huang [1],has pursued this direction: it studies optimal consumption, investment, and healthcare spendingunder Epstein-Zin preferences over an agent’s random lifetime; namely, τ is the death time of theagent, which need not depend on the financial market.The rest of the paper is organized as follows. Section 2 establishes the existence and unique-ness of the Epstein-Zin utility process, for a given consumption stream. Section 3 introduces theconsumption-investment problem, derives candidate optimal strategies by dynamic programming,and proves that they are indeed optimal. Section 4 connects our BSDE framework to a Markoviansetting. It shows theoretically that a random horizon can drastically change optimal strategies,and demonstrates it through a numerical example. Appendices collect all the proofs. Let (Ω , F , P ) be a probability space that supports a d -dimensional Brownian motion ( B t ) t ≥ . Let F = ( F t ) t ≥ be the P -augmentation of the natural filtration generated by B , and T be the set ofall F -stopping times. We consider a random horizon τ ∈ T .An agent obtains utility from a consumption stream c = ( c t ) ≤ t ≤ τ , a nonnegative progressivelymeasurable process, defined on the random horizon [0 , τ ]. Here, c t represents the consumption rateat time t for all 0 ≤ t < τ , while c τ stands for a lump-sum consumption at time τ . Let δ > γ > (cid:54) = 1 be the relative risk aversion, and ψ > U ( c ) = c − γ − γ . Then,given a consumption stream c , the Epstein-Zin utility on the random time horizon τ is a process V c = ( V ct ) t ≥ that satisfies V ct = E t (cid:20) (cid:90) τt ∧ τ f ( c s , V cs ) ds + c − γτ − γ (cid:21) , ∀ t ≥ , (2.1)where E t [ · ] stands for E [ · | F t ] and f : [0 , ∞ ) × ( −∞ , → R , the Epstein-Zin aggregator, is definedby f ( c, v ) := δ (1 − γ ) v − ψ (cid:32)(cid:18) c ((1 − γ ) v ) − γ (cid:19) − ψ − (cid:33) = δ c − ψ − ψ (cid:0) (1 − γ ) v (cid:1) − θ − δθv, with θ := 1 − γ − ψ . (2.2)In this paper, we focus on the specification γ, ψ >
1, which is the empirically relevant case, asdiscussed in the introduction. Note that this implies θ <
0, which will be used frequently.The goal of this section is to establish existence and uniqueness of the Epstein-Zin utility V c in (2.1). This has been done only for the fixed-horizon case (i.e. τ ≡ T for a fixed T >
0) or the θ > V c in (2.1) via the BSDE V ct = c − γτ − γ + (cid:90) τt ∧ τ f ( c s , V cs ) ds − (cid:90) τt ∧ τ Z cs dB s , ∀ t ≥ . (2.3)As observed in [46], with γ, ψ > f ( c, v ) has superlinear growth in v and is thus non-Lipschitz.Following the transformation in [46, Section 2.1], we consider ( Y t , Z t ) := e − δθt (1 − γ )( V ct , Z ct ), withthe corresponding BSDE Y t = e − δθτ c − γτ + (cid:90) τt ∧ τ F ( s, c s , Y s ) ds − (cid:90) τt ∧ τ Z s dB s , ∀ t ≥ , (2.4)where F ( t, c, y ) := δθe − δt c − ψ y | y | − θ . It is expected that (2.4) is more manageable as it satisfies the monotonicity condition : F ( t, c, y ) isdecreasing in y , thanks to θ < y ≥ C := (cid:26) c ∈ R + : E (cid:20) (cid:90) τ e − δs c − ψ ) s ds (cid:21) < ∞ and E (cid:20) e − δθ (2 − θ ) τ c − θ )(1 − γ ) τ (cid:21) < ∞ (cid:27) , (2.5)where R + is the set of all nonnegative progressively measurable processes. Remark 2.1.
Our admissible set C is larger when compared to those used in prior studies onEpstein-Zin utilities (where “ τ ≡ T for a fixed T > ” is normally assumed). The commonly-usedadmissible set (see e.g. [42]) requires the more stringent integrability condition E (cid:20) (cid:90) T c (cid:96)t dt (cid:21) < ∞ and E [ c (cid:96)T ] < ∞ , for all (cid:96) ∈ R . ecently, [46] proposed a much weaker condition E (cid:20) (cid:90) T e − δs c − /ψs ds (cid:21) < ∞ and E (cid:2) c − γT (cid:3) < ∞ . The integrability imposed in C is stronger than this. As will be seen below, the additional integrabilityhelps extend results in [46] from a fixed horizon to a random one. To state the main result of this section, let us introduce some notation. • For any q >
1, let S q denote the set of R -valued progressively measurable processes Y suchthat (cid:107) Y (cid:107) q S q := E [sup t ≥ | Y t ∧ τ | q ] < ∞ . • Let S ∞ denote the set of R -valued progressively measurable processes Y such that (cid:107) Y (cid:107) ∞ :=inf { C ≥ | Y t | ≤ C ∀ t ≥ } < ∞ . • For any q >
1, let M q denote the set of R d -valued progressively measurable processes Z suchthat (cid:107) Z (cid:107) q M q := E [( (cid:82) ∞ (cid:107) Z t (cid:107) dt ) q ] < ∞ . • For any q > B q := S q × M q , with the norm (cid:107) ( Y, Z ) (cid:107) q B q := (cid:107) Y (cid:107) q S q + (cid:107) Z (cid:107) q M q . Proposition 2.1.
Suppose γ, ψ > and c ∈ C . Then, (2.4) admits a unique solution ( Y, Z ) in B with Y ≥ a.s. The proof of Proposition 2.1 is relegated to Appendix A.1.
Remark 2.2.
Contrary to what it may seem, Proposition 2.1 does not follow from [11, Theorem5.2]. Indeed, our condition c ∈ C is weaker than the integrability required in [11]. For [11, Theorem5.2] to be applicable here, we would need a stronger condition on c : there exists (cid:15) > such that E (cid:20) (cid:90) τ e − δ − (cid:15) ) s c − ψ ) s ds (cid:21) < ∞ and E (cid:20) e − δ + (cid:15) ) θ (2 − θ ) τ c − θ )(1 − γ ) τ (cid:21) < ∞ . Remark 2.3.
To prove Proposition 2.1, Appendix A.1 constructs a sequence of solutions, usingthe finite-horizon construction in [46], whose limit solves (2.4) . Alternatively, one could constructa sequence of solutions via bounded consumption streams, using [41, Theorem 3.1], whose limitsolves (2.4) by a monotonicity argument. The proof, however, is no simpler than Appendix A.1,and conditions similar to (2.5) must be imposed on c for the monotonicity argument to hold. The Epstein-Zin utility process can now be constructed.
Theorem 2.1.
Suppose γ, ψ > . For any c ∈ C , let ( Y, Z ) be the unique solution of (2.4) in B .Then, ( V ct , Z ct ) := e δθt − γ ( Y t , Z t ) is the unique solution to (2.3) in B that satisfies (2.1) a.s. The proof of Theorem 2.1 is relegated to Appendix A.2.
Remark 2.4.
It is worth noting that
Y, V c ∈ S particularly implies that they are of class D. In this section, we introduce, under the framework of Section 2, the consumption-investment prob-lem in an incomplete market. By dynamic programming, we derive a BSDE on a random horizon τ ,from which candidate optimal strategies can be deduced. Under appropriate conditions on marketcoefficients (i.e. Assumption 1), a solution to the BSDE exists; see Proposition 3.1. On strengthof certain exponential moment conditions on τ (i.e. Assumption 2), the candidate strategies, givenin (3.9) below, are indeed optimal among an appropriate class of strategies; see Theorem 3.1.5 .1 The Setup We take up the framework in Section 2, with B = ( W, ˆ W ) being a two-dimensional Brownianmotion, i.e. d = 2. Let E be an open domain in R , and consider an E -valued state process dY t = a ( t, Y t ) dt + b ( t, Y t ) dW t , Y = y ∈ E, (3.1)where a, b : R + × E → R are given Borel measurable functions. The market consists of a riskfreeasset S and a risky asset S t , satisfying the dynamics dS t = r ( t, Y t ) S t dt,dS t = S t (cid:16) ( r ( t, Y t ) + λ ( t, Y t )) dt + σ ( t, Y t ) (cid:16) ρ ( t, Y t ) dW t + ˆ ρ ( t, Y t ) d ˆ W t (cid:17)(cid:17) , (3.2)with r, λ, σ, ρ, ˆ ρ : R + × E → R given Borel measurable functions. In particular, ρ and ˆ ρ , called thecorrelation functions, satisfy ρ ( t, y ) + ˆ ρ ( t, y ) = 1 for all ( t, y ) ∈ R + × E .An agent, with initial wealth x >
0, must decide a proportion π t ∈ R of wealth to invest in therisky asset and a consumption rate c t ≥
0, at every moment t ≥ τ . The corresponding wealth process X π,c is given by dX t = X t (cid:104) ( r t + π t λ t )) dt + π t σ t (cid:0) ρ t dW t + ˆ ρ t d ˆ W t (cid:1)(cid:105) − c t dt, = X t [( r t + π t λ t )) dt + π t σ t dW ρt ] − c t dt, X = x, (3.3)where r t , λ t , σ t , ρ t , ˆ ρ t represent r ( t, Y t ), λ ( t, Y t ), σ ( t, Y t ), ρ ( t, Y t ), ˆ ρ ( t, Y t ), respectively, and W ρt := (cid:90) t ρ s dW s + (cid:90) t ˆ ρ s d ˆ W s , t ≥ , is again a Brownian motion. We enforce the following conditions on the market coefficients. Assumption 1.
