arXiv subjects

Nizar Touzi

Publications and source records attributed to Nizar Touzi.

At least 19 recordsLinked to original sources

arXiv2026

Backward SDE characterization of the finite horizon Principal-Agent problem

We consider the finite horizon continuous-time Principal--Agent problem under deterministic discount factors. Following the Sannikov reduction to a stochastic control problem, we provide a further characterization of the Principal's value function in terms of a backward SDE inducing the corresponding optimal contract. In particular, this allows to bypass the fully nonlinear HJB equation satisfied by the Principal value function in the Markovian setting. This new approach allows to handle a new class of Principal-Agent problems which was not accessible with the existing method, namely the setting where the Agent faces a regime-switching control problem.

math.OC↗

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 Cvitanić, Possamaï, 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 $ψ$ 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 $ψ$: 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

Optimal resource allocation for maintaining system solvency

We study two optimal allocation problems for a system of independent Brownian agents whose states evolve under a limited shared control. At each time, a unit of resource can be divided and allocated across components to increase their drifts, with the objective of maximizing either (i) the probability that all components avoid ruin, or (ii) the expected number of components that avoid ruin. We identify drift thresholds separating trivial and nontrivial regimes, and derive the associated Hamilton-Jacobi-Bellman equations on the positive orthant with mixed boundary conditions at the absorbing boundary and at infinity. We also establish the existence, uniqueness, and regularity of a bounded classical solution and a verification theorem linking the PDE to the stochastic control value function. Finally, we prove a conjecture on the optimality of a socialistic allocation rule : the push-the-laggard strategy. It is optimal for the all-survive value function, while it is suboptimal for the count-survivors criterion.

math.OC↗

arXiv2026

Forward Hedging Reshapes Incentive Provision

We study how forward hedging reshapes incentive provision inside the firm. We consider a risk-averse producer facing demand and production risk that can either operate in-house or delegate production to a risk-averse agent under moral hazard, while hedging output in a competitive forward market with a rational market maker. Within a tractable continuous-time CARA framework, we jointly characterize optimal production, compensation, and static hedging in equilibrium. Delegation and external hedging are partial substitutes because both create value through risk sharing. Delegation can increase firm value even when the agent uses the same technology and is more risk averse than the principal, while access to forward hedging reduces the need to provide incentives through risk exposure. This mechanism delivers two main results. First, the principal hedges less under delegation than under in-house production. Second, this lower hedging demand under delegation raises the equilibrium forward price relative to the integrated benchmark. In the constant-demand case, we show that access to hedging lowers the agent's expected compensation under delegation. Numerical results indicate that this mechanism remains robust in the presence of demand uncertainty. More broadly, our results show that external risk transfer through financial markets feeds back into internal organizational design.

q-fin.MF↗

arXiv2026

Path-Dependent Ergodic Optimal Control and Backward Stochastic Differential Equations

We investigate a new class of infinite-horizon backward stochastic differential equations for ergodic optimal control where the cost and state dynamics are time and path-dependent. The state process is defined on an unbounded underlying domain and satisfies an extended dissipativity condition. In contrast with the time-homogeneous Markovian setting, the optimal ergodic cost in our framework is characterized by the asymptotic behavior of a deterministic function, rather than by a single real constant. We obtain well-posedness, verification and stability properties, which extend the previous results in the literature on the Markov case.

math.PR↗

arXiv2026

Sensitivity Analysis of Distributionally Robust BSDEs and RBSDEs

We examine the sensitivity properties of backward stochastic differential equations and reflected backward stochastic differential equations, which naturally arise in the context of optimal control and optimal stopping problems. Motivated by issues of sensitivity analysis in distributionally robust optimization (DRO) control and optimal stopping problems, we establish explicit formulas for the corresponding sensitivities under drift reference measure uncertainty. Our work is closely related to \citeauthor{bartl2023sensitivity} \cite{bartl2023sensitivity}. In contrast to the existing literature, our analysis is carried out within a general non-Markovian framework.

math.OC↗

arXiv2026

Bridging Schrödinger and Bass: A Semimartingale Optimal Transport Problem with Diffusion Control

We study a semimartingale optimal transport problem interpolating between the Schrödinger bridge and the stretched Brownian motion associated with the Bass solution of the Skorokhod embedding problem. The cost combines an entropy term on the drift with a quadratic penalization of the diffusion coefficient, leading to a stochastic control problem over drift and volatility. We establish a complete duality theory for this problem, despite the lack of coercivity in the diffusion component. In particular, we prove strong duality and dual attainment, and derive an equivalent reduced dual formulation in terms of a variational problem over terminal potentials. Optimal solutions are characterized by a coupled Schrödinger-Bass bridge system, involving a backward heat potential and a transport map given by the gradient of a $β$-convex function. This system interpolates between the classical Schrödinger system and the Bass martingale transport. Our results furnish a unified framework encompassing entropic and martingale optimal transport, and yield a variational foundation for data-driven diffusion models.

