Uniqueness classification and well-posedness of the infinite-horizon stochastic differential utility PDE in the superlinear regime (mu < -1) and under degenerate or unbounded coefficients

Duffie and Lions (1992) reduced the Markovian stochastic differential utility (SDU) process to nonlinear partial differential equations and proved existence and uniqueness for the finite-horizon (parabolic) problem for general variance multipliers A <= 0 and for the infinite-horizon (elliptic) problem in the Kreps-Porteus case A(v) = mu/v with -1 <= mu < 0. For mu < -1 they obtained only extremal solutions, with uniqueness guaranteed solely by the sufficient Conditions E, F, G and the spectral condition lambda_1(D) < gamma, under uniform ellipticity and bounded Lipschitz coefficients. However, no necessary-and-sufficient classification of uniqueness for the elliptic SDU PDE in terms of the diffusion operator and the running reward exists, and the degenerate/unbounded-coefficient elliptic case remains untreated.

Background

Stochastic differential utility (SDU), introduced in continuous time by Duffie and Epstein (1992), models intertemporal preferences by a utility process $V=(V_t)$ defined by a backward stochastic integral equation: current utility aggregates current consumption with future utility through an 'aggregator', and a 'variance multiplier' $A(v)$ applied to the rate of change of the quadratic variation $[V]$ (the pathwise accumulated squared randomness of a semimartingale) separates risk attitudes from intertemporal substitution. This generalizes time-additive expected utility in the direction of the recursive preferences of Kreps-Porteus and Epstein-Zin, in which relative risk aversion and the elasticity of intertemporal substitution (EIS) are independent parameters; it is the workhorse preference class of modern asset pricing, most prominently the long-run-risk models of macro-finance. When the consumption-relevant state is a Markov diffusion $dX_t=b(X_t)\,dt+\sigma(X_t)\,dB_t$ on $\mathbb{R}^n$ with $a=\tfrac12\sigma\sigma^{\mathrm T}$, the ansatz $V_t=J(X_t)$ converts the SDU definition, via Itô's formula, into a nonlinear second-order PDE for $J$: on the infinite horizon, $\mathscr{D}J(x)-J(x)+A(J(x))\sum_{i,j}a_{ij}(x)\,\partial_i J(x)\,\partial_j J(x)+u(x)=0\quad\text{in }\mathbb{R}^n,$ where $\mathscr{D}=\sum_{i,j}a_{ij}\partial_{ij}+\sum_i b_i\partial_i$ is the diffusion generator, $u\ge 0$ is the running reward (utility of current consumption), and the $-J$ term encodes the rate of time preference; the finite-horizon version is a parabolic equation with terminal condition $J(\cdot,T)=g$. The term $A(J)|\nabla J|_a^2$ (abbreviating the quadratic form $\sum_{i,j}a_{ij}\,\partial_iJ\,\partial_jJ$ above) makes the equation quasilinear, and for the economically important homothetic specifications — those invariant under proportional rescaling of consumption and utility — $A$ is singular at $J=0$.

