Epstein–Zin Recursive Utility

• Epstein–Zin recursive utility is a non-time-separable model that decouples relative risk aversion from the elasticity of intertemporal substitution, addressing key asset pricing puzzles.
• The model is formulated via a backward stochastic differential equation (BSDE) with a non-Lipschitz, monotonic aggregator to ensure existence and uniqueness of solutions.
• It underpins optimal consumption and portfolio strategies in incomplete markets, explaining phenomena like large equity risk premia and low risk-free rates.

Epstein–Zin recursive utility is a class of non-time-separable utility models originating in macroeconomics and asset pricing theory, designed to decouple relative risk aversion from the elasticity of intertemporal substitution. Unlike classical expected utility, which forces these two aspects into an inverse relationship, the Epstein–Zin functional—parameterized by risk aversion γ and elasticity ψ—permits their independent specification. This flexibility is now a cornerstone in continuous-time financial models, allowing for empirically relevant calibrations and resolution of asset pricing puzzles.

1. Mathematical Formulation and Core Properties

The continuous-time Epstein–Zin utility functional is defined recursively and most rigorously expressed as a solution to a backward stochastic differential equation (BSDE). For a consumption process $c = (c_t)_{t \in [0,T]}$, the agent’s utility is given by

$$V_t^c = \mathbb{E}_t \Big\{ U(c_T) + \int_t^T f(c_s, V_s^c) ds \Big\},$$

where %%%%1%%%% is the terminal (bequest) utility and $f$ is the Epstein–Zin aggregator. In the empirically relevant case where both risk aversion $\gamma$ and elasticity of intertemporal substitution $\psi$ satisfy $\gamma, \psi > 1$, define

$\theta = \frac{1-\gamma}{1-1/\psi}\,,$

so that $\theta<0$. The aggregator takes the form (see (Xing, 2015))

$$f(c, v) = \delta \frac{(1-\gamma)v}{1-1/\psi} \left[ \left( \frac{c}{[(1-\gamma)v]^{1/(1-\gamma)}} \right)^{1-1/\psi} - 1 \right],$$

where $\delta>0$ is the subjective discount rate.

Due to this formulation, the aggregator is non-Lipschitz for $\gamma,\psi>1$, exhibiting superlinear growth and lacking joint concavity. The practical tractability of Epstein–Zin recursive utility is achieved via a nonlinear transformation to a BSDE for a process $(Y_t, Z_t)$: $$Y_t = \xi + \int_t^T F(s, c_s, Y_s) ds - \int_t^T Z_s dB_s,$$ with explicit generator $F$ inheriting a monotonicity property crucial for establishing existence and uniqueness despite the non-Lipschitz nature. The solution $V^c$, and consequently the indirect utility, are then recovered from $Y_t$ through inversion of the transformation (Xing, 2015).

2. Recursive Utility Optimization in Incomplete Markets

In incomplete markets with stochastic investment opportunities, the agent’s wealth process $W_t$ and state variable $X_t$ evolve according to

$$\begin{aligned} dW_t &= W_t \left[ r(X_t) + \pi_t' \mu(X_t) \right] dt + W_t \pi_t' \sigma(X_t) dW_t^\rho - c_t dt, \ dX_t &= b(X_t) dt + a(X_t) dW_t, \end{aligned}$$

where $\pi_t$ denotes the portfolio weights and $c_t$ the consumption rate (Xing, 2015). The indirect utility is conjectured to take the homothetic, exponentiated form

$$V^*_t = \frac{(W^*_t)^{1-\gamma}}{1-\gamma} e^{Y_t},$$

with $Y_t$ solving a BSDE whose generator $H(t,y,z)$ incorporates the market drift, volatility, risk aversion, and EIS as

$$\begin{aligned} H(t,y,z) &= \frac{1}{2} z M_t z' + \frac{1-\gamma}{\gamma} \mu'_t \Sigma_t^{-1} \sigma_t \rho_t z' \ &\quad + \theta \frac{\delta^{\psi}}{\psi} e^{-\psi y/\theta} + h_t - \delta\theta, \end{aligned}$$

with $M_t$, $h_t$, and $\Sigma_t$ derived from market coefficients.

The associated optimal controls (candidate consumption and portfolio strategies) are expressed in terms of the BSDE solution: $$\pi^*_t = \frac{1}{\gamma} \Sigma_t^{-1} \left[ \mu_t + \sigma_t \rho_t' Z_t \right], \qquad \frac{c^*_t}{W^*_t} = \delta^{\psi} e^{-\psi Y_t/\theta}.$$ Verification of optimality relies on martingale and comparison results for the BSDE (Xing, 2015).