The coefficients σ, r, λ, ρ, ˆ ρ, a, and b are locally Lipschitz in E ; the process Y doesnot reach the boundary of E in finite time a.s.; { r t ∧ τ } t ≥ and { λ t ∧ τ σ t ∧ τ } t ≥ are bounded processes (i.e.belong to S ∞ ); inf K σ ( t, y ) > and inf K b ( t, y ) > for any compact subset K of R + × E . A strategy ( π, c ) is called admissible if it belongs to A := { ( π, c ) : c ∈ C , c τ = X π,cτ , X π,ct > ≤ t ≤ τ a.s. } . The agent intends to maximize her Epstein-Zin utility V c by choosing a pair ( π ∗ , c ∗ ) from someappropriate collection P ⊆ A . That is, the goal is to find the optimal value V ∗ := sup ( π,c ) ∈P V c , (3.4)where V c is the solution to (2.1) with c τ = X π,cτ , and the corresponding optimal ( π ∗ , c ∗ ) ∈ P .The collection P ⊆ A is up to the agent’s choice. In this paper, we will take P to be the set of permissible strategies, defined precisely in (3.11) below.6 .2 The Ansatz Motivated by the classical decomposition of time-separable power utilities (see e.g. [40, Section 3])and the decomposition of the Epstein-Zin utility in [46, (2.9)] on a fixed horizon, we suspect thatthe optimal utility process V ∗ can be decomposed into V ∗ t = X − γt ∧ τ − γ e D t ∧ τ t ≥ , (3.5)where D is a process satisfying the BSDE D t = (cid:90) τt ∧ τ H ( s, D s , Z s , ˆ Z s ) ds − (cid:90) τt ∧ τ Z s dW s − (cid:90) τt ∧ τ ˆ Z s d ˆ W s , t ≥ , (3.6)for some generator H to be determined. Note that (3.5) and (2.1) suggest that the process t (cid:55)→ X − γt ∧ τ − γ e D t ∧ τ + (cid:90) t ∧ τ f (cid:32) c s , X − γs − γ e D s (cid:33) ds (3.7)should be a supermartingale for any ( π, c ) ∈ P , and a martingale for an optimal strategy ( π ∗ , c ∗ ).Detailed calculations, similar to those in [46, p. 234], yield the drift term of the above process: X − γt − γ e D t (cid:18) − H ( t, D t , Z t , ˆ Z t ) + Z t + ˆ Z t − γ )( r t − ˜ c t + π t ( λ t + σ t ρ t Z t + σ t ˆ ρ t ˆ Z t )) − γ (1 − γ )2 ( π t σ t ) + δθ ˜ c − ψ t e − Dtθ − δθ (cid:19) , where ˜ c t := c t /X t is the proportion of wealth consumed per unit of time. This indicates that H ( t, D t , Z t , ˆ Z t ) = (1 − γ ) r t + Z t + ˆ Z t − δθ + inf ˜ c ≥ (cid:16) − (1 − γ )˜ c + δθ ˜ c − ψ e − Dtθ (cid:17) + inf π ∈ R (cid:18) (1 − γ ) π (cid:16) λ t + σ t ρ t Z t + σ t ˆ ρ t ˆ Z t (cid:17) − γ (1 − γ )2 π σ t (cid:19) . (3.8)Solving the involved minimization problems yields the candidate optimal strategies ( π ∗ , ˜ c ∗ ): π ∗ t = λ t + σ t ( ρ t Z t + ˆ ρ t ˆ Z t ) γσ t and c ∗ t X ∗ t = ˜ c ∗ t = δ ψ e − ψθ D t ∀ t ∈ [0 , τ ) , (3.9)where X ∗ := X π ∗ ,c ∗ is the candidate optimal wealth process. Plugging these into (3.8), we have H ( t, D t , Z t , ˆ Z t ) = Z t + ˆ Z t − γ ) r t − δθ + δ ψ θψ e − ψθ D t + (1 − γ )( λ t + σ t ρ t Z t + σ t ˆ ρ t ˆ Z t ) γσ t . Rearranging and simplifying terms gives H ( t, D t , Z t , ˆ Z t ) = Z t (cid:18) − γ ) γ ρ t (cid:19) + ˆ Z t (cid:18) − γ ) γ ˆ ρ t (cid:19) + (1 − γ ) λ t γσ t ρ t Z t + (1 − γ ) λ t γσ t ˆ ρ t ˆ Z t + (1 − γ ) γ ρ t ˆ ρ t Z t ˆ Z t + δ ψ θψ e − ψθ D t + (1 − γ ) (cid:18) r t + λ t γσ t (cid:19) − δθ. (3.10)7 emark 3.1. A significant departure from the standard fixed-horizon case is the involvement of ˆ Z in (3.10) and (3.9) . Indeed, the formulas in [46, p.235] can be obtained by taking ˆ Z ≡ in (3.10) and (3.9) , leading to a simpler generator and a more straightforward investment strategy. Thissimplification does not work in our case: the randomness of τ , in general, can be fully capturedonly with the additional process ˆ Z . This will be explained in detail in Section 4, where we compareour results with those on a fixed horizon in a Markovian setting; see Remark 4.2 for details. To make sense of the heuristic derivations above, the first task is to show the existence of asolution to the BSDE (3.6), with the generator H given in (3.10). This can be rather tricky: H has quadratic growth in both Z and ˆ Z , and exponential growth in D (as θ < D prohibits us from effectively doing so. While there is a truncation technique in [46]to tame the exponential growth in D , it requires bounded time horizons. Ultimately, we carefullydevise a new truncation technique, to curb the exponential growth in D on the possibly unboundedrandom horizon; see Remark B.1 for details. With exponential growth contained, a delicate use of[10] and [29] in sequence (see Remark B.2 for details) yields the following. Proposition 3.1.
Suppose γ, ψ > and Assumption 1 holds. If τ < ∞ a.s., there exists a solution ( D, Z, ˆ Z ) ∈ S ∞ × M to (3.6) , with H given in (3.10) . The proof of Proposition 3.1 is relegated to Appendix B.1.
Remark 3.2.
Besides (3.5) , another useful decomposition is V t = X − γt − γ P kt , for some process P and k ∈ R ; see [48] and [40]. This ansatz can potentially generate a simpler BSDE, without quadraticor exponential growth as seen in (3.10) . This simplification, however, only works when ˆ Z ≡ andthe correlation function is constant, i.e. ρ ( t, y ) ≡ ρ . We therefore do not proceed with this ansatz. Remark 3.3.
On a fixed horizon, [46, Proposition 2.9], analogous to Proposition 3.1, is establishedwithout boundedness of market price of risk λ/σ . Indeed, by a change of measure using Girsanov’stheorem, the generator in [46, (2.13)], analogous to H in (3.10) , is simplified specifically to mitigatethe effect of λ/σ . Applying Girsanov’s theorem however requires a fixed horizon T > . As ourrandom horizon τ can be unbounded (i.e. P ( τ > T ) > for all T > ), it is unclear how the sametechnique in [46] can be applied here. Hence, we still impose boundedness of λ/σ in Assumption 1.Note that this is not an uncommon assumption, even for the fixed-horizon case; see [30, 42]. With ( π ∗ , c ∗ ) in (3.9) well-defined, thanks to Proposition 3.1, it remains to show its optimalityamong an appropriate set of strategies. A strategy ( π, c ) is called permissible if it belongs to P := { ( π, c ) ∈ A : ( X π,c · ) − γ is of class D } . (3.11)This collection of strategies was used in [14] for time-separable power utilities with γ >
1. It isalso in line with the set of permissible strategies in [46]: there, it is required that ( X π,c · ) − γ e D · isof class D (see the paragraph under [46, Proposition 2.9]), which is equivalent to (3.11) as D is abounded process in our setting (by Proposition 3.1). The aim of this subsection is to establish theoptimality within P of the candidate ( π ∗ , c ∗ ).To show that ( π ∗ , c ∗ ) is permissible, the random horizon poses nontrivial challenges. As opposedto the standard case with a fixed horizon, the boundedness of D in Proposition 3.1 does not directly8mply that (cid:82) · Z t dW t and (cid:82) · ˆ Z t d ˆ W t are BMO-martingales: the constants used in estimating the BMOnorms are time-dependent and can easily blow up on our potentially unbounded horizon (see e.g.[37, Lemma 3.1] and the comment below [25, (2)]). In other words, the BMO arguments, veryuseful in establishing permissibility of strategies on a fixed horizon (see e.g. [46, Lemma B.2] and[37, Lemma 3.1]), do not apply in our random-horizon case.To proceed, we need to impose appropriate integrability conditions on the random horizon τ ,from which the permissibility of ( π ∗ , c ∗ ) can be extracted. To this end, hereon we set C λ/σ := (cid:13)(cid:13)(cid:13)(cid:13) λ t ∧ τ σ t ∧ τ (cid:13)(cid:13)(cid:13)(cid:13) ∞ , r := ess sup (cid:18) sup t ≥ r t ∧ τ (cid:19) , r := ess inf (cid:18) inf t ≥ r t ∧ τ (cid:19) . (3.12)These constants are finite because of Assumption 1. Also, consider p + := 2 (cid:18) − ψ (cid:19) > p − := 2 (cid:18) − θ (cid:19) (1 − γ ) < . (3.13) Assumption 2.
Let ( D, Z, ˆ Z ) ∈ S ∞ × M be the solution to (3.6) in Proposition 3.1, and set (cid:101) Z t := ρ t Z t + ˆ ρ t ˆ Z t . We assume that there exists q > such that E (cid:20) exp (cid:18) q (cid:18) rp + + (cid:18) p + + 4 p γ (cid:19) C λ/σ (cid:19) τ + 4 qp γ (cid:90) τ (cid:101) Z s ds (cid:19)(cid:21) < ∞ and E (cid:20) exp (cid:18)(cid:18) − p − δ − /ψ + 2 rp − − p − δ ψ e − ψθ (cid:101) C + (cid:18) | p − | + 4 p − γ (cid:19) C λ/σ (cid:19) τ + 4 p − − p − γ (cid:90) τ (cid:101) Z s ds (cid:19)(cid:21) < ∞ where (cid:101) C := ess sup (cid:0) sup t ≥ D t (cid:1) < ∞ . Remark 3.4.
Assumption 2, seemingly complicated, is not restrictive in view of the literature.First, prior studies on the consumption-investment problem under a random horizon τ (with time-separable utilities) all require that τ ≤ T a.s. for a fixed T > ; see e.g. [7], [8], [20], [28], and[26]. Assumption 2 covers this case trivially, and allows for much more general unbounded τ . Inaddition, this type of exponential moment condition is common for random-horizon BSDEs, suchas [9, (A4)], [39, (c), Section 4], and [16, (25)]. Remark 3.5.
Assumption 2 can be relaxed to some extent, depending on the specific market modelemployed. For instance, in the practical model investigated in Section 4.2 below, (cid:101) Z is actually abounded process, which largely simplifies the exponential moment conditions. With the aid of Assumption 2, we are able to derive the permissibility of ( π ∗ , c ∗ ) in (3.9). Lemma 3.1.
Suppose γ, ψ > and Assumptions 1 and 2 hold. Then, ( π ∗ , c ∗ ) defined in (3.9) belongs to P . The proof of Lemma 3.1 is relegated to Section B.2.With ( π ∗ , c ∗ ) ∈ P , it remains to show that ( π ∗ , c ∗ ) is optimal within P ; namely, it solves (3.4).Recall from the arguments in Section 3.2 that this boils down to showing that the process in (3.7) isa supermartingale for each ( π, c ) ∈ P , and a martingale for ( π ∗ , c ∗ ). This can be done by modifyingthe arguments in [46, Theorem 2.14]. Theorem 3.1.
Suppose γ, ψ > and Assumptions 1 and 2 hold. Then, ( π ∗ , c ∗ ) defined in (3.9) ,with ( D, Z, ˆ Z ) ∈ S ∞ × M a solution to (3.6) , is a maximizer of (3.4) . Moreover, for any initialwealth x > , the optimal Epstein-Zin utility is given by V ∗ = x − γ − γ e D . The proof of Theorem 3.1 is relegated to Section B.3.9
The Markovian Framework
In this section, we take the random horizon τ to be the first hitting time (or exit time) of someappropriate state processes. This additional Markovian structure allows us to connect the generalBSDE (3.6) to a specific elliptic PDE with a Dirichlet boundary value condition.The purpose is twofold. First, this facilitates a detailed comparison between our random-horizonresults and classical ones on a fixed horizon, as many prior studies rely on the PDE approach. Thiscomparison particularly reveals how the involvement of ˆ Z in (3.10) and (3.9) is indispensable on arandom horizon, while it is superfluous for the fixed-horizon case; recall Remark 3.1. Second, theMarkovian framework facilitates numerical computation. In Section 4.2, we will demonstrate ourtheoretic results numerically in the Heston model of stochastic volatility. As we will see, optimalstrategies on a random horizon differ drastically from those on a fixed horizon.Let us first recall the notation from [22] for elliptic equations. Consider an open subset D of R n , k ∈ N , and ν ∈ (0 , C k,ν ( D ) (resp. C k,ν ( D )) are defined as the subspaceof C k ( D ) consisting of functions whose k th-order partial derivatives are uniformly (resp. locally)H¨older continuous with exponent ν in D . Recall the setup in Section 3.1. In addition to Y in (3.1), we introduce an additional state process W given by d W t = α ( W t , Y t ) dt + β ( W t , Y t ) dW t + Γ( W t , Y t ) d ˆ W t , W = w ∈ R , (4.1)for some given Borel measurable α, β, Γ : R × E → R . In Section 4.2 below, we will take W to bethe zero-mean return process of S (see [33, 34]) in the Heston model. As in [16], [29], and [10], wetake the random horizon as the exit time of ( W , Y ) from some open set D ⊂ R × E , i.e. τ w,y := inf { t ≥ W wt , Y yt ) / ∈ D} . To ensure the existence of a strong solution to (4.1) and sufficient regularity for subsequent analysis,we impose on the states ( W , Y ) the following conditions, inspired by those in [29, Section 6]. Assumption 3.
