A General Model for Continuous Time Principal-Agent Problem Under Hidden Action

Jaeyoung Sung; Jianfeng Zhang; Zimu Zhu

REVIEW 3 major objections 5 minor 23 references

This paper claims that a new sufficient condition solves the agent's problem in a general continuous-time principal-agent model with hidden action, making the resulting coupled forward-backward SDE well-posed without a separate verification

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2026-08-01 18:42 UTC pith:THC6NELN

load-bearing objection A genuinely broader continuous-time contracting framework with diffusion in continuous payments, a new sufficient condition, and a clean PPS=1 example; the main flagged issue is a notation slip, not a gap. the 3 major comments →

arxiv 2607.17212 v1 pith:THC6NELN submitted 2026-07-19 q-fin.MF

A General Model for Continuous Time Principal-Agent Problem Under Hidden Action

Jaeyoung Sung , Jianfeng Zhang , Zimu Zhu This is my paper

classification q-fin.MF MSC 91B4191B4393E20

keywords principal-agenthidden actioncontinuous-time contractingpay-to-performance sensitivityforward-backward SDEfirst-order approachmoral hazardstochastic maximum principle

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper generalizes continuous-time contracting to a model where the agent privately chooses both effort and consumption, and where pay comes in lump sums, a continuous flow, and a diffusion tied to performance (pay-to-performance sensitivity). Its central contribution is a sufficient concavity condition under which the agent's value function is strictly concave and has a unique optimal consumption, so the first-order approach is fully justified: the coupled FBSDE that characterizes the optimum has a unique solution, and no separate verification argument is needed. This matters because the well-posedness of that FBSDE is generally very hard, and because the paper shows in an explicit example that allowing the continuous payment to have a diffusion component is essential for the existence of an optimal contract. The example also yields a concrete benchmark: the optimal pay-to-performance sensitivity is exactly one, independent of the risk-averse felicity functions.

Core claim

The central claim is Theorem 4.4: under a separable-exponential felicity form and a quantitative smallness condition, the agent's value function J is strictly concave on a bounded interval constructed from the marginal-utility bounds, the agent has a unique optimal consumption, and the coupled FBSDE (4.3) has a unique solution. The proof uses the control problem to establish FBSDE well-posedness, the reverse of the usual order. In the solvable example of Section 6, the principal's optimal value is C0T−R, the optimal continuous payment has pay-to-performance sensitivity equal to one, and restricting to drift-only continuous payment (β=0) leaves the principal's value unchanged but destroys the

What carries the argument

The load-bearing machinery is the coupled forward-backward SDE (4.3) for (m, Y, Z, Y, Z), together with the inverse consumption function I_c(t,y)= (1/λ2(t)) ln(λ1(t)λ2(t)/y). The sufficient condition Assumption 4.1—in particular the separable-exponential felicity fA = f1 + f2 − λ1(t)e^{−λ2(t)c} and the smallness inequality 4LT|γ_T|^2 < λ2 γ_T—makes the agent's value function strictly concave on the interval [K, K̄], which yields both the unique optimal consumption and, by reversing the usual logic, the well-posedness of the FBSDE.

Load-bearing premise

The load-bearing premise is Assumption 4.1(iv), the quantitative smallness inequality 4LT|γ_T|^2 < λ2γ_T, together with the separable-exponential felicity form (4.1); if that inequality or that functional form fails, the proof of strict concavity and uniqueness of the agent's optimal consumption collapses.

What would settle it

Compare the printed Assumption 4.1(iv) with the inequality actually used in the proof of Proposition 4.3: the proof uses λ2 γ_T (the lower bound γ_T in both the exponential cost term and the bound on ∂m gA), while the stated assumption writes |γ_T|^2 in the first factor. A reader could construct parameters satisfying the printed assumption but violating the proof's required inequality and then test numerically whether J admits two distinct optimal consumption controls.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