math.PR↗

arXiv2026

Tweedie's Formulae and Diffusion Generative Models Beyond Gaussian

Diffusion models have achieved remarkable success in generating samples from unknown data distributions. Most popular stochastic differential equation-based diffusion models perturb the target distribution by adding Gaussian noise, transforming it into a simple prior, and then use denoising score matching, a consequence of Tweedie's formula, to learn the score function and generate clean samples from noise. However, non-Gaussian diffusion models with state-dependent diffusion coefficient have been largely underexplored, as have the corresponding Tweedie's formulae. In this work, we extend Tweedie's formula to important non-Gaussian processes, including geometric Brownian motion (GBM), squared Bessel (BESQ) processes, and Cox-Ingersoll-Ross (CIR) processes, thereby yielding the corresponding denoising score-matching objectives. We then apply the derived formulae to image and financial time series generation using GBM- and CIR-based diffusion models, and to empirical Bayes estimation under the BESQ setting. The reported experimental results demonstrate the potential of non-Gaussian models.

stat.ML↗

arXiv2026

LightSBB-M: Bridging Schrödinger and Bass for Generative Diffusion Modeling

The Schrodinger Bridge and Bass (SBB) formulation, which jointly controls drift and volatility, is an established extension of the classical Schrodinger Bridge (SB). Building on this framework, we introduce LightSBB-M, an algorithm that computes the optimal SBB transport plan in only a few iterations. The method exploits a dual representation of the SBB objective to obtain analytic expressions for the optimal drift and volatility, and it incorporates a tunable parameter beta greater than zero that interpolates between pure drift (the Schrodinger Bridge) and pure volatility (Bass martingale transport). We show that LightSBB-M achieves the lowest 2-Wasserstein distance on synthetic datasets against state-of-the-art SB and diffusion baselines with up to 32 percent improvement. We also illustrate the generative capability of the framework on an unpaired image-to-image translation task (adult to child faces in FFHQ). These findings demonstrate that LightSBB-M provides a scalable, high-fidelity SBB solver that outperforms existing SB and diffusion baselines across both synthetic and real-world generative tasks. The code is available at https://github.com/alexouadi/LightSBB-M.

cs.LG↗

arXiv2026

Particle system approximation of Nash equilibria in large games

We develop a probabilistic framework to approximate Nash equilibria in symmetric $N$-player games in the large population regime, via the analysis of associated mean field games (MFGs). The approximation is achieved through the analysis of a McKean-Vlasov type Langevin dynamics and their associated particle systems, with convergence to the MFG solution established in the limit of vanishing temperature parameter. Relying on displacement monotonicity or Lasry-Lions monotonicity of the cost function, we prove contractility of the McKean-Vlasov process and uniform-in-time propagation of chaos for the particle system. Our results contribute to the general theory of interacting diffusions by showing that monotonicity can ensure convergence without requiring small interaction assumptions or functional inequalities.

math.PR↗

arXiv2026

On Approximate Nash Equilibria in Mean Field Games

In the context of large population symmetric games, approximate Nash equilibria are introduced through equilibrium solutions of the corresponding mean field game in the sense that the individual gain from optimal unilateral deviation under such strategies converges to zero in the large population size asymptotic. We show that these strategies satisfy an $Ł^\infty$ notion of approximate Nash equilibrium which guarantees that the individual gain from optimal unilateral deviation is small uniformly among players and uniformly on their initial characteristics. We establish these results in the context of static models and in the dynamic continuous time setting, and we cover situations where the agents' criteria depend on the conditional law of the controlled state process.

cs.GT↗

arXiv2026

A PDE Derivation of the Schrödinger--Bass Bridge

This short paper announces the main results of \cite{SBB2026}, where the Schrödinger--Bass Bridge (SBB) problem is introduced and studied in full generality. Here we provide a direct PDE derivation of the SBB system in dimension one, showing how the optimal coupling problem that interpolates between the classical Schrödinger bridge and the Bass martingale transport can be solved explicitly via Legendre transforms and the heat equation. A key insight is that the optimal SBB process is a Stretched Schrödinger Bridge: the composition of a monotone transport map with a Schrödinger bridge. This extends the stretched Brownian motion representation of Bass martingales to the semimartingale setting and provides a unified framework that recovers both the Sinkhorn algorithm (in the limit $β\to \infty$) and the Bass construction (as $β\to 0$). We refer to \cite{SBB2026} for complete proofs, the multidimensional setting, strong duality, dual attainment, and further developments.