D ⊂ R × E is an open bounded set with ∂ D ∈ C ,ν for some ν ∈ (0 , . Thereexists an open set U ⊂ R × E containing D such that(i) α, β, Γ are Lipschitz on U , inf U β ( w, y ) > , inf U Γ( w, y ) > , and β, Γ ∈ C ( U ) ;(ii) σ, r, λ, ρ, ˆ ρ, a , and b depend only on y , inf U b ( y ) > , and b ∈ C ( U ) . The ellipticity conditions in Assumptions 1 and 3 guarantee the non-degeneracy of ( W , Y ) in D , implying τ w,y < ∞ a.s. In view of (3.1) and (4.1), the infinitesimal generator of ( W , Y ) is L := a ∂∂y + α ∂∂w + b ∂ ∂y + 12 (cid:0) β + Γ (cid:1) ∂ ∂w + bβ ∂ ∂y∂w , and the corresponding elliptic boundary value problem is L u ( w, y ) + G (cid:18) y, u, (cid:18) b ∂u∂y + β ∂u∂w (cid:19) , Γ ∂u∂w (cid:19) = 0 , ( w, y ) ∈ D ,u ( w, y ) = 0 , ( w, y ) ∈ ∂ D , (4.2)10here G ( y, d, z, ˆ z ) := (cid:18) − γ ) γ ρ ( y ) (cid:19) z (cid:18) − γ ) γ ˆ ρ ( y ) (cid:19) ˆ z − γ ) λ ( y ) γσ ( y ) ρ ( y ) z + (1 − γ ) λ ( y ) γσ ( y ) ˆ ρ ( y )ˆ z + (1 − γ ) γ ρ ( y ) ˆ ρ ( y ) z ˆ z + δ ψ θψ e − ψθ d + (1 − γ ) (cid:18) r ( y ) + λ ( y ) γσ ( y ) (cid:19) − δθ. (4.3) Theorem 4.1.
Suppose Assumptions 1 and 3 hold. Then, (4.2) has a unique solution u ∈ C ,ν ( D ) ,with sup D |∇ u | < ∞ . The proof of Theorem 4.1 is relegated to Appendix C.1.Now, let (
D, Z, ˆ Z ) be the solution to the BSDE (3.6) on the random horizon τ w,y , obtainedin Proposition 3.1. The connection between (3.6) and (4.2) can be stated precisely through thefollowing P -a.s. representation: for all t ≥ D t = u ( W wt , Y yt ) , Z t = b ∂u∂y ( W wt , Y yt ) + β ∂u∂w ( W wt , Y yt ) , ˆ Z t = Γ ∂u∂w ( W wt , Y yt ) , (4.4)where u ∈ C ,ν ( D ) is the solution to (4.2). This is shown by applying Itˆo’s formula to u . Remark 4.1.
When Assumption 3 fails to hold, a smooth solution to (4.2) may not exist. Yet, theconnection between (3.6) and (4.2) , stated in (4.4) , may still hold. Specifically, by the idea of [29], (4.4) can be established, when u is only a weak solution to (4.2) in the Sobolev space W ,p ( D ) , forsome p ≥ , with its derivatives taken in the distributional sense; see e.g. [32, Theorem 1]. An important message of (4.4) is that ˆ Z can be dropped completely on a fixed horizon, but isindispensable in general when a random horizon is considered. Remark 4.2. If Γ( w, y ) ≡ in (4.1) , the randomness of τ w,y comes exclusively from W . Then, (4.4) indicates ˆ Z ≡ , implying that one can drop ˆ Z completely in Section 3.2. This particularlycovers the standard case with a fixed horizon T > , by taking α ≡ , β ≡ , Γ ≡ in (4.1) and D = ( w − ε, w + T ) × E , for any ε > . With ˆ Z ≡ , the formulation in Section 3.2 is consistentwith those in [46],[40],[35],[31], and [30]. In particular, ˆ Z ≡ gives much simpler ( π ∗ , c ∗ ) and H in (3.9) and (3.10) , which recover [30, Theorem 5.1] and [46, (2.12), (2.13)].When the randomness of τ w,y comes jointly from W and ˆ W (and thus Γ( w, y ) (cid:54)≡ ), (4.4) indicates that ˆ Z is not identically zero, and thus cannot be omitted in general. Consider a specific example of the general financial model in Section 3.1 as follows. Let the corre-lation between W and ˆ W be constant, i.e. ρ ( t, y ) ≡ ρ ∈ [ − , dY t = − α ( Y t − m ) dt + k (cid:112) Y t dW t , Y = y > ,dS t = rS t dt, S = 1 ,dS t = S t (cid:104)(cid:0) r + λ · (cid:0) Y t + ε (cid:1)(cid:1) dt + (cid:112) Y t + εdW ρt (cid:105) , S = s > , where α , r , k , m , λ and ε are given nonnegative constants, with 2 αm > k satisfied such that Y t > ∀ t ≥ ε = 0, this is the standard Heston model of stochastic volatility, whichhas been investigated in [31], [30], and [46]. For ε >
0, this is an ε -modification of the Hestonmodel. Using an ε -modification is often of practical necessity. For instance, it is used in [49] for11he Scott and Stein-Stein models of stochastic volatility, to obtain bounds of the market price ofrisk. Similarly, [40] uses an ε -modification to ensure uniform ellipticity in the Hull-White model.We will focus on a random horizon that is related to the first exit time of the zero-mean returnof S (defined precisely below). For the Heston model (i.e. ε = 0), such an exit time has beenstudied in detail in [33, 34], under the assumption ρ = 0. In the following, we will follow [33, 34]to take ρ = 0, so that W ρ = ˆ W . This is supported by empirical analysis in [44] and [17], althoughthere exist other estimates of ρ in the literature (such as [38] and [15]).The zero-mean return process of S , denoted by W , is defined as the return of S (i.e. dS t /S t )minus its drift, i.e. d W t = (cid:112) Y t + εd ˆ W t , W = w ∈ R . For any w ∈ R and y >
0, consider the random horizon τ w,y := inf (cid:26) t ≥ W t , Y t ) (cid:54)∈ (cid:18) − L , L (cid:19) × ( y , y ) (cid:27) , where L > < y < y are chosen by an agent a priori. These constants reflect the agent’stolerance of extreme market situations: she carries out consumption-investment optimization untilthe zero-mean return W deviates too far away from 0 or the volatility Y reaches extreme values.It is straightforward to check that Assumption 1 is satisfied under the current setting. Showingthat τ w,y fulfills Assumption 2 demands more involved analysis. Instead of dealing with τ w,y directly,we will study in detail the density of τ := inf (cid:26) t ≥ W t (cid:54)∈ (cid:18) − L , L (cid:19)(cid:27) . (4.5)Since τ w,y ≤ τ , whenever τ satisfies Assumption 2, so does τ w,y . Lemma 4.1.
For any
L > , the density of τ in (4.5) admits the explicit formula P ( τ ∈ ds | W = w, Y = y ) = ∞ (cid:88) n =0 − n π (2 n + 1) (cid:32) αk ( m − y ) B n ( s ) + 2 αk y (cid:18) (cid:18) β n L (cid:19) − B n ( s ) (cid:19) + ε n (cid:33) · exp (cid:18) − A n ( s ) − αk yB n ( s ) (cid:19) cos (cid:18) (2 n + 1) πwL (cid:19) , (4.6) where β n := kα (2 n + 1) π , ε n := (cid:16) (2 n +1) π √ αL (cid:17) ε , A n ( s ) := 2 αm k ln (cid:18) (∆ n + 1) + (∆ n − e − ∆ n s n (cid:19) + (cid:18) αm (∆ n − k + ε n (cid:19) s, (4.7) B n ( s ) := β n L (cid:20) − e − ∆ n s (∆ n + 1) + (∆ n − e − ∆ n s (cid:21) , with ∆ n := (cid:112) β n /L ) . (4.8)The proof of Lemma 4.1 is relegated to Appendix C.2. Following the model parameters in [30] and [46], we take γ = 2 , ψ = 1 . , δ = 0 . , r = 0 . , α = 5 , k = 0 . , m = 0 . , λ = 0 . y = 0 .
001 and y = 1, leaving L > τ in Assumption 2, we need: (i) the densityof τ , (ii) an upper bound of (cid:101) Z = ˆ Z (recall ρ = 0), and (iii) an estimate of (cid:101) C . For (i), we use theexplicit formula in (4.6). For (ii), recall from (4.4) that ˆ Z t = ( Y t + ε ) u w ( W t , Y t ), for which an upperbound can be found by numerically solving (4.2). For (iii), recall from (4.4) that D t = u ( W t , Y t ),so that (cid:101) C can be estimated again by numerically solving (4.2), which shows (cid:101) C = 0. Note that thiscan also be proved theoretically by employing the maximum principle in [4].Numerical computation shows that, with ε = 0, Assumption 2 is satisfied by τ , and thus by τ w,y (as τ w,y ≤ τ ), for all 0 < L ≤ . An ε -modification enables us to enlarge the range ofallowable L . For instance, with ε = 0 .
05, Assumption 2 is satisfied by τ , and thus by τ w,y , for all0 < L ≤ . π ∗ ( w, y ) = λγ + Γ( w, y ) u w ( w, y ) γ √ y + ε = λ + u w ( w, y ) γ and ˜ c ∗ = δ ψ e − ψθ u ( w,y ) , (4.9)where u is the unique solution to (4.2). With ε = 0, we compute ˜ c ∗ and π ∗ , for L = 0 .
02 in τ w,y ;see Figure 1. With ε = 0 .
05, we compute ˜ c ∗ and π ∗ , for L = 0 .
02 and L = 0 .
08 in τ w,y ; see Figure2. This exhibits a significant contrast to optimal strategies on a fixed horizon.On a fixed horizon T >
0, following [30, Theorem 5.1] and [46, (2.14)], the optimal portfolioallocation in the current setting is π ∗ f = λ t γσ t ≡ λγ . This can also be derived from (3.9), by noting ˆ Z ≡ τ w,y , theoptimal portfolio allocation π ∗ ( w, y ), given in (4.9), changes continuously as the market evolves.Specifically, from Figure 1, one should hold the risky asset S when the zero-mean return W t = w is positive (i.e. S performs well relative to its mean return) and the volatility Y t = y is low, andshort the risky asset when W t = w is negative (i.e. S performs poorly relative to its mean return)and Y t = y is low. This makes economic sense, as it is reasonable to expect that, in the former(resp. latter) case, S will continue to perform well (resp. poorly) for some period after time t .Similarly, by [30, Theorem 5.1] and [46, (2.14)], the optimal consumption-wealth ratio, on afixed horizon T >
0, is given by ˜ c ∗ f ( t, y ) = δ ψ e − ψθ v ( t,y ) , where v is the solution to a Cauchy problem.Clearly, ˜ c ∗ f differs from ˜ c ∗ in (4.9), as u and v are solutions to different differential equations. A Proofs for Section 2
Let us first present a useful estimation and a fundamental result. Recall the spaces of processesintroduced above Proposition 2.1. For any
T >
0, define the spaces S q ([0 , T ]), M q ([0 , T ]), and B q ([0 , T ]) similarly, with Y t ∧ τ , Z t ∧ τ , and t ≥ Y t , Z t , and t ∈ [0 , T ]. We numerically solve (4.2) via finite element methods, using a triangular mesh with maximal edge length takento be 0.005. Also, a suitable mollification of D = (cid:0) − L , L (cid:1) × ( y , y ) is employed to ensure the boundary regularityin Assumption 3. For
L > .
