A risk based approach to the principal-agent problem

Djehiche, Boualem; Helgesson, Peter Article A risk based approach to the principal-agent problem Asian Journal of Economics and Banking (AJEB) Provided in Cooperation with: Ho Chi Minh University of Banking (HUB), Ho Chi Minh City Suggested Citation: Djehiche, Boualem; Helgesson, Peter (2024) : A risk based approach to the principal-agent problem, Asian Journal of Economics and Banking (AJEB), ISSN 2633-7991, Emerald, Leeds, Vol. 8, Iss. 3, pp. 310-334, https://doi.org/10.1108/AJEB-05-2024-0065 This Version is available at: https://hdl.handle.net/10419/334126 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ A risk based approach to the principal–agent problem Boualem Djehiche Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden, and Peter Helgesson Department of Mathematics, Chalmers University of Technology, Gothenburg, Sweden Abstract Purpose –We aim to generalize the continuous-time principal–agent problem to incorporate time-inconsistent utility functions, such as those of mean-variance type, which are prevalent in risk management and finance. Design/methodology/approach –We use recent advancements of the Pontryagin maximum principle for forward-backward stochastic differential equations (FBSDEs) to develop a method for characterizing optimal contracts in such models. This approach addresses the challenges posed by the non-applicability of the classical Hamilton–Jacobi–Bellman equation due to time inconsistency. Findings –We provide a framework for deriving optimal contracts in the principal–agent problem under hidden action, specifically tailored for time-inconsistent utilities. This is illustrated through a fully solved example in the linear-quadratic setting, demonstrating the practical applicability of the method. Originality/value –The work contributes to the existing literature by presenting a novel mathematical approach to a class of continuous time principal–agent problems, particularly under hidden action with timeinconsistent utilities, a scenario not previously addressed. The results offer potential insights for both theoretical development and practical applications in finance and economics. Keywords Principal–agent problem, Stochastic maximum principle, Pontryagin’s maximum principle, Mean-variance, Time inconsistent utility functions Paper type Research paper 1. Introduction Risk management or the problem of finding an optimal balance between expected returns and risk taking is a central topic of research within banking, economics and finance. Applications such as portfolio optimization, optimal stopping and liquidation problems have been of particular interest in the literature. In such applications it is common to consider utility functions of mean-variance type. Mean-variance utility functions constitute an important subclass of the so called time inconsistent utility functions for which the Bellman principle of dynamic programming does not hold. Problems involving such utilities can therefore not be approached by the classical Hamilton–Jacobi–Bellman equation. In this paper we develop a method of studying a mean-variance setting of the celebrated principal– agent problem by means of the stochastic generalization of Pontryagin’s maximum principle. The precise structure of the principal–agent problem goes as follows. The principal employs an agent to manage a certain well-defined noisy asset over a fixed period of time. AJEB 8,3 310 JEL Classification — B41, C00, C61, C70, C72 2010 Mathematics Subject Classification — 93E20, 49N70, 49N90 © Boualem Djehiche and Peter Helgesson. Published in Asian Journal of Economics and Banking. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http:// creativecommons.org/licences/by/4.0/legalcode The current issue and full text archive of this journal is available on Emerald Insight at: https://www.emerald.com/insight/2615-9821.htm Received 27 May 2024 Revised 9 August 2024 Accepted 27 August 2024 Asian Journal of Economics and Banking Vol. 8 No. 3, 2024 pp. 310-334 Emerald Publishing Limited e-ISSN: 2633-7991 p-ISSN: 2615-9821 DOI 10.1108/AJEB-05-2024-0065 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 In return for his/her effort the agent receives a compensation according to some agreement, set before the period starts. It could for instance involve a lump-sum payment at the end of the period, a continuously paying cash-flow during the period or both. Depending on what information the principle has at hand to form an agreement, one distinguishes between two cases; the full information and the hidden action-problem. The full information case differs from the hidden action case in that the principal can observe the actions of the agent in addition to the evolution of the asset. Therefore, under full information the principal is allowed to tailor a contract based on both outcome and effort, not only outcome as for hidden actions. In both cases the contract is constrained by the agent via a so called participation constraint, clarifying the minimum requirements of the agent to engage in the project. Under hidden action the contract is further constrained by the incentive compatibility condition, meaning that as soon as a contract is