arXiv subjects

Dylan Possamaï

Publications and source records attributed to Dylan Possamaï.

At least 19 recordsLinked to original sources

arXiv2026

Forcing and duality-corrected contracts for volatility control

In this paper, we revisit the construction of optimal incentives in continuous-time principal-agent problems with drift and volatility control. Originally, a general approach relying on dynamic programming and second-order backward stochastic differential equations (2BSDEs) was developed by Cvitani\'c, Possama\"i, and Touzi (2018) [8] to determine the optimal form of contracts in this setting. More recently, Chiusolo and Hubert (2026) [5] proposed a BSDE-based approach by introducing an alternative `contractible-volatility' problem for the principal. In addition to the proposed new method, this work highlights that the optimality result of [8] actually hinges on an assumption, stated below as Assumption 2.3, which may not hold in general. Motivated by this, we introduce in this paper a more general class of contracts, parametrised by a function $\psi$ subject to conditions that make the contract revealing for the agent and without loss of generality for the principal. We further provide two natural specifications of $\psi$: one, inspired by the BSDE approach, yielding a forcing-type contract; the other, motivated by the 2BSDE approach, correcting the duality gap when Assumption 2.3 is not satisfied.

math.OC↗

arXiv2026

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

cs.CL↗

arXiv2026

Closed-loop equilibria for Stackelberg games: a story about stochastic targets

We provide a general approach to reformulating any continuous-time stochastic Stackelberg differential game under closed-loop strategies as a single-level optimisation problem with target constraints. More precisely, we consider a Stackelberg game in which the leader and the follower can both control the drift and the volatility of a stochastic output process, in order to maximise their respective expected utility. The aim is to characterise the Stackelberg equilibrium when the players adopt 'closed-loop strategies', i.e. their decisions are based solely on the historical information of the output process, excluding especially any direct dependence on the underlying driving noise, often unobservable in real-world applications. We first show that, by considering the second-order backward stochastic differential equation associated with the continuation utility of the follower as a controlled state variable for the leader, the latter's unconventional optimisation problem can be reformulated as a more standard stochastic control problem with target constraints. Thereafter, adapting the methodology developed by Soner and Touzi (2002a) or Bouchard, Elie and Imbert (2010), the optimal strategies, as well as the corresponding value of the Stackelberg equilibrium, can be characterised through the solution of a well-specified system of Hamilton- Jacobi-Bellman equations. For a more comprehensive insight, we illustrate our approach through a simple example, facilitating both theoretical and numerical detailed comparisons with the solutions under different information structures studied in the literature.

math.OC↗

arXiv2026

Optimal incentive scheme for ESG disclosure

This paper characterises optimal incentive schemes for ESG disclosure in a continuous-time principal-agent setting. We model a risk-averse principal (e.g., a platform or standard-setter) contracting with a team of heterogeneous agents whose disclosure signals are each correlated with a traded climate risk factor. The optimal contract balances incentive provision against the variance of aggregate payouts by leveraging three instruments: own-signal loading, cross-signal loadings across agents, and hedging tilts on the traded asset. We derive closed-form linear optimal controls in a tractable linear-quadratic-Gaussian framework. When the principal is nearly risk-neutral, the contract uses the traded asset purely to hedge the specific `enforcement risk' generated by high-powered incentives. As the principal's risk aversion increases, the optimal scheme converges to a `market-neutral' regime where aggregate asset exposure is eliminated and the cross-signal structure tightens to an `identity pooling' constraint. We characterise this limit analytically as a constrained quadratic program governed by an M-matrix. In the high-risk-aversion regime, heterogeneity creates genuinely new effects absent under symmetry: the cross-section of S-tilts must change sign (unless degenerate), and an agent's own-signal diagonal can turn negative when that row is too strongly exposed to the common traded factor relative to the rest of the group. The results provide a theoretical foundation for `mixed' compensation structures in Regenerative Finance (ReFi), rationalising the use of both stable payments and volatile governance tokens to optimise risk-sharing.

econ.GN↗

arXiv2026

Optimal control of Volterra integral diffusions and application to contract theory

This paper focuses on the optimal control of a class of stochastic Volterra integral equations. Here the coefficients are regular and not assumed to be of convolution type. We show that, under mild regularity assumptions, these equations can be lifted in a Sobolev space, whose Hilbertian structure allows us to attack the problem through a dynamic programming approach. We are then able to use the theory of viscosity solutions on Hilbert spaces to characterise the value function of the control problem as the unique solution of a parabolic equation on Sobolev space. We provide applications and examples to illustrate the usefulness of our theory, in particular for a certain class of time inconsistent principal agent problems. As a byproduct of our analysis, we introduce a new Markovian approximation for Volterra type dynamics.

math.PR↗

arXiv2026

Here, there and everywhere: state-dependent time-inconsistent stochastic control

This paper addresses the challenge of time-inconsistent stochastic control within a continuous-time framework. Its primary focus lies in uncovering a probabilistic representation, specifically in the shape of a system of backward stochastic differential equations (BSDEs). These equations encapsulate the equilibrium value function essential for resolving cases where the present state affecting the target functional triggers the inconsistency. Additionally, the paper offers an application exemplifying this theory through the time-inconsistent linear--quadratic regulator.