02, Assumption 2 could still be satisfied by τ w,y , because our computation involves the use of severalupper bounds that may not be the sharpest. ε = 0 and L = 0 . (a) (b) Figure 2: Optimal portfolio for ε = 0 .
05 on different return thresholds.14or any ( Y t , Z t ) ∈ B ([0 , T ]) and q >
0, by Burkh¨older-Davis-Gundy’s inequality, there exists
K >
0, independent of Y , Z , and T , such that q · E (cid:20) sup t ∈ [0 ,T ] (cid:90) t (cid:104) Y s , Z s dB s (cid:105) (cid:21) ≤ qK E (cid:20) ( (cid:90) T | Y s | (cid:107) Z s (cid:107) ds ) / (cid:21) ≤ E (cid:20)(cid:16) sup ≤ t ≤ T | Y t | (cid:17) / (cid:16) q K (cid:90) T (cid:107) Z s (cid:107) ds (cid:17) / (cid:21) ≤ E (cid:20) sup t ∈ [0 ,T ] | Y t | (cid:21) + q K E (cid:20) (cid:90) T (cid:107) Z s (cid:107) ds (cid:21) < ∞ , (A.1)where the third inequality follows from ab ≤ a + b for all a, b ∈ R , and the finiteness is due to( Y t , Z t ) ∈ B ([0 , T ]). Lemma A.1.
Fix
T < ∞ . Given ( Y t , Z t ) ∈ B ([0 , T ]) , the continuous local martingale (cid:82) t (cid:104) Y s , Z s dB s (cid:105) , t ∈ [0 , T ] , is a uniformly integrable martingale.Proof. It suffices to show that E (cid:2) sup t ∈ [0 ,T ] (cid:82) t (cid:104) Y s , Z s dB s (cid:105) (cid:3) < ∞ , which is true by (A.1). A.1 Proof of Proposition 2.1
Motivated by [9] and [39], we will construct a sequence of solutions that is Cauchy in B , and showthat its limit solves (2.4). In the rest of the proof, we set ξ := e − δθτ c − γτ and p := 1 − /θ > γ, ψ > . (A.2) Step 1: Construct a sequence of solutions ( Y n , Z n ) n ∈ N in B . For each n ∈ N , we aimto construct a solution ( Y nt , Z nt ) t ≥ ∈ B to the BSDE Y nt = ξ + (cid:90) τt ∧ τ [0 ,n ] ( s ) F ( s, c s , Y ns ) ds − (cid:90) τt ∧ τ Z ns dB s , t ≥ . (A.3)Note that (A.3) is a random-horizon BSDE, and a solution ( Y nt , Z nt ) t ≥ will be constructed byfinite-horizon results in [46] and a proper extension to the random horizon. Specifically, for thefixed time horizon [0 , n ], thanks to the construction in [46, Proposition 2.2], there exists a uniquesolution ( Y t , Z t ) t ∈ [0 ,n ] ∈ B ([0 , n ]) to the fixed-horizon BSDE Y t = E [ ξ | F n ] + (cid:90) nt [0 ,τ ] ( s ) F ( s, c s , Y s ) ds − (cid:90) nt Z s dB s , t ∈ [0 , n ] . (A.4)Specifically, Y is continuous, 0 < Y t ≤ E [ ξ | F t ] a.s. for all t ∈ [0 , n ] (hence, Y is of class D ). On the other hand, thanks to c ∈ C , (2.5) implies E [ ξ ] = E [ e − δθτ c − γ ) τ ] < ∞ , i.e. ξ is squareintegrable. Thus, by the martingale representation theorem, there exists η ∈ M such that E [ ξ | F t ] = ξ − (cid:90) τt η s dB s and η t = 0 for t > τ . (A.5) In general, the solution derived in [46, Proposition 2.2] need not lie in B ([0 , n ]). This is because [46] assumesonly integrability on the terminal condition ξ , instead of the standard square-integrability. In our case, as c ∈C , E (cid:2) E [ ξ |F n ] (cid:3) ≤ E [ ξ ] = E [ e − δθτ c − γ ) τ ] < ∞ . With E [ ξ |F n ] being square integrable, the construction in [46,Proposition 2.2] then yields a solution in B ([0 , n ]). Specifically, we obtain from Step 1 of its proof (see [46, AppendixA]) the desired solution, with no need of Step 2 therein. Y nt , Z nt ) t ≥ as follows: ( Y nt , Z nt ) := ( Y t , Z t ) for 0 ≤ t ≤ n , and Y nt := E [ ξ |F t ] and Z nt := η t for all t > n . By (A.4) and (A.5), it can be checked directly that ( Y nt , Z nt ) t ≥ ∈ B is asolution to (A.3); a similar construction can be found in [39, Theorem 4.1]. Step 2: Show that the sequence ( Y n , Z n ) n ∈ N is Cauchy in B . For any m, n ∈ N with m > n , consider ∆ Y t := Y mt − Y nt , ∆ Z t := Z mt − Z nt , and ∆ F ( t, c t , Y t ) := F ( t, c t , Y mt ) − F ( t, c t , Y nt ).We intend to show that (cid:107) (∆ Y t , ∆ Z t ) (cid:107) B → m, n → ∞ .For 0 ≤ t ≤ n , observe from (A.3) that∆ Y t = (cid:90) n ∧ τt ∧ τ ∆ F ( s, c s , Y s ) ds − (cid:90) τt ∧ τ ∆ Z s dB s + (cid:90) m ∧ τn ∧ τ F ( s, c s , Y ms ) ds = ∆ Y n ∧ τ + (cid:90) n ∧ τt ∧ τ ∆ F ( s, c s , Y s ) ds − (cid:90) n ∧ τt ∧ τ ∆ Z s dB s . (A.6)Recall p > | ∆ Y t | , with ∆ Y t as in (A.6), yields | ∆ Y t ∧ τ | + (cid:90) n ∧ τt ∧ τ (cid:107) ∆ Z t (cid:107) ds = | ∆ Y n ∧ τ | + 2 (cid:90) n ∧ τt ∧ τ ∆ Y s ∆ F ( s, c s , Y s ) ds − (cid:90) n ∧ τt ∧ τ (cid:104) ∆ Y s , ∆ Z s dB s (cid:105) = | ∆ Y n ∧ τ | + (cid:90) n ∧ τt ∧ τ (2 δθ ∆ Y s e − δs c − /ψs (( Y ms ) p − ( Y ns ) p )) ds − (cid:90) n ∧ τt ∧ τ (cid:104) ∆ Y s , ∆ Z s dB s (cid:105) . (A.7)Since ∆ Y s = Y ms − Y ns , the sign of ∆ Y s must be the same as that of ( Y ms ) p − ( Y ns ) p . This, togetherwith θ <
0, gives 2 δθ ∆ Y s e − δs c − /ψs (( Y ms ) p − ( Y ns ) p ) ≤
0. We then conclude from (A.7) that | ∆ Y t ∧ τ | + (cid:90) n ∧ τt ∧ τ (cid:107) ∆ Z t (cid:107) ds ≤ | ∆ Y n ∧ τ | − (cid:90) n ∧ τt ∧ τ (cid:104) ∆ Y s , ∆ Z s dB s (cid:105) . (A.8)This, together with Lemma A.1, gives E (cid:20) (cid:90) n ∧ τ (cid:107) ∆ Z t (cid:107) ds (cid:21) ≤ E [ | ∆ Y n ∧ τ | ] − E (cid:20) (cid:90) n ∧ τ (cid:104) ∆ Y s , ∆ Z s dB s (cid:105) (cid:21) = E [ | ∆ Y n ∧ τ | ] . Moreover, by using (A.8) and (A.1), with q = 2, E (cid:20) sup ≤ t ≤ n | ∆ Y t ∧ τ | (cid:21) ≤ E (cid:2) | ∆ Y n ∧ τ | (cid:3) + 12 E (cid:20) sup ≤ t ≤ n | ∆ Y t ∧ τ | (cid:21) + 2 K E (cid:20)(cid:90) n ∧ τ (cid:107) ∆ Z s (cid:107) ds (cid:21) , for some K >
0, independent of m and n . By the previous two inequalities, there exists K > m and n , such that E (cid:20) sup ≤ t ≤ n | ∆ Y t ∧ τ | + (cid:90) n ∧ τ (cid:107) ∆ Z t (cid:107) ds (cid:21) ≤ K E [ | ∆ Y n ∧ τ | ] . (A.9)Next, for n < t ≤ m , observe from (A.3) that∆ Y t = (cid:90) m ∧ τt ∧ τ F ( s, c s , Y ms ) ds − (cid:90) τt ∧ τ ∆ Z s dB s = (cid:90) m ∧ τt ∧ τ F ( s, c s , Y ms ) ds − (cid:90) m ∧ τt ∧ τ ∆ Z s dB s , where the second equality follows from (cid:82) τm ∧ τ ∆ Z s dB s = 0, as Z ms = Z ns = η s for all s > m ∧ τ bythe construction in Step 1. Applying Itˆo’s formula to | ∆ Y t | , with ∆ Y t as above, gives | ∆ Y t ∧ τ | + (cid:90) m ∧ τt ∧ τ (cid:107) ∆ Z s (cid:107) ds = 2 (cid:90) m ∧ τt ∧ τ ∆ Y s F ( s, c s , Y ms ) ds − (cid:90) m ∧ τt ∧ τ (cid:104) ∆ Y s , ∆ Z s dB s (cid:105)≤ δ | θ | (cid:90) m ∧ τt ∧ τ e − δs c − /ψs ( E [ ξ | F s ]) p +1 ds − (cid:90) m ∧ τt ∧ τ (cid:104) ∆ Y s , ∆ Z s dB s (cid:105) , (A.10)16here the inequality follows from∆ Y t F ( s, c s , Y ms ) = δθe − δs c − /ψs ( Y ms ) p ( Y ms − Y ns ) ≤ − δθe − δs c − /ψs ( Y ms ) p Y ns ≤ − δθe − δs c − /ψs ( E [ ξ | F s ]) p +1 , thanks to 0 ≤ Y ms , Y ns ≤ E [ ξ | F s ] and θ <