If the central claim is correct, the first-order approach for this class of problems is fully justified: a solution of the coupled FBSDE directly gives the agent's optimal effort and consumption without a separate verification step.
The paper's well-posedness result for this FBSDE is claimed as new in the FBSDE literature, so it provides a template for proving existence and uniqueness in other contracting problems with consumption control.
In the explicit example, the optimal pay-to-performance sensitivity equals one regardless of the risk-averse felicity functions, giving a sharp benchmark for when full performance linkage is optimal.
The diffusion component of continuous payment is essential for existence: with β=0 the principal's value is unchanged but no optimal contract exists, so stochastic continuous pay serves a role beyond risk sharing.
The model unifies lump-sum and continuous-payment contracting, covering signing bonuses, flow compensation, and severance pay in a single framework.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Editorial inference: The sufficient-condition strategy suggests a general recipe—if one can prove strict concavity of the agent's value on an a priori bounded interval, the first-order FBSDE is well-posed; the quantitative difficulty is isolating the interval and the smallness bound.
Editorial inference: The β=0 versus β=1 result hints that the diffusion component acts as a compactifier of the admissible control set, not as an efficiency gain, which could inform why observed contracts often include performance-linked stochastic pay.
Editorial inference: A testable extension is to relax the separable-exponential felicity to other concave forms and check whether the smallness condition becomes a curvature-growth condition on the Hamiltonian, which would clarify how special the exponential form is.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

A genuinely broader continuous-time contracting framework with diffusion in continuous payments, a new sufficient condition, and a clean PPS=1 example; the main flagged issue is a notation slip, not a gap.

read the letter

Quick take: this is a real step forward for continuous-time principal-agent problems. The paper lets continuous payments have their own diffusion (β dX_t), adds an upfront signing bonus, and proves a sufficient condition that solves the agent's consumption/effort problem without a separate verification step. The key FBSDE well-posedness result (Theorem 4.4) is derived from concavity rather than imported. That is a substantive methodological contribution. The explicit example with PPS=1, and the non-existence when you force β=0, is clean and makes a sharp point: the diffusion term can be necessary for existence of an optimal contract.

The paper is honest and mostly careful. The derivations are detailed, the example checks out, and the limitations are stated plainly. I would send it to a serious referee.

Soft spots, in order of importance. First, Assumption 4.1(iv) is printed with γT where the proof of Proposition 4.3 needs the upper/lower bars tracked carefully. The stress-test note is right: if you read the bars as they are used, the inequality 4LT γ̄_T² < λ2 γ̄_T is what the proof actually requires, and Assumption 4.1(iv) is that. But the equality '=4γ_T²T' in the proof is sloppy when the bounds differ. This is a fixable notation issue, not a load-bearing flaw, but it will confuse referees and should be corrected.

Second, Theorem 6.3's upper-bound step 'one can easily see' hides a Girsanov change of measure and a linear BSDE comparison. It's standard, but for a paper that prides itself on rigor, that step deserves a few lines.

Third, the explicit example uses linear terminal utility gA=m, so the agent's problem is solved by comparison rather than by the new sufficient condition. The example and the Section 4 theorem therefore do not exercise each other. The paper is upfront about this, but a reader should know the two central results live in somewhat separate regimes.

The HJB material in Section 5 is explicitly partial — heavy assumptions, face-lifting, no full DPP. The authors say so and do not overclaim. Citations are appropriate; self-citations are for context and standard results.

Recommendation: send to peer review. It deserves a careful referee, with the expectation that the notation mess and the terse step get fixed before publication.

Referee Report

3 major / 5 minor

Summary. The paper develops a continuous-time principal–agent model with hidden action in which the agent controls both effort and consumption and receives payments through a continuous process with both drift and diffusion components, as well as optional initial and terminal lump sums. The central methodological claim is a new sufficient condition, based on separable-exponential felicity and a quantitative smallness condition, under which the agent's value function is strictly concave, the agent's optimal consumption is unique, and the coupled FBSDE arising from the first-order approach is well-posed, so that no separate verification step is needed. The paper also formulates the principal's problem through a dynamic programming/HJB approach under strong assumptions, and closes with a fully solved example in which the optimal pay-to-performance sensitivity is identically one and restricting to absolutely continuous payments destroys existence of an optimal contract.