assigned the agent will act as to maximize his/her own utility and not necessarily that of the principal. The pioneering paper in which the principal–agent problem first appears is Holmstr€ om and Milgrom (1987). They study a continuous time model over a finite period in which the principle and the agent both optimize exponential utility functions. The principal rewards the agent at the end of the period by a lump-sum payment. As a result they find that the optimal contract is linear with respect to output. The paper Holmstr€ om and Milgrom (1987) is generalized in Sch€ attler and Sung (1993) to a mathematical framework that uses methods from dynamic programming and martingale theory to characterize contract optimality. The interest in continuous time models of the principal–agent problem has grown substantially since the first studies appeared. In Cvitani� cet al. (2009),Sannikov (2008), Westerfield (2006) and Williams (2013) (only to mention a few) the authors analyze continuous time models in a classical setting, i.e. having one principal and one agent. Such models are also covered in the recent book Cvitani� c and Zhang (2013). Other models such as various multiplayer versions have been studied for instance in Kang (2013) and Koo et al. (2008). Our goal is to characterize optimal contracts in the classical setting of principal–agent problem under hidden action for time inconsistent utility functions. We consider two different modeling possibilities; hidden action in the weak formulation and hidden contract in the strong formulation. In the first model the agent has full information of the mechanisms behind the cash-flow and the principal wishes to minimize his/her mean-variance utility. In the latter model the agent does not know the structure of the cash-flow and has to protect him-/her-self from high levels of risk by an additional participation constraint of variance type. To the best of our knowledge this has not previously been addressed in the literature. In order to carry the program through we use recent generalizations of Pontryagin’s stochastic maximum principle. The idea is to consider the principal–agent problem as a sequential optimization problem. We first consider the agent’s problem of characterizing optimal choice of effort. Then we proceed to the Principal’s problem which, by incentive compatibility, becomes a constrained optimal control problem of a forward-backward stochastic differential equation (from now on FBSDE). A similar scheme was considered in Djehiche and Helgesson (2014) but without the non-standard mean-variance consideration. Optimal control with respect to mean-variance utility functions has previously been studied in for instance Li (2000), and Andersson and Djehiche (2011). Optimal portfolios based on time inconsistent utilities have been addressed in Bj€ ork and Murgoci (2010),Bj€ ork et al. (2014),Djehiche and Huang (2014),Ekeland and Lazrak (2006) and Ekeland and Pirvu (2008). See also the recent book Bj€ ork et al. (2021). A discrete version of this class of problems boils down to study a system of forward-backward time series whose analysis follows the same lines of reasoning as the time-continuous version but the formulas are a bit clumsy. Asian Journal of Economics and Banking 311 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 In the present literature of the principal–agent problem the paper closest to ours is Williams (2013), in which a similar maximum principle approach is used. The setting is classical (without time inconsistent utility functions) and the author finds a characterization for optimal choice of effort in the agent’s problem. The full model involving the constrained principal’s problem, however, is not considered. The main results of our study are presented in Theorem 4.3 and Theorem 4.4 in which a full characterization of optimal contracts is stated for two different models. In practical terms, the assumptions made in our study reflect typical scenarios in financial decision-making. For example, the mean-variance utility function is a standard way to represent an investor’s desire to balance expected returns with risk. This assumption helps simplify the complex reality of financial markets into a manageable model. However, it might break down in situations where investor preferences are not stable over time or when there are abrupt changes in market conditions. Our results imply that, under these assumptions, it is possible to design contracts that align the interests of both the principal (e.g. an employer) and the agent (e.g. an employee) even when the agent’s actions are not directly observable. The paper is organized as follows. In Section 2 we introduce the mathematical machinery from stochastic optimal control theory that is necessary for our purposes. Mean-variance maximum principles are then derived in Section 3 by results from Section 2 in two different but related cases. Section 4 is devoted to fit the methods from the previous sections into a principle–agent framework. We consider two different models under hidden action and find necessary conditions for optimality. Finally in Section 5 we make the general scheme of Section 4 concrete by a simple and fully solved example in the linear-quadratic (LQ)-setting. 