math.OC↗

arXiv2025

Variance strikes back: sub-game--perfect Nash equilibria in time-inconsistent $N$-player games, and their mean-field sequel

We investigate a time-inconsistent, non-Markovian finite-player game in continuous time, where each player's objective functional depends non-linearly on the expected value of the state process. As a result, the classical Bellman optimality principle no longer applies. To address this, we adopt a two-layer game-theoretic framework and seek sub-game--perfect Nash equilibria both at the intra-personal level, which accounts for time inconsistency, and at the inter-personal level, which captures strategic interactions among players. We first characterise sub-game--perfect Nash equilibria and the corresponding value processes of all players through a system of coupled backward stochastic differential equations. We then analyse the mean-field counterpart and its sub-game--perfect mean-field equilibria, described by a system of McKean-Vlasov backward stochastic differential equations. Building on this representation, we finally prove the convergence of sub-game--perfect Nash equilibria and their corresponding value processes in the $N$-player game to their mean-field counterparts.

math.PR↗

arXiv2025

Randomisation with moral hazard: a path to existence of optimal contracts

We study a generic principal-agent problem in continuous time on a finite time horizon. We introduce a framework in which the agent is allowed to employ measure-valued controls and characterise the continuation utility as a solution to a specific form of a backward stochastic differential equation driven by a martingale measure. We leverage this characterisation to prove that, under appropriate conditions, an optimal solution to the principal's problem exists, even when constraints on the contract are imposed. In doing so, we employ compactification techniques and, as a result, circumvent the typical challenge of showing well-posedness for a degenerate partial differential equation with potential boundary conditions, where regularity problems often arise.

math.PR↗

arXiv2025

One model to solve them all: 2BSDE families via neural operators

We introduce a mild generative variant of the classical neural operator model, which leverages Kolmogorov--Arnold networks to solve infinite families of second-order backward stochastic differential equations ($2$BSDEs) on regular bounded Euclidean domains with random terminal time. Our first main result shows that the solution operator associated with a broad range of $2$BSDE families is approximable by appropriate neural operator models. We then identify a structured subclass of (infinite) families of $2$BSDEs whose neural operator approximation requires only a polynomial number of parameters in the reciprocal approximation rate, as opposed to the exponential requirement in general worst-case neural operator guarantees.

cs.LG↗

arXiv2025

Mean-field games of optimal stopping: master equation and weak equilibria

We are interested in the study of stochastic games for which each player faces an optimal stopping problem. In our setting, the players may interact through the criterion to optimise as well as through their dynamics. After briefly discussing the N-player game, we formulate the corresponding mean-field problem. In particular, we introduce a weak formulation of the game for which we are able to prove existence of Nash equilibria for a large class of criteria. We also prove that equilibria for the mean-field problem provide approximated Nash equilibria for the N-player game, and we formally derive the master equation associated with our mean-field game.

math.PR↗

arXiv2025

Mind the jumps: when 2BSDEs meet semi-martingales

We construct an aggregated version of the value processes associated with stochastic control problems, where the criterion to optimise is given by solutions to semi-martingale backward stochastic differential equations (BSDEs). The results can be applied to control problems where the triplet of semi-martingale characteristics is controlled in a possibly non-dominated case or where uncertainty about the characteristics is present in the optimisation. The construction also provides a time-consistent system of fully nonlinear conditional expectations on the Skorokhod space. We find the semi-martingale decomposition of the value function and characterise it as the solution to a semi-martingale second-order BSDE. The generality we seek allows for the treatment of controlled diffusions, pure-jump processes, and discrete-time processes in a unified setting.

math.PR↗

arXiv2024

Golden parachutes under the threat of accidents

This paper addresses a continuous-time contracting model that extends the problem introduced by Sannikov and later rigorously analysed by Possamaï and Touzi. In our model, a principal hires a risk-averse agent to carry out a project. Specifically, the agent can perform two different tasks, namely to increase the instantaneous growth rate of the project's value, and to reduce the likelihood of accidents occurring. In order to compensate for these costly actions, the principal offers a continuous stream of payments throughout the entire duration of a contract, which concludes at a random time, potentially resulting in a lump-sum payment. We examine the consequences stemming from the introduction of accidents, modelled by a compound Poisson process that negatively impact the project's value. Furthermore, we investigate whether certain economic scenarii are still characterised by a golden parachute as in Sannikov's model. A golden parachute refers to a situation where the agent stops working and subsequently receives a compensation, which may be either a lump-sum payment leading to termination of the contract or a continuous stream of payments, thereby corresponding to a pension.

math.PR↗

arXiv2024

Reflections on BSDEs

We prove well-posedness results for backward stochastic differential equations (BSDEs) and reflected BSDEs with an optional obstacle process in the case of appropriately weighted $\mathbb{L}^2$-data when the generator is integrated with respect to a possibly purely discontinuous process. This leads to a unified treatment of discrete-time and continuous-time (reflected) BSDEs. We compare our well-posedness results with the current literature and highlight that our results are sharp and cannot be improved within the framework presented here. Finally, we provide sufficient conditions for a comparison principle.