0. Observe that E (cid:20) (cid:90) m ∧ τn ∧ τ e − δs c − ψ s ( E [ ξ |F s ]) p +1 ds (cid:21) ≤ E (cid:34)(cid:18)(cid:90) m ∧ τn ∧ τ E [ ξ |F s ] p +1) ds (cid:19) (cid:18)(cid:90) m ∧ τn ∧ τ e − δs c − ψ ) s ds (cid:19) (cid:35) ≤ E (cid:20)(cid:90) m ∧ τn ∧ τ E [ ξ p +1 |F s ] ds (cid:21) E (cid:20)(cid:90) m ∧ τn ∧ τ e − δs c − ψ ) s ds (cid:21) , (A.11)where the first inequality follows from applying the Cauchy-Schwarz inequality to the integral insidethe expectation, and the second inequality follows from applying the Cauchy-Schwarz inequality tothe expectation and then using Jensen’s inequality. By (A.2) and the fact that c ∈ C , E [ ξ p +1) ] = E (cid:104) e − p +1) δθτ c p +1)(1 − γ ) τ (cid:105) < ∞ . (A.12)Thus, the martingale representation theorem entails E [ ξ p +1 |F s ] = E [ ξ p +1 ] + (cid:82) t ∧ τ ν s dB s for someadapted process ν . This allows the use of [16, Lemma 4.1], which yields the finiteness of C := E (cid:2) (cid:82) τ e δs E (cid:2) ξ p +1 |F s (cid:3) ds (cid:3) / . We then obtain from (A.11) that E (cid:20) (cid:90) m ∧ τn ∧ τ e − δs c − ψ s ( E [ ξ |F s ]) p +1 ds (cid:21) ≤ C E (cid:20)(cid:90) m ∧ τn ∧ τ e − δs c − ψ ) s ds (cid:21) . (A.13)In view of (A.10) and Lemma A.1, this directly implies E (cid:20)(cid:90) m ∧ τn ∧ τ (cid:107) ∆ Z s (cid:107) ds (cid:21) ≤ δ | θ | C E (cid:20)(cid:90) m ∧ τn ∧ τ e − δs c − ψ ) s ds (cid:21) . Moreover, by using (A.10) and (A.1), with q = 2, E (cid:20) sup n ≤ t ≤ m | ∆ Y t ∧ τ | (cid:21) ≤ δ | θ | E (cid:20)(cid:90) m ∧ τn ∧ τ e − δs c − /ψs ( E [ ξ | F s ]) p +1 ds (cid:21) + 12 E (cid:20) sup n ≤ t ≤ m | ∆ Y t ∧ τ | (cid:21) + 2 K E (cid:20)(cid:90) m ∧ τn ∧ τ (cid:107) ∆ Z s (cid:107) ds (cid:21) , for some K >
0, independent of m and n . By combining the previous two inequalities and using(A.13), there exists K >
0, independent of m and n , such that E (cid:20) sup n ≤ t ≤ m | ∆ Y t ∧ τ | + (cid:90) m ∧ τn ∧ τ (cid:107) ∆ Z t (cid:107) ds (cid:21) ≤ K δ | θ | E (cid:20)(cid:90) m ∧ τn ∧ τ e − δs c − ψ ) s ds (cid:21) . (A.14)By (A.9), (A.14), and recalling that Y ms = Y ns and Z ms = Z ns for all s > m ∧ τ , we have E (cid:20) sup t ≥ | ∆ Y t ∧ τ | + (cid:90) ∞ (cid:107) ∆ Z t (cid:107) ds (cid:21) ≤ K E [ | ∆ Y n ∧ τ | ] + K δ | θ | E (cid:20)(cid:90) m ∧ τn ∧ τ e − δs c − ψ ) s ds (cid:21) . (A.15)17e know from Step 1 that 0 ≤ Y mn ∧ τ , Y nn ∧ τ ≤ E [ ξ | F n ∧ τ ], which imply | ∆ Y n ∧ τ | = | Y mn ∧ τ − Y nn ∧ τ | ≤ ( Y mn ∧ τ ) + ( Y nn ∧ τ ) ≤ E [ ξ | F n ∧ τ ] ≤ E [ ξ | F n ∧ τ ] . (A.16)Since (A.12) implies that { E [ ξ | F k ] : k ≥ } is uniformly integrable, we deduce from (A.16) thatlim m,n →∞ E [ | ∆ Y n ∧ τ | ] = lim m,n →∞ E (cid:2) | Y mn ∧ τ − Y nn ∧ τ | (cid:3) = E (cid:20) lim m,n →∞ | Y mn ∧ τ − Y nn ∧ τ | (cid:21) = E (cid:2) | ξ − ξ | (cid:3) = 0 . Finally, thanks to (2.5), the second term in (A.15) vanishes as m, n → ∞ . Therefore, we concludefrom (A.15) that (cid:107) (∆ Y, ∆ Z ) (cid:107) B → m, n → ∞ , i.e. { ( Y n , Z n ) } n ∈ N is Cauchy in B . Since B is complete , lim n →∞ ( Y n , Z n ) = ( Y, Z ) ∈ B exists. Step 3: The limit ( Y, Z ) solves (2.4) , and Y is of class D . For any n ∈ N , since( Y nt , Z nt ) t ≥ ∈ B solves (A.3), Y nt = ξ + (cid:90) τt ∧ τ [0 ,n ] ( s ) F ( s, c s , Y ns ) ds − (cid:90) τt ∧ τ Z ns dB s , t ≥ . (A.17)We intend to prove that each term in (A.17) converges to a corresponding term in (2.4) ∀ t ≥ P -a.s., as n → ∞ . This then implies that ( Y, Z ) satisfies (2.4) ∀ t ≥ P -a.s., as desired.First, Y n → Y in S already implies Y nt → Y t ∀ t ≥ P -a.s. To show that (cid:82) τt ∧ τ [0 ,n ] ( s ) F ( s, c s , Y ns ) ds → (cid:82) τt ∧ τ F ( s, c s , Y s ) ds ∀ t ≥ P -a.s. (possibly up to a subsequence), it suffices to prove that E (cid:20) sup ≤ t< ∞ (cid:90) τt ∧ τ | [0 ,n ] ( s ) F ( s, c s , Y ns ) − F ( s, c s , Y s ) | ds (cid:21) = E (cid:20)(cid:90) τ | [0 ,n ] ( s ) F ( s, c s , Y ns ) − F ( s, c s , Y s ) | ds (cid:21) → n → ∞ , which is equivalent to [0 ,n ] ( · ) F ( · , c · , Y n · ) → F ( · , c · , Y · ) in L ( µ ), for the finite measure µ := { ≤ t ≤ τ } dt × d P . Since Y nt → Y t ∀ t ≥ P -a.s., the continuity of F implies [0 ,n ] ( t ) F ( t, c t , Y nt ) → F ( t, c t , Y t ) ∀ t ≥ P -a.s. Also, in view of 0 ≤ Y ns , Y s ≤ E [ ξ |F s ] for all s ≥ n ∈ N , | F ( s, c s , Y ns ) | ≤ δ | θ | e − δs c − /ψs E [ ξ | F s ] p ∀ s ≥ n ∈ N ∪ { } , with Y := Y . Hence, if we can show that e − δ · c − /ψ · E [ ξ | F · ] p is µ -integrable, the dominatedconvergence theorem will give the desired convergence [0 ,n ] ( · ) F ( · , c · , Y · ) → F ( · , c · , Y n · ) in L ( µ ).To this end, observe that E (cid:20)(cid:90) τ e − δs c − /ψs E [ ξ | F s ] p ds (cid:21) ≤ E (cid:20)(cid:90) τ e − δs c − ψ ) s ds (cid:21) E (cid:20)(cid:90) τ E [ ξ p | F s ] ds (cid:21) , (A.18)where the first inequality follows from applying the Cauchy-Schwartz inequality twice and then theJensen inequality (similarly to (A.11)). Note that E (cid:2) (cid:82) τ e − δs c − /ψ ) s ds (cid:3) < ∞ , as c ∈ C ; see (2.5).By the arguments similar to (A.12) and the discussion below it, we get E (cid:2)(cid:82) τ E [ ξ p | F s ] ds (cid:3) < ∞ .We then conclude from (A.18) the µ -integrability of e − δ · c − /ψ · E [ ξ | F · ] p , as desired.It remains to show that (cid:82) τt ∧ τ Z ns dB s → (cid:82) τt ∧ τ Z s dB s ∀ t ≥ P -a.s. Let ∆ Z ns := Z ns − Z s . Since Z n → Z in M , by the Itˆo isometry, E (cid:20) (cid:18)(cid:90) τt ∧ τ ∆ Z ns dB s (cid:19) (cid:21) ≤ E (cid:20)(cid:90) τ (cid:107) ∆ Z ns (cid:107) ds (cid:21) = (cid:107) ∆ Z n (cid:107) M → , for each t ≥ . This is a standard result that can be found in [9] and [39]. It is used explicitly in [9] and [39], as well as implicitlyin the proofs of [16] and [19]. (cid:82) τt ∧ τ Z ns dB s → (cid:82) τt ∧ τ Z s dB s in probability, for each t ≥
0. Because every other term in(A.17) (either on the left or right hand side) converges ∀ t ≥ P -a.s., (cid:82) τt ∧ τ Z ns dB s must also converge ∀ t ≥ P -a.s. Hence, we have (cid:82) τt ∧ τ Z ns dB s → (cid:82) τt ∧ τ Z s dB s ∀ t ≥ P -a.s.Finally, since it holds P -a.s. that 0 ≤ Y nt ≤ E [ ξ |F t ] ∀ t ≥ n ∈ N , and that Y nt → Y t ∀ t ≥
0, we have 0 ≤ Y t ≤ E [ ξ |F t ] ∀ t ≥ P -a.s. Thus, Y is of class D . Step 4: ( Y, Z ) is the unique solution to (2.4) in B . This follow from an immediateapplication of Step 3 in the proof of [46, Proposition 2.2].