2. Preliminaries Let T> 0 be a fixed time horizon and ðΩ;F;F;PÞbe a filtered probability space satisfying the usual conditions on which a 1-dimensional Brownian motion W¼ fWtgt≥0is defined. We let Fbe the natural filtration generated by Waugmented by all P-null sets NP, i.e. F¼ Ft∨NPwhere Ftd σ ðfWsg:0≤s≤tÞ. Consider the following control system of forward stochastic differential equations (SDEs) of mean-field type: dxðtÞ ¼ bðt;xðtÞ;E½xðtÞ�;sðtÞÞdt þ σ ðt;xðtÞ;E½xðtÞ�ÞdWt;t∈ð0;T� xð0Þ ¼ x0 � with a cost functional of the form Jðsð$ÞÞdEZT 0 fðt;xðtÞ;E½xðtÞ�;sðtÞÞdt þhðxðTÞ;E½xðTÞ�Þ � �;(2.1) where b:½0;T�3R3R3S→R, σ :½0;T�3R3R→R,f:½0;T�3R3R3S→Rand h:R3R→Rand S⊂Ris a non-empty subset. The control s($) is admissible if it is an F-adapted and square-integrable process taking values in S. We denote the set of all such admissible controls by S½0;T�. In order to avoid technicalities in regularity that are irrelevant for our purposes we state the following assumption. Assumption 1. The functions b, σ ,fand hare C 1 with respect to xand ~ x, where ~ xdenotes the explicit dependence of E½xð$Þ�. Moreover, b, σ ,fand hand their first order derivatives with respect to xand ~ xare bounded and continuous in x, ~ xand s. AJEB 8,3 312 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 We are interested in the following optimal control problem: Problem (S). Minimize (2.1) over S½0;T�. Any sð$Þ∈S½0;T�satisfying J�sð$Þ�¼inf sð$Þ∈S½0;T�Jðsð$ÞÞ is called an optimal control and the corresponding xð$Þis called the optimal state process. We will refer to ðxð$Þ;sð$ÞÞas an optimal pair. The following stochastic maximum principle for characterizing optimal pairs in problem (S) was found in Buckdahn et al. (2011). Theorem 2.1. The stochastic maximum principle. Let the conditions in Assumption 1 hold and consider an optimal pair ðxð$Þ;sð$ÞÞof problem (S). Then there exists a pair of processes ðpð$Þ;qð$ÞÞ∈L2 Fð0;T;RÞ3ðL2 Fð0;T;RÞÞ satisfying the adjoint equation dpðtÞ ¼ −bx�t;xðtÞ;EhxðtÞi;sðtÞ�pðtÞþEhb~ x�t;xðtÞ;EhxðtÞi;sðtÞ�pðtÞin þ σ x�t;xðtÞ;EhxðtÞi�qðtÞþEh σ ~ x�t;xðtÞ;EhxðtÞi�qðtÞi �fx�t;xðtÞ;EhxðtÞi�;sðtÞ��Ef~ x�t;xðtÞ;EhxðtÞi�;sðtÞÞ h iodt þqðtÞdWt; pðTÞ ¼ −hx�xðTÞ;EhxðTÞi��Ehh~ x�xðTÞ;EhxðTÞi�i; 8 > > > > > > > > < > > > > > > > > : (2.2) such that sðtÞ ¼ argmax s∈SH�t;xðtÞ;s;pðtÞ;qðtÞ�;a:e:t∈½0;T�;P�a:s:(2.3) where the Hamiltonian function His given by Hðt;x;s;p;qÞdbðt;x;E½x�;sÞ$pþ σ ðt;x;E½x�Þ$q�fðt;x;E½x�;sÞ(2.4) for ðt;x;s;p;qÞ∈½0;T�3R3S3R3R. Remark 2.2. It is important to remember that Theorem 2.1 merely states a set of necessary conditions for optimality in (S). It does not claim the existence of an optimal control. Existence theory of stochastic optimal controls (both in the strong and the weak sense) has been a subject of study since the sixties (see e.g. Kushner (1965)) and, at least in the case of strong solutions, the results seem to depend a lot upon the statement of the problem. In the weak sense an account of existence results is to be found in Yong and Zhou (1999) (Theorem 5.3, p. 71). Remark 2.3. Restricting the space Uto be convex allows for a diffusion coefficient of the form σ ðt;x;E½x�;sÞ, without changing the conclusion of Theorem 2.1. In the case of a non-convex control space the stochastic maximum principle with controlled diffusion was proven in Peng (1990) and requires the solution of an additional adjoint BSDE. We choose to leave this most general maximum principle as reference in order to keep the presentation clear. Asian Journal of Economics and Banking 313 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 As pointed out in Remark 2.2 it is a non-trivial task to prove the existence of an optimal pair ðxð$Þ;sð$ÞÞin a general stochastic control model under the additional assumptions. Assumption 2. The control domain Sis a convex body in R. The maps b, σ and fare locally Lipschitz in uand their derivatives in xand ~ xare continuous in x,~ x and s, the following theorem provides sufficient conditions for optimality in (S). Theorem 2.4. Sufficient conditions for optimality. Under Assumptions 1 and 2 let ðxð$Þ;sð$Þ;pð$Þ;qð$ÞÞbe an admissible 4-tuple. Suppose that h is convex and further that Hðt;$;$;$;pðtÞ;qðtÞÞis concave for all t∈½0;T�P-a.s. and sðtÞ ¼ argmax s∈SH�t;xðtÞ;EhxðtÞi;s;pðtÞ;qðtÞ�;a:e:t∈½0;T�;P�a:s: Then ðxð$Þ;sð$ÞÞis an optimal pair for problem (S). The stochastic maximum principle has since the early days of the subject (in pioneering papers by, e.g. Bismut (1978) and Bensoussan (1982)) developed a lot and does by now apply to a wide range of problems more general than (S) (see for instance Peng (1990),Andersson and Djehiche (2011),Buckdahn et al. (2011),Djehiche et al. (2014)). For our purposes we need a refined version of Theorem 2.1, characterizing optimal controls in a FBSDE-dynamical setting under state constraints. More precisely we wish to consider a stochastic control system of the form dxðtÞ ¼ bðt;ΘðtÞ;sðtÞÞdt þ σ ðt;ΘðtÞÞdWt dyðtÞ ¼ −cðt;ΘðtÞ;sðtÞÞdt þzðtÞdWt xð0Þ ¼ x0;yðTÞ ¼ φðxðTÞÞ; 8 < :(2.5) where b; σ ;c:½0;T�3R63S→Rand φ:R→R, with respect to a cost-functional of the form Jðsð$ÞÞdEZT 0 fðt;ΘðtÞ;sðtÞÞdt þhðxðTÞ;E½xðTÞ�Þþgðyð0ÞÞ � �;(2.6) and a set of state constraints EZT 0 Fðt;ΘðtÞ;sðtÞÞdt