math.PR↗

arXiv2024

A policy iteration algorithm for non-Markovian control problems

In this paper, we propose a new policy iteration algorithm to compute the value function and the optimal controls of continuous time stochastic control problems. The algorithm relies on successive approximations using linear-quadratic control problems which can all be solved explicitly, and only require to solve recursively linear PDEs in the Markovian case. Though our procedure fails in general to produce a non-decreasing sequence like the standard algorithm, it can be made arbitrarily close to being monotone. More importantly, we recover the standard exponential speed of convergence for both the value and the controls, through purely probabilistic arguments which are significantly simpler than in the classical case. Our proof also accommodates non-Markovian dynamics as well as volatility control, allowing us to obtain the first convergence results in the latter case for a state process in multi-dimensions.

math.OC↗

arXiv2023

Non-asymptotic convergence rates for mean-field games: weak formulation and McKean-Vlasov BSDEs

This work is mainly concerned with the so-called limit theory for mean-field games. Adopting the weak formulation paradigm put forward by Carmona and Lacker, we consider a fully non-Markovian setting allowing for drift control and interactions through the joint distribution of players' states and controls. We provide first a characterisation of mean-field equilibria as arising from solutions to a novel kind of McKean-Vlasov backward stochastic differential equations, for which we provide a well-posedness theory. We incidentally obtain there unusual existence and uniqueness results for mean-field equilibria, which do not require short time horizon, separability assumptions on the coefficients, nor Lasry and Lions's monotonicity conditions, but rather smallness, or alternatively regularity, conditions on the terminal reward and a dissipativity condition on the drift. We then take advantage of this characterisation to provide non-asymptotic rates of convergence for the value functions and the Nash-equilibria of the N-player version to their mean-field counterparts, for general open-loop equilibria. An appropriate reformulation of our approach also allows us to treat closed-loop equilibria, and to obtain convergence results for the master equation associated to the problem.

math.PR↗

arXiv2023

On the population size in stochastic differential games

Commuters looking for the shortest path to their destinations, the security of networked computers, hedge funds trading on the same stocks, governments and populations acting to mitigate an epidemic, or employers and employees agreeing on a contact, are all examples of (dynamic) stochastic differential games. In essence, game theory deals with the analysis of strategic interactions among multiple decision-makers. The theory has had enormous impact in a wide variety of fields, but its rigorous mathematical analysis is rather recent. It started with the pioneering work of von Neumann and Morgenstern published in 1944. Since then, game theory has taken centre stage in applied mathematics and related areas. Game theory has also played an important role in unsuspected areas: for instance in military applications, when the analysis of guided interceptor missiles in the 1950s motivated the study of games evolving dynamically in time. Such games (when possibly subject to randomness) are called stochastic differential games. Their study started with the work of Issacs, who crucially recognised the importance of (stochastic) control theory in the area. Over the past few decades since Isaacs's work, a rich theory of stochastic differential game has emerged and branched into several directions. This paper will review recent advances in the study of solvability of stochastic differential games, with a focus on a purely probabilistic technique to approach the problem. Unsurprisingly, the number of players involved in the game is a major factor of the analysis. We will explain how the size of the population impacts the analyses and solvability of the problem, and discuss mean field games as well as the convergence of finite player games to mean field games.

math.PR↗

arXiv2023

Randomness and early termination: what makes a game exciting?

In this paper we revisit an open problem posed by Aldous on the max-entropy win-probability martingale: given two players of equal strength, such that the win-probability is a martingale diffusion, which of these processes has maximum entropy and hence gives the most excitement for the spectators? We study a terminal-boundary value problem for the nonlinear parabolic PDE $2\partial_te(t,x)=\log(-\partial_{xx}e(t,x))$ derived by Aldous and prove its wellposedness and regularity of its solution by combining PDE analysis and probabilistic tools, in particular the reformulation as a stochastic control problem with restricted control set, which allows us to deduce strict ellipticity. We establish key qualitative properties of the solution including concavity, monotonicity, convergence to a steady state for long remaining time and the asymptotic behaviour shortly before the terminal time. Moreover, we construct convergent numerical approximations. The analytical and numerical results allow us to highlight the behaviour of the win-probability process in the present case where the match may end early, in contrast to recent work by Backhoff-Veraguas and Beiglböck where the match always runs the full length.

math.PR↗

arXiv2023

Stability of backward stochastic differential equations: the general case

In this paper, we obtain stability results for backward stochastic differential equations with jumps (BSDEs) in a very general framework. More specifically, we consider a convergent sequence of standard data, each associated to their own filtration, and we prove that the associated sequence of (unique) solutions is also convergent. The current result extends earlier contributions in the literature of stability of BSDEs and unifies several frameworks for numerical approximations of BSDEs and their implementations.

math.PR↗