A.2 Proof of Theorem 2.1
By Proposition 2.1, a direct calculation shows that ( V c , Z c ) is the unique solution to (2.3) in B ,and V c is of class D . To show the last assertion that V c satisfies (2.1) a.s., we first note that t (cid:55)→ V ct + (cid:82) t ∧ τ f ( c s , V cs ) ds is a martingale. Indeed, For any 0 ≤ u ≤ t , E u (cid:20) V ct + (cid:90) t ∧ τ f ( c s , V cs ) ds (cid:21) = E u (cid:20) c − γτ − γ + (cid:90) τ f ( c s , V cs ) ds − (cid:90) τt ∧ τ Z cs dB s (cid:21) = (cid:90) u ∧ τ f ( c s , V cs ) ds + E u (cid:20) c − γτ − γ + (cid:90) τu ∧ τ f ( c s , V cs ) ds − (cid:90) τu ∧ τ Z cs dB s + (cid:90) t ∧ τu ∧ τ Z cs dB s (cid:21) = (cid:90) u ∧ τ f ( c s , V cs ) ds + V cu , where the last equality follows from Z c ∈ M . Fix t ≥
0. By the above martingale property, V ct = E t (cid:20) V cm + (cid:90) m ∧ τt ∧ τ f ( c s , V cs ) ds (cid:21) , ∀ m ≥ t. As m → ∞ , similarly to [46, (A.5)], we may apply the monotone convergence theorem to get V ct + δθ E t (cid:20) (cid:90) τt ∧ τ V cs ds (cid:21) = E t (cid:20) V τ + (cid:90) τt ∧ τ δ c − /ψs − ψ (cid:0) (1 − γ ) V cs (cid:1) − θ ds (cid:21) , (A.19)thanks to the definition of f in (2.2) and the fact that V c ≤ D . To show thatthe conditional expectation E t (cid:2)(cid:82) τt ∧ τ V cs ds (cid:3) above is well-defined, note that 0 ≥ V cs = e δθs − γ Y s ≥ − γ E [ ξ | F s ] for all s ≥
0, and thus 0 ≥ E t (cid:2)(cid:82) τt ∧ τ V cs ds (cid:3) ≥ − γ E t (cid:2)(cid:82) τt ∧ τ E [ ξ | F s ] ds (cid:3) > −∞ , wherethe finiteness in the last inequality follows from an argument similar to (A.12) and the discussionbelow it. Finally, observe that (A.19) readily gives (2.1). B Proofs for Section 3
B.1 Derivation of Proposition 3.1
As discussed above Proposition 3.1, the challenge of constructing a solution to (3.6) stems fromthe generator H in (3.10): it has quadratic growth in Z and ˆ Z , and exponential growth in D . Wewill tackle this below in two steps. First, we will construct a sequence of approximating generators { H n } n ∈ N , each of which has only linear growth in D , such that a solution ( D n , Z n , ˆ Z n ) exists bystandard results of quadratic BSDEs. Second, we will show that the sequence { D n } n ∈ N is uniformlybounded from above, such that lim n →∞ ( D n , Z n , ˆ Z n ) is well-defined and actually solves (3.6).19 roof of Proposition 3.1. For simplicity, throughout the proof we will write Z t = ( Z t , ˆ Z t ), M t = (cid:18) − γ ) γ ρ t (cid:19) , ˆ M t = (cid:18) − γ ) γ ˆ ρ t (cid:19) , h t = (1 − γ ) (cid:18) r t + λ t γσ t (cid:19) . (B.1) Step 1: Construct an approximating sequence of solutions { ( D n , Z n ) } n ∈ N in S ∞ ×M . For each n ∈ N , consider the BSDE D nt = (cid:90) τt ∧ τ H n ( s, D ns , Z ns , ˆ Z ns ) ds − (cid:90) τt ∧ τ Z ns dB s t ≥ , (B.2)where the generator H n is defined by H n ( s, d, z, ˆ z ) = M s z M s ˆ z − γ ) λ s γσ s ρ s z + (1 − γ ) λ s γσ s ˆ ρ s ˆ z + (1 − γ ) γ ρ s ˆ ρ s z ˆ z + h s − δθ + θ δ ψ ψ (cid:18) {| d |≤ n } e − ψθ d + {| d | >n } (cid:16) − ψθ d + (cid:0) e − ψθ n + ψθ n (cid:1)(cid:17)(cid:19) . (B.3)Comparing H n with H in (3.10), the exponential term e − ψθ d is now replaced by J ( d ) := {| d |≤ n } e − ψθ d + {| d | >n } (cid:0) − ψθ d + (cid:0) e − ψθ n + ψθ n (cid:1)(cid:1) . (B.4)This ensures that d grows exponentially only on [ − n, n ], and linearly otherwise with a strictlypositive slope − ψθ . As such, d (cid:55)→ J ( d ) is by construction continuous, strictly increasing, and oflinear growth on R . This, together with Assumption 1, implies that BSDE (B.2) satisfies [10,Definition 3.1 and Assumption A.1]. Hence, by [10, Theorem 3.3], there exists a unique solution( D n , Z n ) ∈ S ∞ × M to (B.2). Step 2: Establish a uniform upper bound for { D n } n ∈ N , and a solution ( D, Z ) to (3.6) . We will construct a generator H such that H n ≤ H for all n ∈ N . With γ > ρ t , ˆ ρ t ∈ [ − , M and ˆ M in (B.1) satisfies M t , ˆ M t ∈ (cid:2) γ , (cid:3) . Also, by the fact that ab ≤ a + b for all a, b ∈ R , (cid:12)(cid:12)(cid:12)(cid:12) (1 − γ ) λ t γσ t η t z (cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 − γ ) λ t γ σ t + z (cid:12)(cid:12)(cid:12)(cid:12) (1 − γ ) γ ρ t ˆ ρ t z ˆ z (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( γ − γ (cid:18) z z (cid:19) , for η t = ρ t , ˆ ρ t . As a result, H n ( t, d, z, ˆ z ) ≤ z + ˆ z )2 + (cid:18) (1 − γ ) λ t γ σ t + (1 − γ ) (cid:18) r t + λ t γσ t (cid:19) − δθ (cid:19) + θ δ ψ ψ J ( d )= 3( z + ˆ z )2 + (1 − γ ) (cid:32) r t + (2 − γ ) λ t γ σ t − δ − ψ (cid:33) + θ δ ψ ψ J ( d ) ≤ z + ˆ z )2 + C + θ δ ψ ψ J ( d ) , (B.5)where C := (1 − γ )( r − δ − ψ ) if γ ∈ (1 , C := (1 − γ )( r − δ − ψ ) + (1 − γ )(2 − γ )2 γ C λ/σ if γ > r , r , and C λ/σ defined in (3.12). Now, define H ( d, z, ˆ z ) := 3( z + ˆ z )2 − δ ψ d + C (B.6)20nd consider the BSDE D = (cid:90) τt ∧ τ H ( D s , Z s ) ds − (cid:90) τt ∧ τ Z s dB s , t ≥ . (B.7)Observe from (B.4) that J ( d ) ≥ − ψθ d + 1 ≥ − ψθ d , for all d ∈ R . With θ <
0, this implies θ δ ψ ψ J ( d ) ≤ − δ ψ d for all d ∈ R . This, together with (B.5), gives H n ( s, d, z, ˆ z ) ≤ H ( d, z, ˆ z ) on [0 , ∞ ) × R , for all n ∈ N . (B.8)Note that the generator H satisfies [10, Definition 3.1 and Assumption A.1]; particularly, it is strictly monotone in d . Thus, we can apply [10, Theorem 3.3] to get a unique solution ( D, Z ) to(B.7) in S ∞ × M . Moreover, the linear dependence of H in d , along with the negative slope − δ ψ ,indicates that [29, Theorem 2.3] can also be applied here (as H satisfies condition (ii) therein).Hence, as ( D, Z ) is the unique solution to (B.7) in S ∞ × M , it is trivially the “maximal solution”in [29, Theorem 2.3] for which a comparison result readily holds. In view of (B.8), this implies D nt ≤ D t ≤ (cid:101) C ∀ t ≥ , for any n ∈ N , (B.9)where (cid:101) C := ess sup (cid:0) sup t ≥ D t (cid:1) < ∞ . Now, for any n > (cid:101) C , since D nt ≤ (cid:101) C for all t ≥ J ( D nt ) = e − ψθ D nt for all t ≥ H n ( t, D nt , Z nt , ˆ Z nt ) = H ( t, D nt , Z nt , ˆ Z nt ) ∀ t ≥ D n , Z n ) satisfies (3.6), for all n > (cid:101) C . Specifically, ( D n , Z n ) = ( D m , Z m ) for all n, m > (cid:101) C ,and ( D, Z ) := ( D n , Z n ), for n > (cid:101) C , is a solution to (3.6) in S ∞ × M . Remark B.1.
In Step 1 of the proof above, one cannot control e − ( ψ/θ ) d in (3.10) by the truncation e − ( ψ/θ ) d ∧ n , as opposed to [46]. For [10, Theorem 3.3] to be applied, the generator needs to bestrictly monotone in d ; see [10, Assumption A.1 (ii)]. Since d (cid:55)→ e − ( ψ/θ ) d ∧ n is only monotone, butnot strictly, the more complicated truncation J ( d ) comes into play, to ensure both linear growth andstrict monotonicity. This challenge is not present in [46]: on a fixed horizon T > (or, τ ≤ T a.s.),one can apply the stronger existence result [29, Theorem 2.3], which only requires the generator tobe monotone, but not strictly, in d . Remark B.2.
For quadratic BSDEs on a random horizon τ , [10, Theorem 3.3] gives both theexistence and uniqueness of solutions, while [29, Theorem 2.3] gives only the existence, along withcomparison results for the maximal and minimal solutions. Note that when τ is unbounded, to apply[29, Theorem 2.3], we need the generator to be asymptotically linear in d ; see [29, (H1)].Hence, in Step 2 of the proof above, we need to use [10, Theorem 3.3] first, to get a uniquesolution. Since a unique solution is trivially the maximal one, the comparison result in [29, Theorem2.3] can then be invoked. This shows that constructing H is crucial: its linear dependence on d ,much simpler than that of H n , is the key to accessing a comparison result from [29, Theorem 2.3]. B.2 Derivation of Lemma 3.1
We will write X ∗ = X π ∗ ,c ∗ for the candidate optimal wealth process, with ( π ∗ , c ∗ ) defined in (3.9).To begin, we investigate the integrability of X ∗ . Recall C λ/σ , r , and r , defined in (3.12). There are two distinct comparison results in [29], i.e. Theorems 2.3 and 2.6 therein. Particularly, Theorem 2.3allows for random horizons (so that we can apply it here), while Theorem 2.6 requires a fixed terminal time. emma B.1. Suppose γ, ψ > and Assumption 1 holds. Let ( D, Z, ˆ Z ) ∈ S ∞ × M be a solutionto (3.6) , and ( π ∗ , c ∗ ) be as in (3.9) . Given x > , X ∗ t > for all t ≥ a.s. Moreover, for p ≥ , E (cid:2) ( X ∗ t ) p { t ≤ τ } (cid:3) ≤ x p E (cid:20) exp (cid:18)(cid:18) pr + (cid:18) p + 4 p γ (cid:19) C λ/σ (cid:19) t + 4 p γ (cid:90) t (cid:101) Z s ds (cid:19) { t ≤ τ } (cid:21) / , ∀ t ≥ for p < , with (cid:101) C := ess sup(sup t ≥ D s ) < ∞ , E [( X ∗ π ) p ] ≤ x p E (cid:20) exp (cid:18)(cid:18) pr − pδ ψ e − ψθ (cid:101) C + (cid:18) | p | + 4 p γ (cid:19) C λ/σ (cid:19) τ + 4 p − pγ (cid:90) τ (cid:101) Z s ds (cid:19)(cid:21) , ∀ π ∈ T . Proof.
In view of (3.3) and (3.9), X ∗ satisfies dX ∗ t = X t (cid:20) ( r t − δ ψ e − ψθ D t + λ t + σ t (cid:101) Z t γσ t λ t ) dt + λ t + σ t (cid:101) Z t γσ t dW ρt (cid:21) , ≤ t ≤ τ, where (cid:101) Z t := ρ t Z t + ˆ ρ t ˆ Z t . It follows from Assumption 1 and ( D, Z, ˆ Z ) ∈ S ∞ × M that X ∗ t > t ≥ p ∈ R ,( X ∗ t ) p = x p exp (cid:18) p (cid:90) t ∧ τ ( r t − δ ψ e − ψθ D t + a s + 12 ( p − b s ) ds (cid:19) E t ∧ τ ( L ) , (B.10)with a t := λ t + σ t (cid:101) Z t γσ t λ t , b t := λ t + σ t (cid:101) Z t γσ t , L t := p (cid:82) t b s dW ρs , and E t ( A ) denoting the stochastic exponen-tial of some process A . First, we look for a bound for exp( p (cid:82) t ∧ τ ( a s − b s ) ds ). If p ≥ p (cid:18) a t − b t (cid:19) = p (cid:18) λ t γ σ t ( γ −
12 ) + λ t γ σ t ( γ − (cid:101) Z t − γ (cid:101) Z t (cid:19) ≤ p (cid:18) λ t γ σ t ( γ −
12 ) + λ t γ σ t ( γ − (cid:19) = p (cid:18) λ t σ t (cid:19) ≤ p C λ/σ , (B.11)where the first line follows from the definitions of a t and b t , and the first inequality is due to λ t γ σ t ( γ − z − γ z = − γ (cid:0) z − λ t σ t ( γ − (cid:1) + λ t γ σ t ( γ − , ∀ z ∈ R . Similarly, if p < p (cid:18) a t − b t (cid:19) = p (cid:18) λ t γ σ t ( γ −
12 ) + λ t γ σ t ( γ − (cid:101) Z t + 12 γ (cid:101) Z t − γ (cid:101) Z t (cid:19) ≤ p (cid:18) − λ t γ σ t ( γ − − γ (cid:101) Z t (cid:19) ≤ − p C λ/σ − pγ (cid:101) Z t , (B.12)where the second line is due to λ t γ σ t ( γ − ) ≥ λ t γ σ t ( γ − z + γ z = γ (cid:0) z + λ t σ t ( γ − (cid:1) − λ t γ σ t ( γ − , ∀ z ∈ R . Next, we look for a bound for the quadratic variation of L . Observe that (cid:104) L (cid:105) t = p (cid:90) t ∧ τ | b s | ds = p (cid:90) t ∧ τ (cid:18) λ s γ σ s + 2 λ s γ σ s (cid:101) Z s + 1 γ (cid:101) Z s (cid:19) ds ≤ p (cid:90) t ∧ τ (cid:18) λ s γ σ s + 1 γ (cid:101) Z s (cid:19) ds ≤ p γ (cid:18) C λ/σ ( t ∧ τ ) + (cid:90) t ∧ τ (cid:101) Z s ds (cid:19) , (B.13)where the first inequality follows from the fact that ab ≤ a + b for any a, b ∈ R (by taking a = √ λ s γσ s and b = √ γ (cid:101) Z s ). 22ow, let us take p ≥
0. For any t ≥
0, thanks to (B.10) and δ ψ e − ψθ D t > E (cid:2) ( X ∗ t ) p { t ≤ τ } (cid:3) ≤ E (cid:20) x p exp (cid:18) p (cid:90) t ∧ τ (cid:0) r t + a s + 12 ( p − b s (cid:1) ds (cid:19) E ( L ) t ∧ τ { t ≤ τ } (cid:21) ≤ x p E (cid:20) exp (cid:18)(cid:16) pr + p C λ/σ (cid:17) t + 12 (cid:104) L (cid:105) t (cid:19) E ( L ) t { t ≤ τ } (cid:21) , where the second inequality follows from (B.11). By direct calculation, E ( L ) t = E (2 L ) t exp (cid:0) (cid:104) L (cid:105) t (cid:1) ,for all t ≥
0. It then follows that E (cid:2) ( X ∗ t ) p { t ≤ τ } (cid:3) ≤ x p E (cid:20) exp (cid:18)(cid:16) pr + p C λ/σ (cid:17) t + 12 (cid:104) L (cid:105) t (cid:19) E (2 L ) t exp (cid:18) (cid:104) L (cid:105) t (cid:19) { t ≤ τ } (cid:21) ≤ x p E (cid:104) exp (cid:16) (cid:16) pr + p C λ/σ (cid:17) t + 2 (cid:104) L (cid:105) t (cid:17) { t ≤ τ } (cid:105) E [ E (2 L ) t ] ≤ x p E (cid:104) exp (cid:16) (cid:16) pr + p C λ/σ (cid:17) t + 2 (cid:104) L (cid:105) t (cid:17) { t ≤ τ } (cid:105) , where the second inequality results from applying H¨older’s inequality, and the third follows from E [ E (2 L ) t ] ≤
1, as E (2 L ) is by definition a nonnegative local martingale, and thus a supermartingale.Finally, applying (B.13) to the above inequality gives the desired result for p ≥ p <
0, by using the same arguments as above for the “ p ≥
0” case, except the term δ ψ e − ψθ D t cannot be dropped, and 1, r , and (B.12) replace { t ≤ τ } , r , and (B.11), respectively, weget the desired result.Now, we are ready to show the permissibility of ( π ∗ , c ∗ ) in (3.9). Proof of Lemma 3.1.