þHðxðTÞ;E½xðTÞ�ÞþGðyð0ÞÞ � �d EZT 0 f1ðt;ΘðtÞ;sðtÞÞdt þh1ðxðTÞ;E½xðTÞ�Þþg1ðyð0ÞÞ � � . . . EZT 0 flðt;ΘðtÞ;sðtÞÞdt þhlðxðTÞ;E½xðTÞ�Þþglðyð0ÞÞ � � 0 B B B B B B B @ 1 C C C C C C C A ∈Λ; (2.7) for some closed and convex set Λ ⊆ Rl. In the above expressions we have introduced AJEB 8,3 314 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 ΘðtÞdðxðtÞ;yðtÞ;zðtÞ;E½xðtÞ�;E½yðtÞ�;E½zðtÞ�Þ; in order to avoid unnecessarily heavy notation. The optimal control problem is: Problem (SC). Minimize (2.6) subject to the state constraints (2.7) over the set S½0;T�. To get a good maximum principle for (SC) we require some further regularity conditions ensuring solvability of (2.5). These conditions are listed in the following assumptions and can be found in Li and Liu (2014). Assumption 3. The functions b, σ ,care continuously differentiable and Lipschitz continuous in Θ, the functions h,g,h i ,g i are continuously differentiable in xand yrespectively, and they are bounded by Cð1þjxjþjyj þjzjþj~ xjþj~ yjþj~ zjþjsjÞ,C(1 þ jxj) and C(1 þ jyj), respectively. Assumption 4. All derivatives in Assumption 4 are Lipschitz continuous and bounded. Assumption 5. For all Θ∈R6,s∈S,Að$;Θ;sÞ∈L2 Fð0;T;R3Þ, where we have A(t,Θ,s) d(c(t,Θ,s), b(t,Θ,s), σ (t,Θ)) and L2 F�0;T;Rk�d ψ :½0;T�3Ω→Rk�� ψ is F�adapted and EZT 0j ψ j2dt � �<∞ � �; and for each x∈R,φðxÞ∈L2 FðΩ;RÞ. Furthermore, there exists a constant C> 0 such that jAðt;Θ1;sÞ�Aðt;Θ2;sÞj≤CjΘ1�Θ2j;P�a:s:and for a:e:t∈½0;T�; jφðx1Þ�φðx2Þj≤Cjx1�x2j;P�a:s; for all Θ1;Θ2∈R6: 8 < : Assumption 6. The functions Aand φsatisfy the following monotonicity conditions: EDAðt;Θ1;sÞ�Aðt;Θ2;sÞ;Θ1�Θ2E≤βEjΘ1�Θ2j2;P�a:s Dφðx1Þ�φðx2Þ;x1�x2E≥ μ jx1�x2j2 8 < : for all Θ1;Θ2∈R6,x1;x2∈R In the spirit of Li and Liu (2014) we are now ready to formulate the state constrained stochastic maximum principle for fully coupled FBSDEs of mean-field type. Theorem 2.5. The state constrained maximum principle. Let Assumptions 3–6hold and assume Λ ⊆ Rlto be a closed and convex set. If ðxð$Þ;yð$Þ;zð$Þ;sð$ÞÞis an optimal 4-tuple of problem (SC), then there exists a vector ðλ0;λÞ∈R1þl such that λ0≥0;jλ0j2þjλj2¼1;(2.8) satisfying the transversality condition �λ;v�EZT 0 F�t;xðtÞ;yðtÞ;zðtÞ;sðtÞ�dt þH�xðTÞ�þG�yð0Þ� � ��≥0;∀v∈Λ(2.9) Asian Journal of Economics and Banking 315 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 and a 3-tuple ðrð$Þ;pð$Þ;qð$ÞÞ∈L2 FðΩ;Cð½0;T�;RÞÞ3L2 FðΩ;Cð½0;T�;RÞÞ3L2 Fð0;T;RÞ of solutions to the adjoint FBSDE drðtÞ ¼ cyðtÞrðtÞ�byðtÞpðtÞ� σ yðtÞqðtÞþX l i¼0 λifi yðtÞ ( þEc~ yðtÞrðtÞ�b~ yðtÞpðtÞ� σ ~ yðtÞqðtÞþX l i¼0 λifi ~ yðtÞ " #)dt þczðtÞrðtÞ�bzðtÞpðtÞ� σ zðtÞqðtÞþX l i¼0 λifi zðtÞ ( þEc~ zðtÞrðtÞ�b~ zðtÞpðtÞ� σ ~ zðtÞqðtÞþX l i¼0 λifi ~ zðtÞ " #)dWt; dpðtÞ ¼ −�cxðtÞrðtÞþbxðtÞpðtÞþ σ xðtÞqðtÞ�X l i¼0 λifi xðtÞ ( þE�c~ xðtÞrðtÞþb~ xðtÞpðtÞþ σ ~ xðtÞqðtÞ�X l i¼0 λifi ~ xðtÞ " #)dt þqðtÞdWt; rð0Þ ¼ X l i¼0 λiEhgi�yð0Þ�i; pðTÞ ¼ −φx�xðTÞ�rðTÞ�X l i¼0 λi�hi x�xðTÞ;EhxðTÞi�þEhhi ~ x�xðTÞ;EhxðTÞi�i�; (2.10) such that sðtÞ ¼ argmax s∈SH�t;ΘðtÞ;s;rðtÞ;pðtÞ;qðtÞ;λ0;λ�a:e:t∈½0;T�;P�a:s: where the Hamiltonian function His given by Hðt;Θ;s;r;p;q;λ0;λÞd �r$cðt;Θ;sÞþp$bðt;Θ;sÞþq$ σ ðt;ΘÞ�X l i¼0 λifiðt;Θ;sÞ: Remark 2.6. As in Remark 2.3, analogue principles also hold in Theorem 2.5. Remark 2.7. The maximum principle in Theorem 2.5 without state constraints is an easy extension of the same result in Li and Liu (2014) and follows the proof mutatis mutandis. Extending the result to allow for state constraints is a standard procedure and can be found for instance in Djehiche and Helgesson (2014). AJEB 8,3 316 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 3. Utilities of mean-variance type The mean-variance utility function we use models the trade-off between risk and reward that is central to many financial decisions, such as in portfolio management. By assuming meanvariance preferences, we are capturing a realistic scenario where investors aim to maximize returns while minimizing risk. For instance, consider a portfolio manager (the agent) who is employed by a fund (the principal). The fund wants the manager to invest in a way that maximizes returns while controlling for risk. The manager’s utility function includes both the expected return and the risk (variance) of the investment portfolio. The optimal contract, derived using our methods, ensures that the manager is incentivized to invest in a way that aligns with the fund’s objectives, despite the manager having private information about investment opportunities and risks. We are now going to fit the methods presented in Section 2 to a mean-variance framework, i.e. we want to control the forward-backward dynamics of mean-field type (2.5) with respect to either of the following two cases: (i) Minimize IðuÞd�EZT 0 Uðt;ΘðtÞ;sðtÞÞdt þVðxðTÞÞ � � þr 2Var ZT 0 Φðt;ΘðtÞ;sðtÞÞdt þΨðxðTÞÞ � �; (3.1) over S½0;T�for some risk aversion r> 0. (ii) Minimize JðuÞdEZT 0 Uðt;ΘðtÞ;sðtÞÞdt þVðxðTÞÞ � � (3.2) over S½0;T�subject to a set of state constraints (compare (2.7)), including statements of the form Var ZT 0 Φðt;ΘðtÞ;sðtÞÞdt þΨðxðTÞÞ � �≤R0:(3.3) In order to carry this through we introduce the auxiliary process η ðtÞdZt 0 Φð τ ;Θð τ Þ;sð τ ÞÞd τ þΨðxðtÞÞ; which by It^ o’s Lemma solves the SDE d η ðtÞ ¼ ΦðtÞþbðtÞ$Ψ0ðxðtÞÞþ σ 2ðtÞ 2$Ψ00ðxðtÞÞ � �dt þ σ ðtÞ$Ψ0ðxðtÞÞdWt; η ð0Þ ¼ 0: 8 > > < > > : (3.4) Asian Journal of Economics and Banking 317 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 The principal’s problem (strong formulation): Given that the agent’s problem has an optimal solution eð$Þin the weak formulation the Principal’s problem is to find a process sð$Þ∈S½0;T�, such that the cost functional JP�sð$Þ�d�EZT 0 Uðt;xðtÞ;sðtÞÞdt þVðxðTÞÞ � � þr 2Var ZT 0 Φðt;xðtÞ;sðtÞÞdt þΨðxðTÞÞ � �; is minimized and JA�eð$Þ;s�¼EZT 0 uðt;xðtÞ;eðtÞ;sðtÞÞdt þvðxðTÞÞ � �≤C0; subject to the dynamics dxðtÞ ¼ σ ðt;xðtÞÞdWt;t∈ð0;T�; xð0Þ ¼ 0: � Remark 4.1. Here we have chosen to formulate the principal’s problem in the strong form rather than in the weak form, which seems to be most common in the literature. However, as pointed out in Cvitani� c and Zhang (2013), because of adaptiveness this approach can be problematic in certain models. This is a fact that one should be aware of. In this context the following definition is natural. Definition 4.2. An optimal contract is a pair ðeð$Þ;sð$ÞÞ∈E½0;T�3S½0;T�obtained by sequentially solving first the agent’s and then the principal’s problem. In game theoretic terminology an optimal contract can thus be thought of as a Stackelberg equilibrium in a two-player non-zero-sum game. It is important to note that even though the principal cannot observe the agent’s effort, he/she can still offer the agent a contract by suggesting a choice of effort e($) and a compensation s($). By incentive compatibility, however, the principal knows that the agent only will follow such a contract if the suggested effort solves the agent’s problem. To find the optimal effort, eð$Þ, the principal must have information of the agent’s preferences, i.e. the functions uand v. The realism of such an assumption is indeed questionable but nevertheless necessary in our formulation due to the participation constraint. In order to make the intuition clear and to avoid any confusion we adopt the convention that the principal has full information of the agent’s preferences uand v. This gives a tractable way of thinking of how actual contracting is realized. Thus, the principal is able to predict the optimal effort eð$Þof the agent’s problem and thereby suggest an optimal contract ðeð$Þ;sð$ÞÞ, if it exists. The idea is to apply the methods from Section 2 to characterize optimal contracts in the general principal–agent model presented above. AJEB 8,3 324 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 However, since the control variable efigures in the diffusion of (4.5) we require the following convexity assumption in order to avoid a second order adjoint process in the maximum principle: Assumtion 7. The set E⊂Ris convex. The agent’s Hamiltonian in the weak formulation is HAðt;x;Γe;e;p;q;sÞdq$Γe$fðt;x;eÞ σ ðt;xÞ�Γe$uðt;x;e;sÞ;(4.10) and by Theorem 2.1 any optimal control eðtÞsolving the agent’s problem must maximize HA pointwise. The pair (p($), q($)) solves the agent’s adjoint BSDE: dpðtÞ ¼ −qðtÞ$ f�t;xðtÞ;eðtÞ� σ ðt;xðtÞÞ �u�t;xðtÞ;eðtÞ;sðtÞ� 8 < :9 = ;dt þqðtÞdWt; pðTÞ ¼ −vxðxðTÞÞ 8 > > > > < > > > > : (4.11) If fand uboth are differentiable in the evariable and we assume that eð$Þ∈intðEÞ, maximizing HAtranslates into the first order condition qðtÞ ¼ σ ðt;xðtÞÞ$ ue�t;xðtÞ;eðtÞ;sðtÞ� fe�t;xðtÞ;eðtÞ�;(4.12) which is in agreement with Williams (2013). Before proceeding to the Principal’s problem we assume solvability of ein (4.12) and we write eðtÞ ¼ e*�t;xðtÞ;qðtÞ;sðtÞ�; where e*:Rþ3R4→Ris a function having sufficient regularity to allow for the existence of a unique solution to the FBSDE (4.13) below. Based on the information given by e* the principal wishes to minimize the cost JPby selecting a process s($) respecting (4.4). The dynamics of the corresponding control problem is, in contrast to the SDE of the agent’s problem, a FBSDE built up by the output SDE coupled to the agent’s adjoint BSDE. More precisely: dxðtÞ ¼ σ ðt;xðtÞÞdWt; dpðtÞ ¼ −qðtÞ$ f�t;xðtÞ;e*ðt;xðtÞ;qðtÞ;sðtÞÞ� σ ðt;xðtÞÞ �u�t;xðtÞ;e*ðt;xðtÞ;qðtÞ;sðtÞÞ;sðtÞ� 8 < :9 = ;dt þqðtÞdWt; xð0Þ ¼ 0;pðTÞ ¼ −vxðxðTÞÞ: 8 > > > > > > > > > > > > < > > > > > > > > > > > > :(4.13) Asian Journal of Economics and Banking 325 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 In order to characterize cash-flow optimality in the Principal’s problem we apply Theorem 3.1. The Hamiltonian reads HPðt;x;q;s;R;P1;P2;Q1;Q2;λP;λAÞd R$�q$ f�t;x;e*ðt;x;q;sÞ� σ ðt;xÞþu�t;x;e*ðt;x;q;sÞ;s� 8 < :9 = ;þP2$Φðt;x;sÞþ σ 2ðt;xÞ 2Ψ00ðxÞ � � þQ1$ σ ðt;xÞþQ2$ σ ðt;xÞΨ0ðxÞ�λA$u�t;x;e*ðt;x;q;sÞ;s�þλP$Uðt;x;sÞ; (4.14) and for any optimal 4-tuple ðxð$Þ;pð$Þ;qð$Þ;sð$ÞÞ we have the existence of Lagrange multipliers λA;λP∈Rsatisfying the conditions in Theorem (3.1). The adjoint processes (R($), P 1 ($), Q 1 ($), P 2 ($), Q 2 ($)) solve the FBSDE (3.7), in which case sðtÞ ¼ argmax s∈SHP�t;xðtÞ;pðtÞ;qðtÞ;s;RðtÞ;P1ðtÞ;Q1ðtÞ;P2ðtÞ;Q2ðtÞ;λP;λA�: Before stating the full characterization of optimal contracts in the Mean-Variance PrincipalAgent problem under Hidden Action we introduce the following technical assumption: (PA1). All functions involved in the Agent’s problem satisfy Assumption 1 from Section 2 and the density of output is a martingale. The