Significance. If the technical gaps are repaired, the paper makes a worthwhile contribution to the continuous-time contract theory literature. It unifies lump-sum and continuous-payment models, allows the continuous payment to have a diffusion component (connecting to PPS), and provides a new sufficient condition that avoids the usual verification step in the first-order approach. The explicit example is valuable: it is solved in closed form, demonstrates the role of stochastic continuous payments for existence, and highlights the face-lifting phenomenon. A particular strength is that the main agent-side argument is self-contained: the FBSDE well-posedness is derived from concavity and bound estimates rather than imported from an external existence theorem. The principal-side results are admittedly partial and rely on strong, hard-to-verify assumptions, but the paper is transparent about this and the example is treated rigorously.

major comments (3)

[§4, Theorem 4.4, Step 3] The proof that any FBSDE solution satisfies c∈[K,\bar K] contains a sign/inequality error. With Δc_t=(c_t-K_t)^+ and c_t>K_t, one has e^{-λ2 c_t}<e^{-λ2 K_t}, so the displayed assertion e^{-λ2 c}-e^{-λ2 K}>0 is false; the preceding inequality also has the wrong direction for the stated choice of K. This step is load-bearing for the uniqueness of FBSDE (4.3). The conclusion is nevertheless recoverable: immediately before, the paper proves γ_T≤Y_t≤γT, and since c_t=I_c(t,Y_t) is decreasing in Y_t, c_t automatically lies in [ln(λ1λ2/γ̄_T)/λ2(t), ln(λ1λ2/γ̲_T)/λ2(t)], which is the interval [K,\bar K] of (4.4). Please replace the erroneous argument by this direct bound, or correct the signs and the choices of K and \bar K.
[§4, Assumption 4.1(iv) and Proposition 4.3] The smallness condition is printed as 4LT|γ_T|^2<λ2γT without visible bars. The proof of Proposition 4.3 requires 4LT(γ̄_T)^2<λ2γ̲_T: the positive term carries the square of the upper bound γ̄_T, while the negative term is linear in the lower bound γ̲_T. As printed, the assumption is ambiguous and one natural reading is too weak for the strict-concavity claim. Please write the assumption with explicit \underline γ_T and \bar γ_T, and check that the intermediate equality '=4γ_T^2T' is replaced by an inequality with γ̄_T^2.
[§6.2, Theorem 6.3] The upper-bound argument is too terse. After the inequality \hat Y^P_t0 ≤ C0(T−t0)+∫(Z̃+β)\hat Z^P ds−∫\hat Z^P dB, the sentence 'one can easily see' skips a nontrivial Girsanov step: the drift term does not vanish under the original measure P. One must change to the measure with density exp(∫(Z̃+β)dB−1/2∫|Z̃+β|²ds) and use the BMO property of Z̃ and boundedness of β to justify the bound. Please expand this step; it is standard but not immediate as written.

minor comments (5)

[Throughout] There are numerous typographical inconsistencies: 'consumtion' in the introduction, 'Thoorem' in the proof of Proposition 3.2, and inconsistent rendering of \underline γ_T and \bar γ_T in Assumption 4.1 and (4.4). The notation for the bounds should be made unambiguous in the displayed equations.
[§2, (2.9)] The set A^{1,2}_P is defined twice: once for bounded (α,β) and once for the terminal payment ξ. The second set should presumably be A^{1,3}_P. Also, Assumption 2.1(iv) uses ΦP but the terminal utility is gP; please align notation.
[§5, (5.11)] In the HJB operator L^V, the term −∂_{m^P}V[α+c] should be −∂_{m^P}V[α+c^P]. Please correct the superscript.
[§6, (6.2)–(6.3)] The Hamiltonian \tilde F is introduced as \tilde F(z) and then used as \tilde F(t,\tilde z) with β_t inside. Please make the dependence on t and β_t explicit to avoid confusion.
[§6, opening paragraph] The paper says the example 'may violate some conditions' in earlier assumptions but is still rigorous because it is solved explicitly. This is acceptable, but it would be helpful to state which assumptions are violated and why the explicit verification bypasses them.

