On existence and uniqueness properties for solutions of stochastic fixed point equations

Arnulf Jentzen; Christian Beck; Lukas Gonon; Martin Hutzenthaler

On existence and uniqueness properties for solutions of stochastic fixed point equations

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Pith's one-line read The paper proves that stochastic fixed point equations of Feynman–Kac type have unique at-most-polynomially growing continuous solutions under Lipschitz and Lyapunov-type hypotheses, covering semilinear Kolmogorov PDEs even without…

desk verdict Solid, self-contained existence and uniqueness results for stochastic fixed-point equations; the abstract weighted-space theorem is the real contribution, and the SDE application checks out despite a minor constant issue in Lemma 3.7. read the letter →

arxiv 1908.03382 v1 pith:3VK5TU23 submitted 2019-08-09 math.PR math.FA

Christian Beck , Lukas Gonon , Martin Hutzenthaler , Arnulf Jentzen This is my paper · ORCID

classification math.PRmath.FA MSC 60H1060H3035K5847H10

keywords stochasticfixedpointequationsFeynman-KacformulasemilinearKolmogorovPDEBanachtheoremLyapunovfunctiondifferentialexistenceanduniquenesspolynomialgrowth

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes existence and uniqueness for stochastic fixed point equations (SFPEs) of the form $u(t,x)=\mathbb{E}[g(X_T^{t,x})+\int_t^T f(s,X_s^{t,x},u(s,X_s^{t,x}))\,ds]$, where $X^{t,x}$ solves an SDE. The main theorem proves this under a uniform Lipschitz condition on $f$ in its last variable, local Lipschitz continuity of the SDE coefficients, polynomial growth of $f$ and $g$, and a coercivity-type bound on the drift and diffusion. The significance is that the result applies to semilinear Kolmogorov PDEs with Lipschitz nonlinearities even when the PDE has no classical solution, so the SFPE representation itself supplies the object of study. The proof is a Banach fixed point argument on a weighted space of continuous functions, with the growth controlled by a Lyapunov weight $V$.

What carries the argument

The central object is the weighted Banach space of continuous functions $u$ on $[0,T]\times O$ with $\lim_{r\to\infty}\sup_{[0,T]\times(O\setminus O_r)}|u|/V=0$, equipped with the exponentially weighted norm $\|u\|_\lambda=\sup e^{\lambda t}|u(t,x)|/V(t,x)$. On this space the SFPE defines a fixed point map $\Phi$; Lemma 2.8 shows $\Phi$ is a contraction with constant $L/\lambda$ using the supermartingale property of $V$ along the SDE, while Corollary 2.7 and Lemma 3.7 ensure that $\Phi$ maps the space into itself and preserves continuity. Banach's fixed point theorem then yields the unique fixed point.

What would settle it

Run the Banach fixed point iteration in an explicit admissible example where the unique solution is known, for instance $d=m=1$, $\mu(x)=x$, $\sigma(x)=1$, $f(t,x,v)=L v$, $g(x)=0$ (the only solution is $u=0$). If two different starting functions produce two different weighted-norm limits, the contraction estimate fails; more decisively, exhibiting any two distinct at-most-polynomial continuous solutions for any coefficient set satisfying the theorem's hypotheses would refute Theorem 1.1.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 1.1 and in greater generality as Theorem 3.8 and Corollary 3.10, is that under the stated hypotheses there exists a unique continuous function $u:[0,T]\times\mathbb{R}^d\to\mathbb{R}$, growing at most polynomially in the spatial variable, satisfying the stochastic fixed point equation (2) for every $(t,x)$. The abstract engine, Theorem 2.9, works on any open domain $O$: given a Lyapunov function $V$ that is a supersolution of the associated Kolmogorov operator, grows to infinity at the boundary, and dominates the terminal and initial nonlinearities outside large balls, the fixed point map $\Phi(u)(t,x)=\mathbb{E}[g(X_T^{t,x})+\int_t^T f(s,X_s^{t,x},u(s,X_s^{t,x}))\,ds]$ is a contraction in the weighted norm $\|u\|_\lambda=\sup_{t,x}e^{\lambda t}|u(t,x)|/V(t,x)$ for $\lambda\ge 2L$. The SDE applications verify the Lyapunov, integrability, and stochastic-continuity hypotheses from the coercivity condition $\max\{\langle x,\mu(x)\rangle,\|\sigma(x)\|^2\}\le L(1+\|x\|^2)$ and polynomial growth, so uniqueness holds within the class of at most polynomially growing continuous functions.

