Recursive Utility in Continuous Time
Introduction
The recursive preferences in a multiperiod economy notebook derived the Epstein-Zin SDF in discrete time by exploiting homotheticity and Euler’s theorem. The key result was m_{t+1} = \left[\beta\!\left(\frac{c_{t+1}}{c_t}\right)^{-1/\psi}\right]^{\!\theta} \left[\frac{1}{R^{w}_{t+1}}\right]^{1-\theta}, \qquad \theta = \frac{1-\gamma}{1-1/\psi}. Under power utility (\psi = 1/\gamma, so \theta = 1) the wealth return drops out entirely; under Epstein-Zin (\theta \neq 1) it enters as an independent pricing factor that captures the agent’s concern for news about future investment opportunities.
Passing this result to continuous time requires some care. The discrete aggregator V_t = [(1-\beta)c_t^\rho + \beta \mu_t^\rho]^{1/\rho} has no direct limit when \Delta t \to 0 because both the discount factor \beta = e^{-\delta\Delta t} \to 1 and the one-period certainty-equivalent \mu_t collapse in a correlated way. Duffie and Epstein (1992) resolve this by showing that Epstein-Zin preferences in continuous time are characterized by a stochastic differential equation for the utility process, governed by a normalized aggregator f(c,V). This object plays the role of the period felicity function in additive utility, but the continuation value V_t itself enters as an argument, separating risk aversion from intertemporal substitution at every instant.
This notebook develops stochastic differential utility (SDU), derives the continuous-time Epstein-Zin aggregator and consumption first-order condition, and explains how recursive utility modifies the stochastic discount factor relative to the additive case. The continuous-time limit is both mathematically elegant and economically important: it makes precise exactly when and how \gamma and \psi produce independent effects on asset prices.
The general theory here stops at the recursive-utility pricing kernel. A separate notebook, Affine Recursive Utility in Continuous Time, applies these results to a one-factor affine-Gaussian state model and derives the local exponential-affine approximation for the value function and the SDF.
Stochastic Differential Utility
The Aggregator
In continuous time, the utility process V_t is defined implicitly by V_t = \operatorname{E}_t\!\left[\int_t^\infty f(c_s, V_s)\,ds\right], \tag{1} where the function f(c, v) is the normalized aggregator. Equation (1) is not a closed-form definition but a functional equation: V_t must be the process such that the stochastic integral representation holds simultaneously for every t. Define M_t = V_t + \int_0^t f(c_s, V_s)\,ds. The integral representation (1) implies M_t = \operatorname{E}_t\!\left[\int_0^\infty f(c_s, V_s)\,ds\right], which is a martingale. In a Brownian filtration, the martingale representation theorem guarantees that any square-integrable martingale is a stochastic integral with respect to \mathbf{B}, so dM_t = \pmb{\sigma}^V_t \cdot d\mathbf{B}_t for some adapted process \pmb{\sigma}^V_t. Rearranging dM_t = dV_t + f(c_t, V_t)\,dt yields the stochastic differential equation dV_t = -f(c_t, V_t)\,dt + \pmb{\sigma}^V_t \cdot d\mathbf{B}_t, \tag{2} where the drift is pinned down by the aggregator and the martingale part captures unpredictable revisions to the continuation value as new information arrives.
For additive utility with felicity u(c) and discount rate \delta, the aggregator is f(c, v) = \delta\bigl(u(c) - v\bigr). To see why (1) then reduces to the standard discounted utility representation, substitute into the functional equation to get the fixed-point condition V_t = \operatorname{E}_t\!\left[\int_t^\infty \delta\bigl(u(c_s) - V_s\bigr)\,ds\right]. One can verify that V_t = \operatorname{E}_t\!\left[\int_t^\infty \delta e^{-\delta(s-t)} u(c_s)\,ds\right] solves this equation: the factor \delta ensures that in steady state V = u(c), so the continuation value has the same units as the felicity function itself.
Substituting f(c,v) = \delta(u(c)-v) into (2) gives the additive utility SDE directly: dV_t = \delta\bigl(V_t - u(c_t)\bigr)\,dt + \pmb{\sigma}^V_t \cdot d\mathbf{B}_t, where the martingale term \pmb{\sigma}^V_t \cdot d\mathbf{B}_t captures revisions to V_t driven by news about the future consumption path. The steady-state condition f = 0 gives V = u(c), consistent with the normalized integral formula.
Epstein-Zin Aggregator
To specialize the aggregator to Epstein-Zin preferences, impose two conditions on f. First, require f(c,v) = 0 if and only if v = c^{1-\gamma}/(1-\gamma): a constant consumption path c is a steady state precisely when the continuation value equals the CRRA value of c enjoyed forever. Inverting this condition defines the certainty-equivalent consumption \hat{c}(v) = \bigl[(1-\gamma)\,v\bigr]^{1/(1-\gamma)}, the unique consumption level whose CRRA lifetime utility equals v. This power transformation requires (1-\gamma)v > 0, so along any admissible path the utility process must have the same sign as (1-\gamma): v > 0 when \gamma < 1 and v < 0 when \gamma > 1.
Second, require f to respect CRRA scale invariance. Scaling all consumption by \lambda scales utility by \lambda^{1-\gamma}, so demand f(\lambda c,\,\lambda^{1-\gamma}v) = \lambda^{1-\gamma}f(c,v). This homogeneity condition forces f to depend on c and v only through the ratio c/\hat{c}(v), with the prefactor (1-\gamma)v supplying the correct scaling: f(c,v) = (1-\gamma)v\cdot g\!\left(\frac{c}{\hat{c}(v)}\right), for some function g with g(1) = 0. The ratio c/\hat{c}(v) measures how current consumption compares to its break-even level, and g determines how sensitively the drift responds to deviations from that benchmark. Parametrizing g as a CES function with elasticity \psi > 0 gives g(x) = \delta(x^{1-1/\psi}-1)/(1-1/\psi) and hence the Epstein-Zin normalized aggregator f(c, v) = \frac{\delta(1-\gamma)v}{1-1/\psi} \left[ \left(\frac{c}{\hat{c}(v)}\right)^{1-1/\psi} - 1 \right]. \tag{3} The two preference parameters enter through entirely separate objects: \gamma determines \hat{c}(v), governing how the agent values uncertainty about future utility, while \psi determines the curvature of g, governing how the drift reacts to gaps between current and break-even consumption. This separation in the aggregator is precisely what additive utility cannot achieve, even though equilibrium allocations and prices still depend jointly on the solution for the value-function coefficient.
In the limit \psi \to 1, the CES function (x^{1-1/\psi}-1)/(1-1/\psi) converges to \ln x, giving f(c, v) \xrightarrow{\psi \to 1} \delta(1-\gamma)v\ln\!\left(\frac{c}{\hat{c}(v)}\right).
For additive utility \psi = 1/\gamma, so 1-1/\psi = 1-\gamma and \hat{c}^{1-\gamma} = (1-\gamma)v. Substituting into (3) gives f(c, v) = \delta\Bigl[\frac{c^{1-\gamma}}{1-\gamma} - v\Bigr] = \delta\bigl(u(c) - v\bigr), recovering the additive aggregator exactly. Under this normalization, the utility process V_t represents the average future felicity, which is the standard representation used to derive the Epstein-Zin SDF in continuous time.