3. Technical Challenges for Parameter Regimes with $\gamma, \psi > 1$

When both risk aversion and EIS exceed one—an empirically typical regime—the Epstein–Zin aggregator is not Lipschitz (indeed, $\theta<0$). Standard existence and uniqueness arguments for the BSDE solution do not apply. Technical resolution involves:

• A monotonicity method: Transforming the BSDE so that the generator is monotonic in $y$, which suffices for generalized comparison theorems.
• Non-integrability and superlinear growth: Application of approximation and truncation techniques ensures the solution domain can be characterized, exploiting properties intrinsic to the transformed equation and exponential moments.
• Lack of joint concavity: The lack of concavity in $(c,v)$ requires care in verification arguments and structural calculations for optimal feedback policies.

This regime is economically significant, as it is associated with “preference for early resolution of uncertainty,” a phenomenon that underpins the ability of the Epstein–Zin model to rationalize large equity premia and low risk-free rates that are unattainable under time-separable utility (Xing, 2015).

4. Superdifferentials, State-Price Densities, and Asset Pricing Implications

A key structural object in the Epstein–Zin framework is the superdifferential (or utility gradient) of the indirect utility, which functions as the pricing kernel in equilibrium asset pricing. In the stochastic market context, this is captured by the process

$$D^*_t = \exp\left\{ \int_0^t \left[ (\theta-1)\delta^{\psi} e^{-{\psi Y_s}/{\theta}} - \delta\theta \right] ds \right\} \left( \frac{W^*_t}{w} \right)^{-\gamma} e^{Y_t - Y_0}.$$

The martingale property of $W^*_t D^*_t + \int_0^t D^*_s c^*_s ds$ verifies the optimality of $(\pi^*, c^*)$, and, in equilibrium, $D^*_t$ determines the state-price density (pricing kernel), thereby directly linking individual optimization to observed asset prices (Xing, 2015).

This identification is central to explaining features such as

• Large equity risk premia,
• Low risk-free rates,
• Strong sensitivity to the separation of risk aversion and intertemporal substitution.

Empirically, γ,ψ>1 is consistent with observed market pricing phenomena, and the theoretical apparatus confirms that such calibrations are feasible only under Epstein–Zin and not under time-separable preferences.

5. Comparative Statics and the Role of the Dynamic Programming Principle

The general framework for recursive optimization under Epstein–Zin utility is unified by the dynamic programming principle (DPP) even when the aggregator is non-Lipschitz. Through backward semigroups and general comparison theorems for BSDEs, the value function representation satisfies

$$u(t,x) = \operatorname{ess\,sup}_{v\in\mathcal{U}} G_{t,t+\Delta t}[u(t+\Delta t, X_{t+\Delta t})],$$

where $G_{t,t+\Delta t}$ propagates the BSDE with non-Lipschitz monotone aggregator (Pu et al., 2015). Consequently, the value function is characterized as the viscosity solution of the associated Hamilton–Jacobi–BeLLMan (HJB) equation, confirming existence and uniqueness under non-classical assumptions.

By these methods, one can explicitly formulate optimal consumption and investment strategies, superdifferentials, and pricing kernels, and analyze their comparative static properties with respect to $\gamma$, $\psi$, and the economic environment (incomplete, stochastic markets, constraints, etc.). These methodologies accommodate a variety of extensions, including robust preferences and model ambiguity (Ji et al., 2016), and generalize to settings with unbounded endowments and noncompact constraints (Hu et al., 2017).

6. Broader Extensions, Further Results, and Applications

Epstein–Zin recursive utility has become a standard modeling device to address asset-pricing anomalies and heterogeneity in time preferences without requiring restrictive parameter identification. The mathematical techniques—core among them BSDEs with monotonic generators, viscosity solution approaches to HJB equations with non-standard Hamiltonians, and duality-based methods for partial information or ambiguity aversion—form a rigorous backbone for analyses in incomplete, unbounded, or ambiguous markets (Hu et al., 2017, Ji et al., 2016).

Moreover, the superdifferential and associated martingale and pricing kernel properties elucidate both the verification arguments for optimality of candidate strategies and the structural link between individual preferences and equilibrium asset returns (Xing, 2015). These results provide benchmark formulas and algorithms for both theoretical asset pricing and applied quantitative finance.

In summary, the Epstein–Zin recursive utility framework, defined rigorously via non-Lipschitz BSDEs, underlies a broad class of optimal consumption–investment problems in continuous time, supporting explicit feedback policies, superdifferential constructions, and the analysis of equilibrium pricing phenomena in incomplete and stochastic environments. This structure supports the empirical relevance and flexibility required by financial economists and asset-pricing theorists.