Load-bearing premise

The load-bearing premise is that a positive Lyapunov weight $V$ exists which the stochastic flow makes decrease on average, which blows up at infinity, and which dominates both the terminal data and the size of $f(\cdot,\cdot,0)$; without such a $V$ the fixed point map has no complete weighted space in which to contract.

Editorial extensions

If this is right

Wherever the Lyapunov hypotheses can be verified, the SFPE has a unique solution directly, without first solving the semilinear Kolmogorov PDE.
For SDEs satisfying the coercivity bound with locally Lipschitz coefficients, the unique solution grows at most polynomially, covering coefficient regimes in which the PDE has no classical solution.
The contraction constant $L/\lambda$ shows that Picard iteration converges in the weighted norm once $\lambda\ge 2L$.
Uniqueness holds within the class of at most polynomially growing continuous functions; no second solution of that growth class can exist.
The abstract theorem applies on general open domains with an appropriate boundary-blowing Lyapunov function, not only on $\mathbb{R}^d$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not in the paper: the contraction estimate could yield explicit convergence rates for full-history recursive multilevel Picard schemes in coefficient regimes where previously only existence was known.
Not in the paper: the Lyapunov-supersolution condition is the flexible input, so one could replace polynomial weights by other weight families to obtain uniqueness in different growth classes.
Not in the paper: because the argument does not rely on PDE regularity, it suggests that the fixed point equation, rather than the PDE, is the more fundamental object for semilinear Kolmogorov equations with rough coefficients.
Not in the paper: a testable extension would be to weaken the uniform Lipschitz condition on $f$ to a local Lipschitz condition with a suitable one-sided bound, though the proof would need a refined contraction estimate.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper deserves a serious referee. The main result, Theorem 2.9, is a clean Banach fixed-point argument in a weighted continuous-function space on an open domain, with a general Lyapunov weight V. The authors spell out every step: integrability, continuity, contractivity, and the SDE verification. The payoff is that SFPE solutions exist without requiring classical solutions of the associated semilinear Kolmogorov PDE, which matters for work on multilevel Picard schemes and Feynman-Kac representations.

What is actually new: the abstract theorem with a general V and open domain is more general than the cited Pazy/Segal/Weissler results and than the authors' own earlier [11, Corollary 3.11]. The polynomial-growth corollary is closer to prior work, and the paper does not quantify exactly how much of Corollary 3.10 was already implicit in [11]. That is a real but minor presentation gap, not a mathematical one.

I checked the chain Theorem 1.1 through Corollary 3.10, Corollary 3.9, Theorem 3.8, and back to Theorem 2.9. No load-bearing gap. The contraction argument is complete: the weighted space is Banach, the map sends the space into itself, and for large enough lambda it is a contraction. The SDE application verifies the Lyapunov supermartingale inequality, the stochastic continuity hypothesis, and provides polynomial weights via Lemma 3.3. The uniqueness step in Corollary 3.10 is handled carefully: any polynomial-growth solution is placed in a heavier weight, then the abstract uniqueness applies.

The only soft spot I see is in Lemma 3.7. In the approximation argument, the constant in (151) may be off by a factor from the epsilon/3 split. But the final infimum over k is still zero, so the conclusion survives. That is a minor blemish, the kind a referee should flag but not reject over.

Citation pattern is honest. The self-citations point to earlier MLP work that this paper extends, and the comparison with Pazy, Segal, and Weissler is legitimate. No fitted parameters, no circularity, no invented entities.

Who gets value from this: researchers working on MLP approximations for high-dimensional semilinear PDEs, and anyone who needs existence and uniqueness for SFPEs under weak regularity. I would send this to peer review and, given the careful proofs, expect acceptance after minor revision. I would also bring it to a reading group, though the heavy detail makes it a slower read.

Referee Report

0 major / 3 minor

Summary. This paper studies stochastic fixed point equations (SFPEs) of the form u(t,x)=E[g(X^{t,x}_T)+∫_t^T f(s,X^{t,x}_s,u(s,X^{t,x}_s))ds] for diffusion processes X. The main abstract result (Theorem 2.9) proves existence and uniqueness of a solution u in a weighted space of continuous functions under a Lyapunov supermartingale condition, a stochastic continuity condition on X, and a uniform Lipschitz condition on f in the solution variable. The proof is a contraction argument on a Banach space with an exponentially weighted norm. The paper then specializes this to SDEs with locally Lipschitz coefficients: Theorem 3.8 verifies the abstract hypotheses using a Lyapunov function satisfying an elliptic-type inequality, and Corollary 3.10 establishes existence and uniqueness of at most polynomially growing continuous solutions under the coercivity condition max{<x,mu(x)>, ||sigma(x)||^2} <= L(1+||x||^2). The advertised consequence is that SFPEs can have unique solutions even when the associated semilinear Kolmogorov PDE lacks a classical solution.