Circularity Check

0 steps flagged

No significant circularity found; derivation is self-contained.

full rationale

The paper's central agent-side claim (Theorem 4.4) is obtained from explicit structural assumptions (4.1) and the quantitative smallness condition Assumption 4.1(iv), not from the target conclusion. Concavity of J is proved by differentiating the value function and performing a BSDE energy estimate; uniqueness of the optimal consumption and of the coupled FBSDE solution then follows from strict concavity and the maximum principle, with no inversion of the desired result. The sufficient condition is an input; the FBSDE well-posedness is an output. The principal-side analysis explicitly stops short of a full dynamic programming principle (Remark 5.4) and states that the conditions of Theorem 5.5 are strong and hard to verify, so it does not pretend to force a result by assumption. In Section 6, the explicit example is solved by upper and lower bounds: the candidate controls give the lower bound and a comparison/BSDE argument gives the upper bound; no fitted quantity is renamed a prediction. Self-citations ([5], [6], [19], [20], [23]) are used for background, standard BSDE/comparison results, or face-lifting context; the sufficiency reference in Remark 3.5 to [6, Theorem 10.3.10] is not load-bearing because Theorem 4.4 establishes optimality through concavity rather than through that cited theorem. There is a notational ambiguity in Assumption 4.1(iv) and in the displayed estimate in Proposition 4.3 ('4γ_T^2 T' versus upper/lower bounds), but tracking the bars explicitly makes the inequality valid, so this is a typesetting or proof-writing issue, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

No free parameters are fitted to data; the model inputs are functions and constants assumed to exist, listed as axioms. No new unobserved entities are postulated; the only objects are contracts, wealth processes, and Brownian noise.

axioms (7)

standard math Standard BSDE theory: existence/uniqueness, comparison principle, Girsanov theorem, Itô calculus.
Used throughout Sections 3–6 to define utilities via BSDEs and to solve the agent's and principal's problems; not proved in the paper.
domain assumption Weak formulation: the agent's action changes the probability measure via Girsanov, while X is fixed with diffusion σ (eqs. (2.1)–(2.3)).
This is the standard moral-hazard modeling device inherited from the cited literature; the paper states it as the setup.
domain assumption Contract space restricted to S0, α dt + β dX, and terminal ξ (eq. (2.4)).
The whole model is built on this parametrization; the claimed generality is relative to this class of contracts.
standard math Zero interest rates are without loss of generality via discounting (eq. (3.1)).
The paper argues discounted wealth reduces nonzero rates to zero; accepted as a standard reduction.
ad hoc to paper Agent felicity takes the separable-exponential form f_A = f1 + f2(a) − λ1(t)e^{−λ2(t)c}, with Assumption 4.1(i)–(iv) including 0≤∂_zz F2≤L and the smallness inequality.
Necessary for the explicit inverse I_c, the bounds K,K, and strict concavity in Section 4; without it Theorem 4.4 is not proved.
ad hoc to paper Theorem 5.5 hypotheses (i)–(iv): existence of boundary value V−, comparison principle, ε-optimal approximate controls, and nonempty DP.
The principal-side characterization is conditional on these strong conditions; the paper admits they are hard to verify.
ad hoc to paper Example felicity forms (6.1): f_A=f(c−½a²), f_P=c^P−κ(t)(1−β)², g_A=m^A, g_P=m^P, and Assumption 6.1 (strict concavity and Inada-like limits of f).
Defines the explicit solvable example and is not a general restriction; results in Section 6 are for this specification.

pith-pipeline@v1.3.0-alltime-deepseek · 21388 in / 27243 out tokens · 233958 ms · 2026-08-01T18:42:39.392474+00:00 · methodology

0 comments

read the original abstract

In this paper, we study a general continuous-time Principal-Agent (PA) problem, where the agent privately makes effort and consumption decisions over time under a contract with payment schemes both in continuous time and in lump sums. In particular, we allow the continuous payment process to be a controlled diffusion, which is directly related to the pay-to-performance sensitivity (PPS) in the empirical literature. In solving the agent's problem, we propose a new sufficient condition that directly yields a solution to the agent's problem, without requiring a separate verification step for the solution obtained from the first-order approach. We also present an example which can be solved explicitly.

discussion (0)

Reference graph

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and Dewatripont, M., 2005

Bolton, P. and Dewatripont, M., 2005. Contract theory. MIT press

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and Soner, H.M., 1998

Broadie, M., Cvitani´ c, J. and Soner, H.M., 1998. Optimal replication of contingent claims under portfolio constraints. The Review of Financial Studies, 11(1), pp.59-79. 1ξ∗ is arbitrary, since it is not involved in this problem. Sinceg P is linear inm P , so the effect ofc P on the continuous part and that on the terminal part cancel each other, and the...