math.PR↗

arXiv2025

Itō and Itō-Wentzell chain rule for flows of conditional laws of continuous semimartingales: an easy approach

We provide a general Itō\,-Wentzell formula for a random field of maps on the Wasserstein space of probability measures, defined by continuous semimartingales, and evaluated along the flow of conditional distributions of another continuous semimartingale. Our method follows standard arguments of Itō calculus, and thus bypasses the approximation by empirical measures commonly used in the existing literature. As an application, we derive the dynamic programming equation for a mean field stochastic control problem with common noise.

math.PR↗

arXiv2025

First order Martingale model risk and semi-static hedging

We investigate model risk distributionally robust sensitivities for functionals on the Wasserstein space when the underlying model is constrained to the martingale class and/or is subject to constraints on the first marginal law. Our results extend the findings of Bartl, Drapeau, Obloj \& Wiesel \cite{bartl2021sensitivity} and Bartl \& Wiesel \cite{bartlsensitivityadapted} by introducing the minimization of the distributionally robust problem with respect to semi-static hedging strategies. We provide explicit characterizations of the model risk (first order) optimal semi-static hedging strategies. The distributional robustness is analyzed both in terms of the adapted Wasserstein metric and the more relevant standard Wasserstein metric.

q-fin.MF↗

arXiv2025

Itô-Wentzell formulas for semimartingale conditional laws with applications to mean-field control

The present paper is an extension of Fadle-Touzi (2024). Following the same methodology, merely based on Taylor expansions, we establish the Itô and Itô-Wentzell formulae for flows of conditional distributions of general semimartingales, thus allowing for discontinuous semimartingales with possibly discontinuous flows of conditional marginals. We apply these results to derive the dynamic programming equations corresponding to mean field control problems with Poisson type common noise and mean field stopping problems with common noise.

math.PR↗

arXiv2025

Viscosity Solutions for HJB Equations on the Process Space

In this paper we investigate a path dependent optimal control problem on the process space with both drift and volatility controls, with possibly degenerate volatility. The dynamic value function is characterized by a fully nonlinear second order path dependent HJB equation on the process space, which is by nature infinite dimensional. In particular, our model covers mean field control problems with common noise as a special case. We shall introduce a new notion of viscosity solutions and establish both the existence and the comparison principle, under merely Lipschitz/Holder continuity assumptions. The main feature of our notion is that, besides the standard smooth part, the test function consists of an extra singular component which allows us to handle the second order derivatives of the smooth test functions without invoking the Crandall-Ishii lemma. We shall use the doubling variable arguments, combined with the Ekeland-Borwein-Preiss variational principle in order to overcome the noncompactness of the state space. A smooth gauge-type function on the path space is crucial for our estimates.

math.OC↗

arXiv2025

Sensitivity of functionals of McKean-Vlasov SDE's with respect to the initial distribution

We examine the sensitivity at the origin of the distributional robust optimization problem in the context of a model generated by a mean field stochastic differential equation. We adapt the finite dimensional argument developed by Bartl, Drapeau, Obloj \& Wiesel to our framework involving the infinite dimensional gradient of the solution of the mean field SDE with respect to its initial data. We revisit the derivation of this gradient process as previously introduced by Buckdahn, Li \& Peng, and we complement the existing properties so as to satisfy the requirement of our main result.

math.PR↗

arXiv2025

A Principal-Agent Model for Optimal Incentives in Renewable Investments

We investigate the optimal regulation of energy production in alignment with the long-term goals of the Paris Climate Agreement. We analyze the optimal regulatory incentives to foster the development of non-emissive electricity generation when the demand for power is met either by a single firm or by two interacting agents. The regulator aims to encourage green investments to limit carbon emissions while simultaneously reducing the intermittency of total energy production. We find that the regulator can achieve a higher certainty equivalent by regulating two interacting firms, each investing in one technology, rather than a single firm managing both technologies. This higher value is achieved thanks to a greater degree of freedom in the incentive mechanisms, which involve cross-subsidies between firms. Moreover, we find that it is optimal to compensate firms for shutting down their emissive production assets. We provide closed-form expressions of the second-best contracts and show that they take a rebate form, involving time-dependent prices for each state variable. A numerical study quantifies the impact of the designed second-best contract in both market structures compared to the business-as-usual scenario.

econ.GN↗