Thanks to Lemma B.1 and Assumption 2, E [( X ∗ π ) p − ] < ∞ for all π ∈ T .With p − = 2(2 − θ )(1 − γ ) and θ <
0, this readily implies that { E [( X ∗ π ) − γ ] } π ∈T is uniformlyintegrable, i.e. ( X ∗ ) − γ is of class D .It remains to show that c ∗ ∈ C , which, in view of (3.13), is equivalent to, E (cid:20) (cid:90) τ e − δs ( c ∗ s ) p + ds (cid:21) < ∞ and E (cid:104) e − p − δθ − γ τ ( c ∗ τ ) p − (cid:105) < ∞ (B.14)By the definitions of c ∗ and p + in (3.9) and (3.13),( c ∗ s ) p + = ( δ ψ e − ψθ D s X ∗ s ) p + = δ ψ − e − ψ − θ D s ( X ∗ s ) p + ≤ δ ψ − e − ψ − θ (cid:101) C ( X ∗ s ) p + , where (cid:101) C := ess sup(sup t ≥ D t ) < ∞ and the inequality is due to δ > ψ >
1, and θ <
0. Hence, E (cid:20) (cid:90) τ e − δs ( c ∗ s ) p + ds (cid:21) ≤ δ (2( ψ − e − ψ − θ (cid:101) C E (cid:20) (cid:90) τ e − δs ( X ∗ s ) p + ds (cid:21) . (B.15)Using Fubini’s theorem, we get E (cid:20) (cid:90) τ e − δs ( X ∗ s ) p + ds (cid:21) = E (cid:20)(cid:90) ∞ e − δs ( X ∗ s ) p + { s ≤ τ } ds (cid:21) = (cid:90) ∞ e − δs E (cid:2) ( X ∗ s ) p + { s ≤ τ } (cid:3) ds ≤ (cid:90) ∞ e − δs E (cid:20) exp (cid:18)(cid:18) rp + + (cid:18) p + + 4 p γ (cid:19) C λ/σ (cid:19) τ + 4 p γ (cid:90) τ (cid:101) Z u du (cid:19) { s ≤ τ } (cid:21) / ds ≤ δ (cid:18)(cid:90) ∞ δe − δs E (cid:20) exp (cid:18)(cid:18) rp + + (cid:18) p + + 4 p γ (cid:19) C λ/σ (cid:19) τ + 4 p γ (cid:90) τ (cid:101) Z u du (cid:19) { s ≤ τ } (cid:21) ds (cid:19) / ≤ √ δ E (cid:20) exp (cid:18)(cid:18) rp + + (cid:18) p + + 4 p γ (cid:19) C λ/σ (cid:19) τ + 4 p γ (cid:90) τ (cid:101) Z u du (cid:19) · τ (cid:21) / , (B.16)23here the first, second, and third inequalities follow from Lemma B.1, Jensen’s inequality, andFubini’s theorem, respectively. Now, by taking q > E (cid:20) exp (cid:18)(cid:18) rp + + (cid:18) p + + 4 p γ (cid:19) C λσ (cid:19) τ + 4 p γ (cid:90) τ (cid:101) Z s ds (cid:19) · τ (cid:21) ≤ E (cid:20) exp (cid:18) q (cid:18) rp + + (cid:18) p + + 4 p γ (cid:19) C λσ (cid:19) τ + 4 qp γ (cid:90) τ (cid:101) Z s ds (cid:19)(cid:21) /q E (cid:104) τ qq − (cid:105) q − q < ∞ , where the finiteness is guaranteed by Assumption 2. This, together with (B.15) and (B.16), estab-lishes the first part of (B.14). On the other hand, a straightforward calculation, using the definitionsof c ∗ and p − and Lemma B.1, shows that the second part of (B.14) holds under Assumption 2. B.3 Proof of Theorem 3.1
The result will be proved by modifying the arguments in [46, Lemma B.1 and Theorem 2.14]. Forany ( π, c ) ∈ P , define R π,ct := ( X π,ct ∧ τ ) − γ − γ e D t ∧ τ and F π,ct := R π,ct + (cid:82) t ∧ τ f ( c s , R π,cs ) ds , for t ≥
0. In viewof H in (3.10) and the calculation in Section 3.2, F π,c is by construction a local supermartingale.By the Doob-Meyer decomposition and the martingale representation theorem, there exists anincreasing processes A π,c and Z π,c such that F π,ct = (cid:82) t ∧ τ Z π,cs dB s − A π,ct ∧ τ , for all t ≥
0. We deducefrom the definition of F π,c and its decomposition that R π,ct = ( X π,cτ ) − γ − γ e D τ + (cid:90) τt ∧ τ f ( c, R π,cs ) ds − (cid:90) τt ∧ τ Z π,cs dB s + ( A π,cτ − A π,ct ∧ τ ) , t ≥ . Noting that D τ = 0 from (3.6), this shows that ( R π,ct , Z π,ct ) t ≥ is a supersolution to (2.3). Recallfrom Theorem 2.1 that ( V c , Z c ) is a solution to (2.3). Then, a comparison result implies R ≥ V c ;such a comparison result can be established by following the arguments in Step 3 of [46, Proposition2.2]). Thus, we obtain x − γ − γ e D ≥ V c , ∀ ( π, c ) ∈ P . Recall from Lemma 3.1 that ( π ∗ , c ∗ ) ∈ P . We will show that the upper bound x − γ − γ e D is achievedby ( π ∗ , c ∗ ). Again, in view of H in (3.10) and the calculation in Section 3.2, F π ∗ ,c ∗ is by constructiona local martingale; hence, there exists Z ∗ such that F π ∗ ,c ∗ t = (cid:82) t ∧ τ Z ∗ s dB s , for all t ≥
0. This gives R π ∗ ,c ∗ t = ( X ∗ τ ) − γ − γ + (cid:90) τt ∧ τ f ( c ∗ s , R π ∗ ,c ∗ s ) ds − (cid:90) τt ∧ τ Z ∗ s dB s , t ≥ , implying x − γ − γ e D = E (cid:20)(cid:90) τ f (cid:18) c ∗ s , ( X ∗ s ) − γ e D s − γ (cid:19) ds + ( X ∗ τ ) − γ − γ (cid:21) = V c ∗ , where the last equality follows from Theorem 2.1. C Proofs for Section 4
C.1 Proof of Theorem 4.1
For any ( w, y ) ∈ D , let ( D w,yt , Z w,yt , ˆ Z w,yt ) ∈ S ∞ × M be a solution to (3.6) under the randomhorizon τ w,y (obtained from Proposition 3.1), and set C := (cid:107) D w,y (cid:107) ∞ < ∞ . By the ellipticity condi-tions in Assumptions 1 and 3, ( W , Y ) is non-degenerate in D , which implies that C is independent24f the choice of ( w, y ). For any ( w, y ) ∈ D , d ∈ R , and p = ( p , p ) ∈ R , define the function G ( w, y, d, p , p ) := a ( y ) p + α ( w, y ) p + G ( y, d, b ( y ) p + β ( w, y ) p , Γ( w, y ) p ) , (C.1)where G is given in (4.3). We aim to show the existence of a solution to (4.1) by using [22, Theorem15.12], which requires G to satisfy d · G ( w, y, d, , ≤ | d | ≥ M, for some M > . (C.2)Note that G ( w, y, d, ,
0) = δ ψ θψ e − ψθ d + (1 − γ ) (cid:0) r ( y ) + λ ( y ) γσ ( y ) (cid:1) − δθ , and thus the above conditionneed not hold in general. To remedy this, define ϕ ∈ C , ( R ) by ϕ ( d ) := d + (cid:0) e − ψθ C − C (cid:1) , d > C,e − ψθ d , − C ≤ d ≤ C,d + (cid:0) e ψθ C + C (cid:1) , d < − C. Define the function G ϕ as G in (4.3), with the term e − ψθ d therein replaced by ϕ ( d ). Consider thecorresponding boundary value problem L g + G ϕ ( y, g, bg y + βg w , Γ g w ) = 0 , for ( w, y ) ∈ D g ( w, y ) = 0 , for ( w, y ) ∈ ∂ D . (C.3)Define G ϕ as in (C.1) with G replaced by G ϕ . Note that G ϕ satisfies (C.2) and is Lipschitz on D × R × R under Assumption 1. Hence, we can apply [22, Theorem 15.12] to obtain a solution g ∈ C ,ν ( D ) to (C.3). Now, define the process D w,yt := g ( W wt , Y yt ). Applying Itˆo’s formula yields dD w,yt = − G ϕ (cid:0) Y yt , D w,yt , Z w,yt , Z w,yt (cid:1) dt + Z w,yt dW t + Z w,yt d ˆ W t (C.4)where Z w,yt := b ∂g∂y ( Y yt , W wt ) + β ∂g∂w ( Y yt , W wt ) and Z w,yt := Γ ∂g∂w ( Y yt , W wt ). On the other hand, define u ( w, y ) := D w,y for all ( w, y ) ∈ D . Thanks to (3.6), dD w,yt = − H (cid:16) t, D w,yt , Z w,yt , ˆ Z w,yt (cid:17) dt + Z w,yt dW t + ˆ Z w,yt d ˆ W t = − G ϕ (cid:16) Y yt , D w,yt , Z w,yt , ˆ Z w,yt (cid:17) dt + Z w,yt dW t + ˆ Z w,yt d ˆ W t , where the second line follows from the definitions of H and G in (3.10) and (4.3), as well as | D w,yt | ≤ C for all t ≥
0. Using the comparison result for quadratic BSDEs in [29, Theorem 2.6],we conclude that g ( w, y ) = D w,y = D w,y = u ( w, y ), for all ( w, y ) ∈ D . Hence, u ∈ C ,ν ( D ) solves(C.3), and thus (4.2) (as | u ( w, y ) | = | D w,y | ≤ C , making G ϕ = G ). The uniqueness follows fromthe comparison principle for PDEs with quadratic growth in [4, Theorem 1.2]. C.2 Proof of Lemma 4.1
Consider the survival probability P ( w, y, t ) := P ( τ > t | W = w, Y = y ). The associatedbackward Fokker-Planck equation is ∂P∂t = − α ( y − m ) ∂P∂y + 12 k y ∂ P∂y + 12 ( y + ε ) ∂ P∂w , (C.5)25ith initial condition P ( w, y,
0) = 1 and boundary condition P ( ± L , y, t ) = 0. Motivated by [33],we conjecture the form of the solution to (C.5) via a Fourier series, i.e. P ( w, y, t ) = ∞ (cid:88) n =0 P n ( y, t ) cos (cid:18) (2 n + 1) πwL (cid:19) . By the change of variables s = αt and v = ( αk ) y , and using the notation µ := αm k and β n , ε n defined in the statement of Lemma 4.1, (C.5) becomes ∂P n ∂s = − ( v − µ ) ∂P n ∂v + v ∂ P n ∂v − (cid:18) β n L (cid:19) vP n + ε n P n , (C.6)with initial condition P n ( v,
0) = − n π (2 n +1) , ∀ n ∈ N . This can readily be solved by the ansatz P n ( v, s ) = 4( − n π (2 n + 1) exp ( − A n ( s ) − B n ( s ) v ) . Differentiating and substituting this back into (C.6), we find A (cid:48) n ( s ) = − v (cid:32) B (cid:48) n ( s ) + B n ( s ) + B n ( s ) − (cid:18) β n L (cid:19) (cid:33) + µB n ( s ) + ε n . Notice that B n ( s ) must solve the Riccati equation B (cid:48) n ( s ) = − B n ( s ) − B n ( s ) + (cid:18) β n L (cid:19) , B n (0) = 0 , (C.7)under which A (cid:48) n ( s ) = µB n ( s ) + ε n , implying A n ( s ) = µ (cid:82) s B n ( t ) dt + ε n s . The solution to (C.7),derived in [33], is given as in (4.8). If follows that one can calculate A n ( s ) as in (4.7). Therefore,the survival probability has the representation P ( w, v, s ) = ∞ (cid:88) n =0 − n π (2 n + 1) exp ( − A n ( s ) − B n ( s ) v ) cos (cid:18) (2 n + 1) πwL (cid:19) , (C.8)with A n and B n specified as above. Since the density of τ is given by P ( τ ∈ ds | W = w, V = v ) = − ∂P ( w, v, s ) ∂s , a direct calculation leads to (4.6). References [1]