Duffie and Lions (1992) proved existence and uniqueness of positive solutions for the parabolic problem for general $A\le 0$ (under growth Conditions A and B on the data), and for the elliptic problem in the Kreps-Porteus family $A(v)=\mu/v$, $\mu\le 0$: for $\mu=-1$ via the logarithmic change of unknown $w=\log J$, and for $-1<\mu<0$ via a power transform, in both of which the transformed nonlinearity is convex and nonincreasing so that comparison and maximum-principle arguments yield uniqueness. For $\mu<-1$ — setting $\gamma=|\mu+1|>0$ and $z=J^{-\gamma}$, the equation becomes the semilinear elliptic equation $\mathscr{D}z+\gamma z=\gamma z^{1+1/\gamma}u\quad\text{in }\mathbb{R}^n,\qquad z>0,$ with a superlinear, non-monotone structure — they obtained a minimal solution for every strictly positive $u$, and uniqueness only under additional sufficient hypotheses: Condition E ($u$ bounded above and slowly oscillating — a standard local bounded-oscillation condition — together with the spectral bound $\lambda_1(\mathscr{D})<\gamma$, where $\lambda_1(\mathscr{D})$ is the generalized principal eigenvalue of $-\mathscr{D}$, i.e., the threshold below which the maximum principle holds on the whole space), Condition F ($u$ growing superexponentially at infinity), and Condition G ($\mathscr{D}$ self-adjoint in divergence form and $u$ bounded below by a power of $1+|x|$ whose exponent satisfies dimension-dependent inequalities). All results assume $a$ uniformly elliptic and $b,\sigma$ bounded and Lipschitz, hypotheses the authors state were imposed for convenience. The paper also verifies, under sufficient integrability of $J(X)$, $u(X)$ and $A(J)|\nabla J|^2$ along the diffusion, that the PDE solution realizes the SDU stochastic integral identities and yields state-price densities (the positive martingale densities that discount future payoffs to present prices in no-arbitrage pricing), and it defers extensions to Hamilton-Jacobi-Bellman (HJB) control equations and to viscosity solutions (the standard weak solution concept for nonlinear PDEs, in which the equation is tested against smooth functions at local extrema) for degenerate or fully nonlinear operators. The superlinear elliptic regime is economically important: the Epstein-Zin specifications treated as empirically central by the modern literature — those with relative risk aversion and EIS both above one — have superlinear-growth aggregators in the backward stochastic differential equation (BSDE; the probabilistic counterpart of a semilinear PDE, in which an adapted solution and its martingale integrand are determined by a terminal condition and a driver) formalization emphasized by Xing (2017) and Aurand and Huang (2023). The PDE-side multiplier $\mu$ and the Epstein-Zin parameters are related by transformations of the utility index and are not identified here; the two regimes are treated separately, and component (1) below concerns the PDE regime $\mu<-1$. For instance, Shigeta (2026) shows that an order-equivalent transformation of the utility index removes the pathological behaviour of infinite-horizon Epstein-Zin utility when risk aversion and EIS both exceed one. The uniqueness question is delicate in both parametrizations: in the separate Epstein-Zin regime $\vartheta>1$ of Herdegen, Hobson and Jerome (with $\vartheta=(1-R)/(1-S)$, where $R$ is the relative risk aversion and $S$ the elasticity of intertemporal complementarity, the reciprocal of EIS, so that the risk-aversion-and-EIS-above-one case is the disjoint region $\vartheta<0$) they prove that every nonzero consumption stream admits an infinite family of utility processes.

Problem Statement

Let $\mathscr{D}=\sum_{i,j}a_{ij}(x)\partial_{ij}+\sum_i b_i(x)\partial_i$ be the generator of a Markov diffusion on $\mathbb{R}^n$ (in the source setting: $a=\tfrac12\sigma\sigma^{\mathrm T}$ uniformly elliptic, $b,\sigma$ bounded and Lipschitz), let $u>0$ be a running reward, and consider the infinite-horizon stochastic-differential-utility PDE of Duffie and Lions (1992) with Kreps-Porteus variance multiplier $A(v)=\mu/v$, $\mu<-1$: $\mathscr{D}J(x)-J(x)+\frac{\mu}{J(x)}\sum_{i,j}a_{ij}(x)\,\partial_i J(x)\,\partial_j J(x)+u(x)=0\quad\text{in }\mathbb{R}^n,$ equivalently, for $\gamma=|\mu+1|>0$ and $z=J^{-\gamma}$, $\mathscr{D}z+\gamma z=\gamma z^{1+1/\gamma}u\quad\text{in }\mathbb{R}^n,\qquad z>0.$ Throughout, a solution means a positive function $J$ in the source paper's local Sobolev class $W^{2,p}_{\mathrm{loc}}(\mathbb{R}^n)$ (which is classical wherever the coefficients permit), and uniqueness refers to this class; in component (2), where the standing ellipticity and regularity hypotheses are dropped, existence, uniqueness and comparison are understood in the solution class appropriate to the relaxed hypotheses (for instance the viscosity class where a viscosity theory is used).

(1) Uniqueness classification. Determine necessary and sufficient conditions on the pair $(\mathscr{D},u)$ under which this elliptic equation has a unique positive solution — a classification going beyond the sufficient Conditions E (bounded, slowly oscillating $u$ with the spectral bound $\lambda_1(\mathscr{D})<\gamma$), F (superexponentially growing $u$) and G (self-adjoint divergence-form $\mathscr{D}$ with polynomially bounded-below $u$) of the source paper; in regimes where uniqueness fails, characterize the family of positive solutions (for example its extremal elements) and how it depends on the data.

(2) Coefficient generality. Establish existence, uniqueness and comparison for the SDU PDEs of the source paper — the elliptic equation above and its parabolic (finite-horizon) counterpart for general variance multipliers $A\le 0$ — once the standing hypotheses of the source paper are dropped: degenerate diffusion matrices $a$, or unbounded or non-Lipschitz $b,\sigma$.

The classification problem is to determine necessary and sufficient conditions on $(\mathscr{D},u)$ under which the elliptic equation has a unique positive solution, and, where uniqueness fails, to characterize the solution family; the coefficient-generality problem is to establish existence, uniqueness and comparison under the relaxed hypotheses. Jointly they support one objective: a well-posedness theory for the SDU PDEs, in the superlinear regime and under general Markovian data, at the standard of classical semilinear elliptic/parabolic theory.