functions defining the Principal’s problem (including composition with the map e*) satisfy Assumptions 2-6, also from Section 2, and Ψ is three times differentiable. Theorem 4.3. Let the statements in (PA1) and Assumption 7 hold and consider the Mean-Variance Principal-Agent problem under Hidden Actions with risk aversion r > 0 and participation constraint defined by C 0 < 0. Then, if ðeð$Þ;sð$ÞÞ is an optimal contract there exist numbers λA;λP∈R such that λP≥0;λ2 Aþλ2 P¼1; a pair ðpð$Þ;qð$ÞÞ∈L2 Fð0;T;RÞ3ðL2 Fð0;T;RÞÞsolving the SDE in (4.11) and a quintuple ðRð$Þ;P1ð$Þ;P2ð$Þ;Q1ð$Þ;Q2ð$ÞÞ∈L2 FðΩ;Cð½0;T�;RÞÞ 3L2 FðΩ;Cð½0;T�;RÞÞ3L2 Fð0;T;RÞsolving the adjoint FBSDE (3.7) defined by (4.13) such that, sequentially, eðtÞ ¼ argmax e∈EHA�t;xðtÞ;ΓeðtÞ;e;qðtÞ;sðtÞ�; and sðtÞ ¼ argmax s∈SHP�t;xðtÞ;qðtÞ;s;RðtÞ;P1ðtÞ;Q1ðtÞ;P2ðtÞ;Q2ðtÞ;λP;λA�; with Hamiltonians HAand HPas in (4.10) and (4.14) respectively. AJEB 8,3 326 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 4.2 Hidden Contract in the strong formulation We are now going to study a different type of mean-variance principal–agent problems called hidden contract models (introduced in Djehiche and Helgesson (2014)). Comparing to the hidden action model in Section 4.1 the hidden contracts differ in two key aspects. First we relax the information set of the principal from Fxto the full filtration generated by the Brownian motion. Secondly we treat the process s($) as hidden, meaning that the Agent reacts to the provided cash-flow given as an F-adapted process, without being aware of the underlying dependence of the output. This explains the name Hidden Contract. The fact that the underlying mathematical structure of s($) is unknown to the Agent in the Hidden Contract model motivates the relevance of a Mean-Variance framework by an extended participation constraint (compared to (4.4)). By requiring an upper bound for the variance of for instance the expected accumulated wealth provided by s($) the Agent can protect him/her-self from undesirable high levels of risk. The setup goes as follows. Consider a Principal-Agent model in which output x(t) is modeled as a risky asset solving the SDE dxðtÞ ¼ fðt;xðtÞ;eðtÞÞdt þ σ ðt;xðtÞÞdWt;t∈ð0;T�; xð0Þ ¼ 0: �(4.15) Here T> 0 and W t is a 1-dimensional standard Brownian motion defined on the filtered probability space ðΩ;F;F;PÞ. The functions fand σ represent production rate and volatility respectively, and we assume both of them to satisfy Assumption 1 from Section 2. Just as for the Hidden Action case we require any admissible effort process e($) to be in E½0;T�. For the admissible cash-flows, however, we enlarge S½0;T�(due to the extended flow of information to the Principal) to S½0;T�dfs:½0;T�3Ω→S;sis F-adaptedg: We consider the cost functionals JAðeð$Þ;sÞdEZT 0 uðt;xðtÞ;eðtÞ;sðtÞÞdt þvðxðTÞÞ � �;(4.16) and JPðsð$ÞÞdEZT 0Uðt;xðtÞ;sðtÞÞdt þVðxðTÞÞ � �;(4.17) and the participation constraint: JA�eð$Þ;s�dEZT 0 u�t;xðtÞ;eðtÞ;sðtÞ�dt þvðxðTÞÞ � �≤C0; IA�eð$Þ;s�dVar ZT 0 f �t;xðtÞ;eðtÞ;sðtÞ�dt þ ψ ðxðTÞÞ � �≤R0: 8 > > > < > > > : (4.18) Just as for the Hidden Action case in Section 4.1 we consider the Agent’sand the Principal’s problem sequentially. The precise statements are: Asian Journal of Economics and Banking 327 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 The Agent’s Problem. Given any sð$Þ∈S½0;T�(fulfilling the participation constraint) the Agent’s problem is to find a process eð$Þ∈E½0;T�minimizing (4.16). The Principal’s Problem. Given that the Agent’s problem has an optimal solution eð$Þ the Principal’s problem is to find a process sð$Þ∈S½0;T�minimizing the cost functional (4.17) subject to the participation constraint (4.18). The mathematical virtue of Hidden Contracts is the possibility of working solely in the strong formulation. For the Agent’s problem we are facing the Hamiltonian HAðt;x;e;p;q;sÞdp$fðt;x;eÞþq$ σ ðt;xÞ�uðt;x;e;sÞ:(4.19) Therefore, by Theorem 2.1 we have for any optimal pair ðxð$Þ;eð$ÞÞthe existence of adjoint processes (p($), q($)) solving the backward stochastic differential equation (BSDE): dpðtÞ ¼ −fx�t;xðtÞ;eðtÞ�pðtÞþ σ x�t;xðtÞ�qðtÞ�ux�t;xðtÞ;eðtÞ�n odt þqðtÞdWt; pðTÞ ¼ −vx�xðTÞ�; 8 < :(4.20) and the characterization eðtÞ ¼ argmax e∈EHA�t;xðtÞ;e;pðtÞ;qðtÞ;sðtÞ�;(4.21) for a.e. t∈[0, T] and P-a.s. As in the hidden contract case we proceed into the principal’s problem by assuming the existence of a function e* such that et¼e*ðt;xðtÞ;pðtÞ;qðtÞ;sðtÞÞ (having sufficient regularity to allow for existence and uniqueness of a solution to (4.22)). The principal is facing the problem of minimizing JPsubject to (4.18) by controlling the following FBSDE: dxt¼f�t;xðtÞ;e*�t;xðtÞ;pðtÞ;qðtÞ;sðtÞ��dt þ σ �t;xðtÞ�dWt; dpðtÞ ¼ −fx�t;xðtÞ;e*�t;xðtÞ;pðtÞ;qðtÞ;sðtÞ��pðtÞþ σ x�t;xðtÞ�qðtÞ n �ux�t;xðtÞ;e*�t;xðtÞ;pðtÞ;qðtÞ;sðtÞ��odt þqðtÞdWt; xð0Þ ¼ 0;pðTÞ ¼ −vx�xðTÞ�: 8 > > > > > > < > > > > > > : (4.22) We now apply Theorem 3.2 in order to characterize optimal cash-flows in the principal’s problem. The associated Hamiltonian is HPðt;x;p;q;s;R;P1;P2;Q1;Q2;λE;λV;λPÞ ¼ R$ðfxðt;x;eðt;x;p;q;sÞÞpþ σ xðt;xÞq�uxðt;x;eðt;x;p;q;sÞ;sÞÞ þP1$fðt;x;eðt;x;p;q;sÞÞþP2$ f ðt;x;eðt;x;p;q;sÞ;sÞf þfðt;x;eðt;x;p;q;sÞÞ ψ 0ðxÞþ σ 2ðt;xÞ 2 ψ 00ðxÞ�þQ1$ σ ðt;xÞþQ2$ σ ðt;xÞ ψ 