Significance. If correct, the result is a clean and useful existence/uniqueness theorem for SFPEs that does not rely on classical PDE theory. The proof is self-contained, detailed, and based on standard tools (Banach fixed point theorem, Ito's formula, Fatou's lemma, Gronwall's inequality, and known SDE existence theory). The paper carefully develops a general weighted-space framework (Theorem 2.9) and verifies its hypotheses for diffusions with locally Lipschitz coefficients and polynomial Lyapunov weights. The application to semilinear Kolmogorov PDEs without classical solutions is a genuine improvement over earlier results and is relevant to full-history recursive multilevel Picard (MLP) methods. The assumptions are explicit and checkable, and the paper includes full proofs of all auxiliary lemmas.

minor comments (3)

[§3.3, Lemma 3.6] The statement uses the symbol t both for the initial time and for a later starting time (e.g., 'for all t ∈ [0,T], t ∈ [t,T], s ∈ [t,T]'), which is confusing; using a different symbol, such as t̄ or t', for the second time parameter would improve readability.
[§2.5, proof of Theorem 2.9] The assertion that W2 is a closed subset of (W1, ||·||_W1) is attributed to Lemma 2.2, but Lemma 2.2 does not explicitly state this closedness; a one-line direct argument (or a reference to the standard fact that uniform limits of functions vanishing at infinity vanish at infinity) would make the proof self-contained.
[§3.4, Corollary 3.10] In the uniqueness part, the parameter q is introduced for the polynomial growth of the candidate solution v, and then the weight V_{max{2q,2p}} is used; the relation between q and the exponent of the weight could be explained more explicitly to avoid the impression that the choice of exponent is arbitrary.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the existence and uniqueness proof is a self-contained contraction argument; self-citations are contextual only.

full rationale

The paper's central claim, Theorem 2.9, is proved by Banach's fixed point theorem on a weighted space. The assumptions—the stochastic supermartingale inequality for the weight V, the relative growth conditions, the local Lipschitz and polynomial-growth conditions, and stochastic continuity—are stated independently of the target solution u. Corollary 2.7 and Lemma 2.8 establish that the fixed-point map is well-defined and a contraction, and the fixed point then satisfies the SFPE by construction. No parameter is fitted to data, and no 'prediction' is renamed input. The later SDE results verify these hypotheses: Lemma 3.1 supplies the Lyapunov inequality from the generator condition, Lemma 3.7 supplies stochastic continuity, and Lemma 3.3 constructs polynomial Lyapunov weights from the coercivity condition. The uniqueness argument for Corollary 3.10 correctly handles arbitrary polynomial-growth solutions by choosing a heavier polynomial weight and applying the already-proved uniqueness statement. References to the authors' earlier MLP papers appear only as background context, not as load-bearing justification for the existence or uniqueness theorem. The cited external results (e.g., Karatzas & Shreve, Gyöngy & Krylov, Liu & Röckner) are standard SDE existence/regularity facts whose assumptions do not include the target SFPE solution. There is no self-definitional step, no fitted-input-called-prediction step, no uniqueness imported from the authors' prior work, and no ansatz smuggled in via citation. The derivation chain is self-contained, so the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters and no invented entities. The central claim rests on standard mathematical tools and explicit structural assumptions (Lyapunov supersolution, stochastic continuity, Lipschitz nonlinearity), all stated as hypotheses rather than derived.