The Bellman Equation
The agent maximizes V_0 by choosing a consumption plan c_t \geq 0 and a portfolio weight vector \pmb{\alpha}_t. The investment opportunity set is time-varying: the short rate r(\mathbf{z}_t), the vector of expected excess returns \pmb{\mu}(\mathbf{z}_t) - r(\mathbf{z}_t)\pmb{\iota}, and the return volatility matrix \pmb{\sigma}(\mathbf{z}_t) all depend on a state vector \mathbf{z}_t. Wealth therefore evolves as dW_t = \left[W_t\left(r(\mathbf{z}_t) + \pmb{\alpha}_t'(\pmb{\mu}(\mathbf{z}_t) - r(\mathbf{z}_t)\pmb{\iota})\right) - c_t\right]dt + W_t\pmb{\alpha}_t'\pmb{\sigma}(\mathbf{z}_t)\,d\mathbf{B}_t, and \mathbf{z}_t follows its own diffusion driven by the same Brownian motion, capturing the covariation between portfolio returns and shifts in investment opportunities: d\mathbf{z}_t = \pmb{\mu}^z(\mathbf{z}_t)\,dt + \pmb{\sigma}^z(\mathbf{z}_t)\,d\mathbf{B}_t. Then the HJB equation for stochastic differential utility is 0 = \sup_{c,\pmb{\alpha}} \left\{ f\!\bigl(c,V(W,\mathbf{z})\bigr) + \mathcal{L}^{c,\pmb{\alpha}}V(W,\mathbf{z}) \right\}, where \mathcal{L}^{c,\pmb{\alpha}} is the Ito generator associated with the joint process (W_t,\mathbf{z}_t). Expanding it explicitly, \begin{aligned} \mathcal{L}^{c,\pmb{\alpha}}V &= V_W\!\bigl[W(r(\mathbf{z})+\pmb{\alpha}'(\pmb{\mu}(\mathbf{z})-r(\mathbf{z})\pmb{\iota}))-c\bigr] + \tfrac{1}{2}V_{WW}W^2\|\pmb{\alpha}'\pmb{\sigma}(\mathbf{z})\|^2 \\ &\quad + (\nabla_z V)'\pmb{\mu}^z + \tfrac{1}{2}\operatorname{tr}\!\bigl(\pmb{\sigma}^z(\pmb{\sigma}^z)' H_z V\bigr) + W(\nabla_z V_W)'\pmb{\sigma}^z\pmb{\sigma}(\mathbf{z})'\pmb{\alpha}, \end{aligned} where H_z V is the Hessian of V with respect to \mathbf{z} and the last term captures the covariation between wealth and the state variables. Conjecturing that the value function is of the form V(W, \mathbf{z}) = h(\mathbf{z})\,W^{1-\gamma}/(1-\gamma) for some function h > 0, homotheticity implies that optimal consumption is proportional to wealth: c/W = \kappa(\mathbf{z}). The key derivatives are V_W(W, \mathbf{z}) = h(\mathbf{z})\,W^{-\gamma}, V_{WW}(W,\mathbf{z}) = -\gamma h(\mathbf{z})W^{-\gamma-1}, and \nabla_z V(W,\mathbf{z}) = \frac{W^{1-\gamma}}{1-\gamma}\nabla_z h(\mathbf{z}). Substituting the homothetic guess into the HJB separates the choice variables from the scale variable W. The first-order condition for current consumption is f_c\!\bigl(c,V(W,\mathbf{z})\bigr) = V_W(W,\mathbf{z}), which says that the marginal gain from an extra unit of current consumption must equal the shadow value of one more unit of wealth. Using the Epstein-Zin aggregator (3) and the identity V = hW^{1-\gamma}/(1-\gamma), this condition simplifies, for \psi \neq 1, to \delta c^{-1/\psi} \bigl[(1-\gamma)V\bigr]^{1-1/\theta} = V_W, \qquad \theta = \frac{1-\gamma}{1-1/\psi}. To verify this, write J = (1-\gamma)V, so \hat c(V) = J^{1/(1-\gamma)}. Then f(c,V) = \frac{\delta J}{1-1/\psi}\left[\left(\frac{c}{J^{1/(1-\gamma)}}\right)^{1-1/\psi} - 1\right], and differentiating with respect to c gives f_c(c,V) = \delta c^{-1/\psi}J^{1-\frac{1-1/\psi}{1-\gamma}} = \delta c^{-1/\psi}J^{1-1/\theta}. Differentiating with respect to v requires the chain rule through \hat{c}(v). Since \hat{c}(v) = [(1-\gamma)v]^{1/(1-\gamma)}, one has d\hat{c}/dv = \hat{c}(v)/[(1-\gamma)v], so dx/dv = -x/[(1-\gamma)v] where x = c/\hat{c}(v). Applying this to (3): f_v(c,v) = \frac{\delta(1-\gamma)}{1-1/\psi}\bigl[x^{1-1/\psi}-1\bigr] - \delta x^{1-1/\psi} = \delta\!\left[(\theta-1)\!\left(\frac{c}{\hat{c}(v)}\right)^{1-1/\psi} - \theta\right]. \tag{4} When \theta = 1 (power utility), f_v = -\delta everywhere. When \theta \neq 1, f_v is state-dependent through the ratio c/\hat{c}(v), which measures how far current consumption deviates from its break-even level.