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Transaction costs, risk aversion, and the choice of contractual ar- rangements

Cheung, S.N., 1969. Transaction costs, risk aversion, and the choice of contractual ar- rangements. The Journal of Law and Economics, 12(1), pp.23-42

1969

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and Kocherlakota, N.R., 2001

Cole, H.L. and Kocherlakota, N.R., 2001. Efficient allocations with hidden income and hidden storage. The Review of Economic Studies, 68(3), pp.523-542

2001

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and Zhang, J., 2009

Cvitani´ c, J., Wan, X. and Zhang, J., 2009. Optimal compensation with hidden action and lump-sum payment in a continuous-time model. Applied Mathematics and Opti- mization, 59(1), pp.99-146

2009

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and Zhang, J., 2012

Cvitanic, J. and Zhang, J., 2012. Contract theory in continuous-time models. Springer Science & Business Media

2012

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and Touzi, N., 2017

Cvitani´ c, J., Possama ¨ ı, D. and Touzi, N., 2017. Moral hazard in dynamic risk manage- ment. Management Science, 63(10), pp.3328-3346

2017

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and Touzi, N., 2018

Cvitani´ c, J., Possama ¨ ı, D. and Touzi, N., 2018. Dynamic programming approach to principal–agent problems. Finance and Stochastics, 22(1), pp.1-37

2018

[9] [9]

and Zhang, J., 2022

Feinstein, Z., Rudloff, B. and Zhang, J., 2022. Dynamic set values for nonzero-sum games with multiple equilibriums. Mathematics of Operations Research, 47(1), pp.616-642

2022

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and Milgrom, P., 1987

Holmstrom, B. and Milgrom, P., 1987. Aggregation and linearity in the provision of intertemporal incentives. Econometrica: Journal of the Econometric Society, pp.303-328

1987

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and Martimort, D., 2001

Laffont, J.J. and Martimort, D., 2001. The theory of incentives: the principal-agent model. Princeton university press

2001

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and Zhu, Z., 2025

Ludkovski, M., Xie, C. and Zhu, Z., 2025. DeepPAAC: A New Deep Galerkin Method for Principal-Agent Problems. arXiv preprint arXiv:2511.04309

Pith/arXiv arXiv 2025

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Functional Analysis, 2nd ed., McGraw–Hill, Hamburg

Rudin, W., 1991. Functional Analysis, 2nd ed., McGraw–Hill, Hamburg

1991

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The economics of contracts: a primer

Salani´ e, B., 2005. The economics of contracts: a primer. MIT press

2005

[15] [15]

A continuous-time version of the principal-agent problem

Sannikov, Y., 2008. A continuous-time version of the principal-agent problem. The Review of Economic Studies, 75(3), pp.957-984

2008

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Incentives and risk sharing in sharecropping

Stiglitz, J.E., 1974. Incentives and risk sharing in sharecropping. The Review of Eco- nomic Studies, 41(2), pp.219-255. 27

1974

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and Sung, J., 1993

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1993

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and Touzi, N., 2000

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2000

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Linearity with project selection and controllable diffusion rate in continuous-time principal-agent problems

Sung, J., 1995. Linearity with project selection and controllable diffusion rate in continuous-time principal-agent problems. The RAND Journal of Economics, pp.720- 743

1995

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Contract theory: discrete-and continuous-time models

Sung, J., 2023. Contract theory: discrete-and continuous-time models. Springer

2023

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A solvable continuous time dynamic principal–agent model

Williams, N., 2015. A solvable continuous time dynamic principal–agent model. Journal of Economic Theory, 159, pp.989-1015

2015

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Backward stochastic differential equations

Zhang, J., 2017. Backward stochastic differential equations. In Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory . New York, NY: Springer New York

2017

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and Zhu, Z., 2025

Zhang, J. and Zhu, Z., 2025. A dynamic principal-agent problem with one-sided com- mitment. Mathematics of Operations Research, 50(4), pp.2600-2632. 28

2025