assumptions (7)

standard math Banach fixed point theorem
Used in the proof of Theorem 2.9 to obtain existence and uniqueness of the fixed point of the contraction mapping Phi.
standard math Ito formula and Ito isometry
Used in Lemma 3.1 to prove the Lyapunov inequality E[V(tau,X_tau)] <= E[V(0,X_0)] and in Lemma 3.5 and Lemma 3.6 estimates.
standard math Strong existence and pathwise uniqueness for SDEs with Lipschitz coefficients (Karatzas-Shreve Theorems 5.2.5 and 5.2.9)
Invoked in Lemmas 3.4, 3.6, and 3.7 to obtain the solution processes X^{t,x} and to compare solutions of truncated SDEs.
standard math Urysohn lemma for the existence of smooth cutoffs
Used in Lemma 2.3 and Lemma 3.7 to construct compactly supported approximations of continuous functions.
standard math Fatou lemma, Fubini theorem, Gronwall inequality, and Vitali convergence theorem
Used throughout Sections 2 and 3 for expectation estimates, continuity arguments, and convergence of function sequences.
domain assumption Existence of a C^{1,2} Lyapunov supersolution V satisfying inequality (153) and the boundary growth conditions of Theorem 3.8
The entire weighted-space contraction argument depends on such a V; in Corollary 3.10 it is constructed from the coercivity condition and polynomial growth.
domain assumption Stochastic continuity in probability of the solution family X^{t,x} in the initial time and position
This is a hypothesis of Theorem 2.9 and is verified for SDEs under local Lipschitz and Lyapunov conditions in Lemma 3.7.

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Pith. "Pith review of On existence and uniqueness properties for solutions of stochastic fixed point equations." pith.science (2026). https://pith.science/paper/3VK5TU23

@misc{pith2026190803382,
  author       = {Pith},
  title        = {Pith review of: On existence and uniqueness properties for solutions of stochastic fixed point equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3VK5TU23}},
  note         = {Machine review of arXiv:1908.03382}
}

read the original abstract

The Feynman-Kac formula implies that every suitable classical solution of a semilinear Kolmogorov partial differential equation (PDE) is also a solution of a certain stochastic fixed point equation (SFPE). In this article we study such and related SFPEs. In particular, the main result of this work proves existence of unique solutions of certain SFPEs in a general setting. As an application of this main result we establish the existence of unique solutions of SFPEs associated with semilinear Kolmogorov PDEs with Lipschitz continuous nonlinearities even in the case where the associated semilinear Kolmogorov PDE does not possess a classical solution.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

Hutzenthaler, M., Jentzen, A., Kruse, T., Nguyen, T. A., and von Wurstem- berger, P. Overcoming the curse of dimensionality in the numerical app roximation of semilinear parabolic partial diﬀerential equations. arXiv:1807.01212 (2018), 30 pages

work page Pith review arXiv 2018

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Beck, C., Hornung, F., Hutzenthaler, M., Jentzen, A., and Kr use, T. Over- coming the curse of dimensionality in the numerical approxi mation of Allen–Cahn partial diﬀerential equations via truncated full-history recursi ve multilevel Picard approximations. arXiv:1907:06729 (2019), 31 pages

work page 2019

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Partial diﬀerential equation representations of derivati ves with bilateral counterparty risk and funding costs

Burgard, C., and Kjaer, M. Partial diﬀerential equation representations of derivati ves with bilateral counterparty risk and funding costs. The Journal of Credit Risk 7 , 3 (2011), 1–19

work page 2011

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Counterparty risk and funding: the four wings of the TV A

Crépey, S., Gerboud, R., Grbac, Z., and Ngor, N. Counterparty risk and funding: the four wings of the TV A. Int. J. Theor. Appl. Finance 16 , 2 (2013), 1350006, 31

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Duffie, D., Schroder, M., and Skiadas, C. Recursive valuation of defaultable securities and the timing of resolution of uncertainty. Ann. Appl. Probab. 6 , 4 (1996), 1075–1090

work page 1996

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Multilevel Picard iterations for solving smooth semilinear parabolic heat equations

E, W., Hutzenthaler, M., Jentzen, A., and Kruse, T. Multilevel Picard iterations for solving smooth semilinear parabolic heat equations. arXiv:1607.03295 (2016), 19 pages. 32

work page Pith review arXiv 2016

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E, W., Hutzenthaler, M., Jentzen, A., and Kruse, T. On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial diﬀerential equations and high-dimensional nonlinear backward stochastic diﬀerent ial equations. J. Sci. Comput. 79 , 3 (2019), 1534–1571

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A proof that artiﬁcial neural networks overcome the curse of dimensiona lity in the numerical approximation of Black-Scholes partial diﬀerential equations

Grohs, P., Hornung, F., Jentzen, A., and Von Wurstemberger, P. A proof that artiﬁcial neural networks overcome the curse of dimensiona lity in the numerical approximation of Black-Scholes partial diﬀerential equations. arXiv:1809.02362 (2018), 124 pages

arXiv 2018