After substituting V = hW^{1-\gamma}/(1-\gamma) and V_W = hW^{-\gamma} into the FOC and collecting powers of W, one obtains, for \psi \neq 1, \frac{c_t}{W_t} = \delta^\psi h(\mathbf{z}_t)^{-\psi/\theta}. The portfolio first-order condition (differentiating the HJB with respect to \pmb{\alpha}) gives the standard Merton decomposition: \pmb{\alpha}^* = \underbrace{\frac{1}{\gamma}(\pmb{\sigma}(\mathbf{z})\pmb{\sigma}(\mathbf{z})')^{-1}(\pmb{\mu}(\mathbf{z})-r(\mathbf{z})\pmb{\iota})}_{\text{myopic demand}} + \underbrace{\frac{1}{\gamma}(\pmb{\sigma}(\mathbf{z})\pmb{\sigma}(\mathbf{z})')^{-1}\pmb{\sigma}(\mathbf{z})(\pmb{\sigma}^z)'\frac{\nabla_z h(\mathbf{z})}{h(\mathbf{z})}}_{\text{hedging demand}}. \tag{5} The myopic component maximizes the instantaneous Sharpe ratio; the hedging component tilts the portfolio toward assets that co-vary with changes in investment opportunities. The two preference parameters enter only through h(\mathbf{z}): any two specifications sharing the same h yield identical portfolios. For pricing, the key point is more subtle than under additive utility. In stochastic differential utility, the intertemporal marginal rate of substitution is not simply e^{-\delta t}V_W. The correct marginal-utility process is \Lambda_t = \Lambda_0 \exp\!\left(\int_0^t f_v(c_s,V_s)\,ds\right) f_c(c_t,V_t), and using the consumption FOC this can be written equivalently as \Lambda_t = \Lambda_0 \exp\!\left(\int_0^t f_v(c_s,V_s)\,ds\right) \frac{V_W(W_t,\mathbf{z}_t)}{V_W(W_0,\mathbf{z}_0)}. \tag{6} Only in the additive case, where f_v(c,v) = -\delta, does this reduce to the familiar formula \Lambda_t = \Lambda_0 e^{-\delta t}\frac{V_W(W_t,\mathbf{z}_t)}{V_W(W_0,\mathbf{z}_0)}. For Epstein-Zin preferences, f_v(c_t,V_t) is state dependent, so recursive utility contributes an additional discounting term through continuation-value sensitivity.