References

1. Darrel Duffie and Pierre-Louis Lions, PDE solutions of stochastic differential utility, Journal of Mathematical Economics 21(6):577-606, 1992. DOI 10.1016/0304-4068(92)90028-6, https://doi.org/10.1016/0304-4068(92)90028-6
2. Darrell Duffie and Larry G. Epstein, Stochastic differential utility, Econometrica 60(2):353-394, 1992. DOI 10.2307/2951600, https://doi.org/10.2307/2951600
3. Mark Schroder and Costis Skiadas, Optimal consumption and portfolio selection with stochastic differential utility, Journal of Economic Theory 89(1-2):68-126, 1999. DOI 10.1006/jeth.1999.2558, https://doi.org/10.1006/jeth.1999.2558
4. Holger Kraft, Thomas Seiferling, and Frank Thomas Seifried, Optimal consumption and investment with Epstein-Zin recursive utility, Finance and Stochastics 21(1):187-226, 2017. DOI 10.1007/s00780-016-0316-0, https://doi.org/10.1007/s00780-016-0316-0
5. Hao Xing, Consumption-investment optimization with Epstein-Zin utility in incomplete markets, Finance and Stochastics 21(1):227-262, 2017. DOI 10.1007/s00780-016-0297-z, https://doi.org/10.1007/s00780-016-0297-z (arXiv:1501.04747)
6. Thomas Seiferling and Frank Thomas Seifried, Epstein-Zin stochastic differential utility: Existence, uniqueness, concavity, and utility gradients, SSRN working paper no. 2625800, 2015 (revised 2016). DOI 10.2139/ssrn.2625800, https://doi.org/10.2139/ssrn.2625800
7. Joshua Aurand and Yu-Jui Huang, Epstein-Zin utility maximization on a random horizon, Mathematical Finance 33(4):1370-1411, 2023. DOI 10.1111/mafi.12404, https://doi.org/10.1111/mafi.12404 (arXiv:1903.08782)
8. Khaled Bahlali and Ludovic Tangpi, BSDEs driven by |z|^2/y and applications to PDEs and decision theory, arXiv preprint, 2018 (v3 2021). arXiv:1810.05664, https://arxiv.org/abs/1810.05664
9. Dariusz Zawisza, On the parabolic equation for portfolio problems, Banach Center Publications 122:287-302, 2020. DOI 10.4064/bc122-16, https://doi.org/10.4064/bc122-16 (arXiv:2003.13317)
10. Martin Herdegen, David Hobson, and Joseph Jerome, The infinite-horizon investment-consumption problem for Epstein-Zin stochastic differential utility. I: Foundations, Finance and Stochastics 27(1):127-158, 2023. DOI 10.1007/s00780-022-00495-6, https://doi.org/10.1007/s00780-022-00495-6 (arXiv:2107.06593)
11. Martin Herdegen, David Hobson, and Joseph Jerome, The infinite-horizon investment-consumption problem for Epstein-Zin stochastic differential utility. II: Existence, uniqueness and verification for vartheta in (0,1), Finance and Stochastics 27(1):159-188, 2023. DOI 10.1007/s00780-022-00496-5, https://doi.org/10.1007/s00780-022-00496-5
12. Martin Herdegen, David Hobson, and Joseph Jerome, Proper solutions for Epstein-Zin stochastic differential utility, Finance and Stochastics 29(3):885-932, 2025. DOI 10.1007/s00780-025-00569-1, https://doi.org/10.1007/s00780-025-00569-1 (preprint arXiv:2112.06708, 2021, https://arxiv.org/abs/2112.06708)
13. Yu Chen, Thomas F. Cosimano, Alex A. Himonas, and Peter W. Kelly, An analytic approach for stochastic differential utility for endowment and production economies, Computational Economics 44(4):397-443, 2014 (online first 2013). DOI 10.1007/s10614-013-9397-4, https://doi.org/10.1007/s10614-013-9397-4
14. Martin Herdegen, David Hobson, and Joseph Jerome, When is recursive utility well-founded?, SSRN working paper no. 4217738, 2022. DOI 10.2139/ssrn.4217738, https://doi.org/10.2139/ssrn.4217738
15. Yuki Shigeta, An economic interpretation and mathematical analysis of Epstein-Zin stochastic differential utility for an infinite horizon when theta < 0, Finance and Stochastics 30(3):765-819, 2026. DOI 10.1007/s00780-026-00594-8, https://doi.org/10.1007/s00780-026-00594-8 (SSRN preprint no. 4701952, 2024)