0ðxÞ �λE$uðt;x;eðt;x;p;q;sÞ;sÞ�λP$Uðt;x;sÞ: (4.23) AJEB 8,3 328 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 For any optimal 4-tuple ðxð$Þ;pð$Þ;qð$Þ;sð$ÞÞ of the principal’s problem we have the existence of Lagrange multipliers λE;λV;λP∈Rsatisfying either of the conditions (i)-(v) in Section 2, with λ P ≥0 and λ2 Eþλ2 Vþλ2 P¼1; and a triple of adjoint processes (R($), P($), Q($)) solving the FBSDE (3.10) so that sðtÞ ¼ argmax s∈SHPðt;xðtÞ;pðtÞ;qðtÞ;s;RðtÞ;P1ðtÞ;P2ðtÞ;Q1ðtÞ;Q2ðtÞ;λE;λPÞ: For the full characterization of optimality we require the following technical assumption: (PA2). All functions involved in the agent’s problem satisfy the Assumption 1 from Section 2. The functions defining the principal’s problem (including composition with the map e*) satisfy the Assumptions 2–6, also from Section 2, and ψ is three times differentiable. Theorem 4.4. Let the statements in (PA2) hold and consider the mean-variance principal– agent problem under hidden contract with participation constraints defined by the given parameters C 0 < 0 and R 0 > 0. Then, if ðeð$Þ;sð$ÞÞis an optimal contract there exist numbers λE;λV;λP∈Rsuch that λP≥0;λ2 Eþλ2 Vþλ2 P¼1; a pair ðpð$Þ;qð$ÞÞ∈L2 Fð0;T;RÞ3ðL2 Fð0;T;RÞÞsolving the BSDE in (4.20) and a quintuple ðRð$Þ;P1ð$Þ;P2ð$Þ;Q1ð$Þ;Q2ð$ÞÞ∈L2 FðΩ;Cð½0;T�;RÞÞ3L2 FðΩ;Cð½0;T�;RÞÞ3L2 Fð0;T; RÞsolving the adjoint FBSDE (3.10) defined by (4.22) such that, sequentially, eðtÞ ¼ argmax e∈EHA�t;xðtÞ;e;pðtÞ;qðtÞ;sðtÞ�; and sðtÞ ¼ argmax s∈SHP�t;xðtÞ;pðtÞ;qðtÞ;s;RðtÞ;P1ðtÞ;P2ðtÞ;Q1ðtÞ;Q2ðtÞ;λE;λV;λP�: with Hamiltonians HAand HPas in (4.19) and (4.23), respectively. 5. A solved example in the case of hidden contracts We now illustrate the method of Section 4 by considering a concrete example of hidden contract type. In order to find explicit solutions we choose a linear-quadratic setup. As a result we get optimal contracts adapted to the filtration generated by output. Consider the following dynamics of production, dxðtÞ ¼ ðaxðtÞþbeðtÞÞdt þ σ dWt;t∈ð0;T�; xð0Þ ¼ 0;a;b∈Rand σ >0; � and let the preferences of the agent and the principal be described by quadratic utility functions: JAðeð$Þ;sÞdEZT 0 ðst�etÞ2 2dt � α $xðTÞ2 2 " #;(5.1) Asian Journal of Economics and Banking 329 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 JPðsð$ÞÞdEZT 0 s2 t 2dt �β$xðTÞ2 2 " #:(5.2) Note that we are following the convention of Section 4 to consider cost-rather than payofffunctionals. Thus, the agent’s utility function should be interpreted as a desire to maintain a level of effort close to the compensation given by the cash-flow. We think of the parameters α > 0 and β> 0 as bonus factors of total production at time T. For the participation constraint we require any admissible cash-flow s(t) to satisfy the following: JA�eð$Þ;s�≤C0; Var xðTÞð Þ <R0; ((5.3) where C 0 < 0, R 0 > 0 and eð$Þdenotes the optimal effort policy of the agent given s($). Assume that the principal offers the agent s($) over the period 0 ≤t≤T. The Hamiltonian function of the agent is HAðx;e;p;q;sÞdp$ðax þbeÞþq$ σ �ðs�eÞ2 2; so vHA ve¼bp þs�e¼0 and eðtÞ ¼ bpðtÞþsðtÞ;(5.4) where the pair (p,q) solves the adjoint equation dpðtÞ ¼ −apðtÞdt þqðtÞdWt; pðTÞ ¼ α xðTÞ: � Turning to the principal’s problem we want to control the FBSDE dxðtÞ ¼ �axðtÞþb2pðtÞþbsðtÞ�dt þ σ dWt; dpðtÞ ¼ −apðtÞdt þqðtÞdWt; xð0Þ ¼ 0;pðTÞ ¼ α xðTÞ; 8 > < > :(5.5) optimally with respect to the cost function (5.2) and the participation constraint (5.3). The principal’s Hamiltonian is HPðx;p;s;R;P1;P2;Q1;Q2;λE;λPÞd �ap$Rþ�ax þb2pþbs�$P1þ�sþax þb2pþbs�$P2þ σ $ðQ1þQ2Þ �λE$b2p2 2�λP$s2 2 (5.6) AJEB 8,3 330 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 so vHP vs¼bP1þð1þbÞP2�λPsand sðtÞ ¼ bP1ðtÞþð1þbÞP2ðtÞ λP ; where the quintuple (R(t), P 1 (t), P 2 (t), Q 1 (t), Q 2 (t)) solves the adjoint FBSDE: dRðtÞ ¼ �aRðtÞ�b2ðP1ðtÞþP2ðtÞÞþλEb2pðtÞ�dt; dP1ðtÞ ¼ −aðP1ðtÞþP2ðtÞÞdt þQ1ðtÞdWt; dP2ðtÞ ¼ Q2ðtÞdWt; Rð0Þ¼0; P1ðTÞ ¼ − α RðTÞþð α λEþβλPÞxðTÞ;P2ðTÞ ¼ 2λVðE½ η ðTÞ�� η ðTÞÞ: 8 > > > > > < > > > > > : (5.7) In this case, however, the auxiliary process η (t) is the same as the output x(t) in which case P2ðTÞ ¼ 2λVðE½xðTÞ�−xðTÞÞ. To solve the BSDE in (5.7) we can make a general linear ansatz: pðtÞ ¼ A11ðtÞxðtÞþB11ðtÞRðtÞþA21ðtÞE½xðtÞ�þB21ðtÞE½RðtÞ�; P1ðtÞ ¼ A12ðtÞxðtÞþB12ðtÞRðtÞþA22ðtÞE½xðtÞ�þB22ðtÞE½RðtÞ�; P2ðtÞ ¼ A13ðtÞxðtÞþB13ðtÞRðtÞþA23ðtÞE½xðtÞ�þB23ðtÞE½RðtÞ�: 8 < :(5.8) Using the standard procedure with It^ o’s lemma it is elementary (but tedious) to derive a set of twelve coupled Riccati equations for the coefficients in (5.8). A numerical example is presented in Figure 2 below. We get the unique semi-explicit solution to the optimal contract feðtÞ;sðtÞg, driven by the optimal dynamics ðxðtÞ;RðtÞÞ. What remains is to find a feasible triple (λ E ,λ V ,λ P ) so that the optimal contract fulfills the participation constraint in (5.3). One way of finding such a triple is for instance by stochastic simulation of ðxðtÞ;RðtÞÞ (e.g. a simple Euler–Maruyama scheme) and then estimate the payoff and the variance in (5.3) by Monte-Carlo techniques for different values of λ P . In Figure 3 we have included the results of such a scheme corresponding to case (iv) of the transversality condition in Corollary 3.2. Note that RðtÞsatisfies the linear ODE. dR dt þ�b2B12 þb2B13 �λEb2B11 �a�RðtÞ ¼ �λEb2A11 �b2A12 �b2A13�xðtÞ; Rð0Þ ¼ 0; 8 > < > :(5.9) so R0 @t1 A¼Zt 0 exp Zs 0 b2B12 þb2B13 �λEb2B11 �a du � �$λEb2A11 �b2A12 �b2A13 � �xds exp Zt 0 b2B12 þb2B13 �λEb2B11 �a ds � � ; Asian Journal of Economics and Banking 331 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 and is by that Fx-adapted. Therefore, in this model the optimal contract feðtÞ;sðtÞg is Fx-adapted and coincides with the corresponding strong solution to the hidden action problem, i.e. when the information set of the principal is generated by output. 