J. Aurand and Y.-J. Huang , Mortality and healthcare: an analysis under Epstein-Zinpreferences , (2020). Preprint, available at https://arxiv.org/abs/2003.01783.[2]
R. Bansal , Long-run risks and financial markets , Review - Federal Reserve Bank of St.Louis,89 (2007), pp. 283–299.[3]
R. Bansal and A. Yaron , Risks for the long run: A potential resolution of asset pricingpuzzles , The Journal of Finance, 59 (2004), pp. 1481–1509.264]
G. Barles and F. Murat , Uniqueness and the maximum principle for quasilinear ellipticequations with quadratic growth conditions , Arch. Rational Mech. Anal., 133 (1995), pp. 77–101.[5]
L. Benzoni, P. Collin-Dufresne, and R. S. Goldstein , Explaining asset pricing puzzlesassociated with the 1987 market crash , Journal of Financial Economics, 101 (2011), pp. 552–573.[6]
H. S. Bhamra, L.-A. Kuehn, and I. A. Strebulaev , The levered equity risk premium andcredit spreads: A unified framework , The Review of Financial Studies, 23 (2010), pp. 645–703.[7]
C. Blanchet-Scalliet, N. El Karoui, M. Jeanblanc, and L. Martellini , Optimalinvestment decisions when time-horizon is uncertain , J. Math. Econom., 44 (2008), pp. 1100–1113.[8]
B. Bouchard and H. Pham , Wealth-path dependent utility maximization in incompletemarkets , Finance Stoch., 8 (2004), pp. 579–603.[9]
P. Briand and R. Carmona , BSDEs with polynomial growth generators , J. Appl. Math.Stochastic Anal., 13 (2000), pp. 207–238.[10]
P. Briand and F. Confortola , Quadratic BSDEs with random terminal time and ellipticPDEs in infinite dimension , Electron. J. Probab., 13 (2008), pp. 1529–1561.[11]
P. Briand, B. Delyon, Y. Hu, E. Pardoux, and L. Stoica , Lp solutions of back-ward stochastic differential equations , Stochastic Processes and their Applications, 108 (2003),pp. 109 – 129.[12]
P. Briand and Y. Hu , BSDE with quadratic growth and unbounded terminal value , Probab.Theory Related Fields, 136 (2006), pp. 604–618.[13] ,
Quadratic BSDEs with convex generators and unbounded terminal conditions , Probab.Theory Related Fields, 141 (2008), pp. 543–567.[14]
P. Cheridito and Y. Hu , Optimal consumption and investment in incomplete markets withgeneral constraints , Stoch. Dyn., 11 (2011), pp. 283–299.[15]
M. Chernov and E. Ghysels , A study towards a unified approach to the joint estimation ofobjective and risk neutral measures for the purpose of options valuation , Journal of FinancialEconomics, 56 (2000), pp. 407–458.[16]
R. W. R. Darling and E. Pardoux , Backwards sde with random terminal time and appli-cations to semilinear elliptic pde , Ann. Probab., 25 (1997), pp. 1135–1159.[17]
A. D. Dr˘agulescu and V. M. Yakovenko , Probability distribution of returns in the Hestonmodel with stochastic volatility , Quant. Finance, 2 (2002), pp. 443–453.[18]
D. Duffie and L. G. Epstein , Stochastic differential utility , Econometrica, 60 (1992),pp. 353–394. With an appendix by the authors and C. Skiadas.[19]
D. Duffie and P.-L. Lions , PDE solutions of stochastic differential utility , J. Math.Econom., 21 (1992), pp. 577–606. 2720]
N. El Karoui, M. Jeanblanc, and Y. Jiao , What happens after a default: the conditionaldensity approach , Stochastic Process. Appl., 120 (2010), pp. 1011–1032.[21]
L. Epstein and S. Zin , Substitution, risk aversion, and the temporal behavior of consumptionand asset returns: A theoretical framework , Econometrica, 57 (1989), pp. 937–69.[22]
D. Gilbarg and N. S. Trudinger , Elliptic partial differential equations of second order ,Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.[23]
P. Guasoni and Y.-J. Huang , Consumption, Investment, and Healthcare with Aging , toappear in Finance and Stochastics, (2019). Available at https://arxiv.org/abs/1901.00424.[24]
L. P. Hansen, J. Heaton, J. Lee, and N. Roussanov , Intertemporal substitution andrisk aversion , in Handbook of Econometrics, J. J. Heckman and E. E. Leamer, eds., vol. 6A,Elsevier, 1 ed., 2007, ch. 61, pp. 3967–4056.[25]
Y. Hu, P. Imkeller, and M. M¨uller , Utility maximization in incomplete markets , Ann.Appl. Probab., 15 (2005), pp. 1691–1712.[26]
M. Jeanblanc, T. Mastrolia, D. Possama¨ı, and A. R´eveillac , Utility maximizationwith random horizon: a BSDE approach , Int. J. Theor. Appl. Finance, 18 (2015), p. 1550045.[27]
I. Karatzas and H. Wang , Utility maximization with discretionary stopping , SIAM J. Con-trol Optim., 39 (2000), pp. 306–329.[28]
I. Kharroubi, T. Lim, and A. Ngoupeyou , Mean-variance hedging on uncertain timehorizon in a market with a jump , Appl. Math. Optim., 68 (2013), pp. 413–444.[29]
M. Kobylanski , Backward stochastic differential equations and partial differential equationswith quadratic growth , Ann. Probab., 28 (2000), pp. 558–602.[30]
H. Kraft, T. Seiferling, and F. T. Seifried , Optimal consumption and investment withEpstein-Zin recursive utility , Finance Stoch., 21 (2017), pp. 187–226.[31]
H. Kraft, F. T. Seifried, and M. Steffensen , Consumption-portfolio optimization withrecursive utility in incomplete markets , Finance Stoch., 17 (2013), pp. 161–196.[32]
N. V. Krylov , Controlled diffusion processes , vol. 14 of Stochastic Modelling and AppliedProbability, Springer-Verlag, Berlin, 2009. Translated from the 1977 Russian original by A. B.Aries, Reprint of the 1980 edition.[33]
J. Masoliver and J. Perell´o , Escape problem under stochastic volatility: the Heston model ,Phys. Rev. E (3), 78 (2008), p. 056104.[34] ,
First-passage and risk evaluation under stochastic volatility , Phys. Rev. E (3), 80 (2009),p. 016108.[35]
A. Matoussi and H. Xing , Convex duality for Epstein-Zin stochastic differential utility ,Math. Finance, 28 (2018), pp. 991–1019.[36]
R. C. Merton , Lifetime portfolio selection under uncertainty: The continuous-time case , TheReview of Economics and Statistics, 51 (1969), pp. 247–257.2837]
M.-A. Morlais , Quadratic BSDEs driven by a continuous martingale and applications to theutility maximization problem , Finance Stoch., 13 (2009), pp. 121–150.[38]
J. Pan , The jump-risk premia implicit in options: evidence from an integrated time-seriesstudy , Journal of Financial Economics, 63 (2002), pp. 3–50.[39]
E. Pardoux , BSDEs, weak convergence and homogenization of semilinear PDEs , in Nonlinearanalysis, differential equations and control (Montreal, QC, 1998), vol. 528 of NATO Sci. Ser.C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 1999, pp. 503–549.[40]
H. Pham , Smooth solutions to optimal investment models with stochastic volatilities and port-folio constraints , Appl. Math. Optim., 46 (2002), pp. 55–78.[41]
M. Royer , Bsdes with a random terminal time driven by a monotone generator and theirlinks with pdes , Stochastics and Stochastic Reports, 76 (2004), pp. 281–307.[42]
M. Schroder and C. Skiadas , Optimal consumption and portfolio selection with stochasticdifferential utility , J. Econom. Theory, 89 (1999), pp. 68–126.[43]
T. Seiferling and F. T. Seifried , Epstein-zin stochastic differential utility: Ex-istence, uniqueness, concavity, and utility gradients , (2016). Preprint. Available athttps://dx.doi.org/10.2139/ssrn.2625800.[44]
A. C. Silva and V. M. Yakovenko , Comparison between the probability distribution ofreturns in the Heston model and empirical data for stock indexes , Phys. A, 324 (2003), pp. 303–310. International Econophysics Conference IEC2002 (Bali).[45]
A. Vissing-Jørgensen and O. Attanasio , Stock-market participation, intertemporal sub-stitution, and risk-aversion , American Economic Review, 93 (2003), pp. 383–391.[46]
H. Xing , Consumption-investment optimization with Epstein-Zin utility in incomplete mar-kets , Finance Stoch., 21 (2017), pp. 227–262.[47]
M. E. Yaari , Uncertain lifetime, life insurance, and the theory of the consumer , The Reviewof Economic Studies, 32 (1965), pp. 137–150.[48]
T. Zariphopoulou , A solution approach to valuation with unhedgeable risks , Finance Stoch.,5 (2001), pp. 61–82.[49]