The Stochastic Discount Factor
General Form
Equation (6) is the correct starting point for asset pricing under SDU. Differentiating \Lambda_t = \Lambda_0 e^{\int_0^t f_v\,ds} V_W(W_t,\mathbf{z}_t)/V_W(W_0,\mathbf{z}_0) gives \frac{d\Lambda_t}{\Lambda_t} = f_v(c_t,V_t)\,dt + \frac{dV_W(W_t,\mathbf{z}_t)}{V_W(W_t,\mathbf{z}_t)}. With the homothetic guess V_W = h(\mathbf{z})W^{-\gamma}, Ito’s lemma extracts the stochastic component of dV_W/V_W: \frac{dV_W}{V_W}\bigg|_{d\mathbf{B}} = -\gamma\pmb{\sigma}^W_t \cdot d\mathbf{B}_t + \frac{(\nabla_z h(\mathbf{z}_t))'\pmb{\sigma}^z(\mathbf{z}_t)}{h(\mathbf{z}_t)} \cdot d\mathbf{B}_t, where \pmb{\sigma}^W_t = \pmb{\sigma}'\pmb{\alpha}^*_t collects the wealth-return volatilities. No-arbitrage pins the drift of \Lambda_t to -r_t, so the SDF satisfies \frac{d\Lambda_t}{\Lambda_t} = -r_t\,dt - \underbrace{\gamma\pmb{\sigma}^W_t}_{\text{wealth risk}} \cdot d\mathbf{B}_t + \underbrace{\frac{(\nabla_z h)'\pmb{\sigma}^z}{h}}_{\text{opportunity risk}} \cdot d\mathbf{B}_t. \tag{7} The first diffusion term prices wealth-return shocks exactly as under power utility. The second prices news about future investment opportunities through \nabla_z h/h, the logarithmic sensitivity of the value-function coefficient to state-variable shocks. The sign of this term is model-dependent: it depends on how the state vector is defined and on how shocks to \mathbf{z} affect the continuation-value coefficient h. In many applications better future opportunities raise h when \gamma < 1 and lower it when \gamma > 1 (because V is then negative), but that interpretation requires additional structure beyond the general SDU derivation.
Power Utility as a Special Case
Under power utility (\psi = 1/\gamma, so \theta = 1), the aggregator becomes additive and f_v(c,v) = -\delta. The general SDU pricing kernel therefore collapses to the familiar form \Lambda_t = \Lambda_0 e^{-\delta t}\frac{V_W(W_t,\mathbf{z}_t)}{V_W(W_0,\mathbf{z}_0)}. If, in addition, the investment opportunity set is constant so that h is constant, then V_W = hW_t^{-\gamma} and the SDF satisfies \frac{d\Lambda_t}{\Lambda_t} = -r\,dt - \gamma\,\pmb{\sigma}^W \cdot d\mathbf{B}, which is exactly the power-utility SDF from the continuous-time SDF notebook.
The Epstein-Zin SDF
Under Epstein-Zin preferences with \psi \neq 1/\gamma (so \theta \neq 1), the opportunity-risk term (\nabla_z h)'\pmb{\sigma}^z/h in (7) does not vanish whenever the investment opportunity set is stochastic. From (4), the state-dependent discount rate along the optimal path satisfies f_v(c_t,V_t) = \delta\!\left[(\theta-1)\!\left(\frac{c_t}{\hat{c}(V_t)}\right)^{1-1/\psi} - \theta\right], which is constant only when h is constant.
Under a constant opportunity set (h constant), \nabla_z h = 0 and f_v is constant, so (7) reduces to d\Lambda/\Lambda = -r\,dt - \gamma\pmb{\sigma}^W \cdot d\mathbf{B} regardless of \psi. In this case EZ and power utility produce the same equity risk premium; the two parameters differ only in their implications for the risk-free rate. The full separation of \gamma and \psi in asset prices requires a time-varying opportunity set so that \nabla_z h \neq 0. That is the continuous-time channel through which recursive utility makes news about future opportunities independently priced — the exact counterpart of the wealth-return factor 1/R^w_{t+1} in the discrete-time SDF.
The companion notebook Affine Recursive Utility in Continuous Time implements this in a one-factor Gaussian model, where h takes an exponential-affine form and (7) yields closed-form risk prices.