6. Conclusion In this paper, we have extended the continuous-time principal–agent problem by incorporating time-inconsistent utility functions, specifically mean-variance utilities. This approach overcomes the limitations of traditional methods that rely on the Bellman principle t t t t t t t t t t t t Source(s): Figure by the authors θθθ λλ λPP P Note(s): Parameter values: a = b = σ = 1, α = 0.2, β = 1, T = 0.03 Source(s): Figure by the authors Figure 2. Solution curves of (5.7) with parameter values chosen as: a5b5 σ 51, α 50.2, β51, λ P 50.1, θ5 π /2, T50.03 Figure 3. Monte-Carlo simulations of JAðeð$Þ;sÞ,JPðsð$ÞÞ and Var(x(T)) as functions of λ P and θ (relating to λ E and λ V via case (iv)) based on 10 6 sample paths at each point AJEB 8,3 332 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025 and the Hamilton–Jacobi–Bellman equation, which are not suitable for time-inconsistent scenarios. By applying the Pontryagin maximum principle for FBSDEs, we have developed a novel method for characterizing optimal contracts under time-inconsistent preferences. This contribution fills a significant gap in the literature. Our framework is particularly relevant for risk management applications in finance and economics, such as portfolio optimization and performance-based compensation. By illustrating the method through a fully solved linear-quadratic example, we demonstrated how our approach can be applied to real-world scenarios involving time-inconsistent preferences. This example underscores the practical value of our results for designing optimal contracts and managing risk. Our study provides a rigorous framework for designing optimal contracts in principal–agent problems with timeinconsistent preferences, which are common in financial settings. By modeling these scenarios, we offer insights into how principals can effectively manage agents who have private information and differing risk preferences. Future research could build on our findings by exploring additional forms of time inconsistency, including hyperbolic-type discounting kernels, and extending the framework to more complex utility functions. Investigating the application of our methods in multi-agent settings or different economic environments may provide further insights. Empirical validation through case studies or simulations would also be beneficial, offering a deeper understanding of the practical performance and robustness of the proposed methods. References Andersson, D. and Djehiche, B. (2011), “A maximum principle for SDEs of mean-field type”, Applied Mathematics and Optimization, Vol. 63 No. 3, pp. 341-356, doi: 10.1007/s00245-010-9123-8. Bensoussan, A. (1982), “Lectures on stochastic control”, in Lecture Notes in Mathematics, Vol. 972, pp. 1-62, doi: 10.1007/bfb0064859. Bismut, J.-M. (1978), “An introductory approach to duality in optimal stochastic control”, SIAM Review, Vol. 20 No. 1, pp. 62-78, doi: 10.1137/1020004. Bj€ ork, T. and Murgoci, A. (2010), “A general theory of Markovian time inconsistent stochastic control problems”, SSRN:1694759. Bj€ ork, T., Murgoci, A. and Zhou, X.Y. (2014), “Mean-variance portfolio optimization with statedependent risk aversion”, Mathematical Finance, Vol. 24 No. 1, pp. 1-24, doi: 10.1111/j.14679965.2011.00515.x. Bj€ ork, T., Khapko, M. and Murgoci, A. (2021), Time-inconsistent Control Theory with Finance Applications, Springer, Berlin, Vol. 732. Buckdahn, R., Djehiche, B. and Li, J. (2011), “A general stochastic maximum principle for sdes of mean-field type”, Applied Mathematics and Optimization, Vol. 64 No. 2, pp. 197-216, doi: 10. 1007/s00245-011-9136-y. Cvitani� c, J. and Zhang, J. (2013), Contract Theory in Continuous-Time Models, Springer, Heidelberg. 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 Optimization, Vol. 59 No. 1, pp. 99-146, doi: 10.1007/s00245-008-9050-0. Djehiche, B. and Helgesson, P. (2014), “The principal-agent problem; a stochastic maximum principle approach”, available at: http://arxiv.org/abs/1410.6392 Djehiche, B. and Huang, M. (2014), “A characterization of sub-game perfect nash equilibria for SDEs of mean field type”, available at: http://arxiv.org/abs/1403.6324 Djehiche, B., Tembine, H. and Tempone, R. (2014), “A stochastic maximum principle for risk-sensitive mean-field type control”, 53rd IEEE Conference on Decision and Control, Vol. 31, pp. 3481-3486, doi: 10.1109/cdc.2014.7039929, available at: http://arxiv.org/abs/1404.1441 Asian Journal of Economics and Banking 333 Downloaded from http://www.emerald.com/ajeb/article-pdf/8/3/310/9503217/ajeb-05-2024-0065.pdf by ZBW German National Library of Economics user on 16 December 2025