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Nizar Touzi

Publications and source records attributed to Nizar Touzi.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2024

Mean Field Game of Mutual Holding with common noise

We consider the mean field game of cross--holding introduced in \citeauthor*{DjeteTouzi} \cite{DjeteTouzi} in the context where the equity value dynamics are affected by a common noise. In contrast with \cite{DjeteTouzi}, the problem exhibits the standard paradigm of mean--variance trade off. Our crucial observation is to search for equilibrium solutions of our mean field game among those models which satisfy an appropriate notion of no--arbitrage. Under this condition, it follows that the representative agent optimization step is reduced to a standard portfolio optimization problem with random endowment.

math.PR↗
arXiv2023

Impact of carbon market on production emissions

The aim of this paper is to address the effect of the carbon emission allowance market on the production policy of a large polluter production firm. We investigate this effect in two cases; when the large polluter cannot affect the risk premium of the allowance market, and when it can change the risk premium by its production. In this simple model, we ignore any possible investment of the firm in pollution reducing technologies. We formulate the problem of optimal production by a stochastic optimization problem. Then, we show that, as expected, the market reduces the optimal production policy in the first case if the firm is not given a generous initial cheap allowance package. However, when the large producer activities can change the market risk premium, the cut on the production and consequently pollution cannot be guaranteed. In fact, there are cases in this model when the optimal production is {\it always} larger than expected, and an increase in production, and thus pollution, can increase the profit of the firm. We conclude that some of the parameters of the market which contribute to this effect can be wisely controlled by the regulators in order to diminish this manipulative behavior of the firm.

math.OC↗
arXiv2023

From finite population optimal stopping to mean field optimal stopping

This paper analyzes the convergence of the finite population optimal stopping problem towards the corresponding mean field limit. Building on the viscosity solution characterization of the mean field optimal stopping problem of our previous papers [Talbi, Touzi & Zhang 2021 & 2022], we prove the convergence of the value functions by adapting the Barles-Souganidis [1991] monotone scheme method to our context. We next characterize the optimal stopping policies of the mean field problem by the accumulation points of the finite population optimal stopping strategies. In particular, if the limiting problem has a unique optimal stopping policy, then the finite population optimal stopping strategies do converge towards this solution. As a by-product of our analysis, we provide an extension of the standard propagation of chaos to the context of stopped McKean-Vlasov diffusions.

math.PR↗
arXiv2023

Entropic optimal planning for path-dependent mean field games

In the context of mean field games, with possible control of the diffusion coefficient, we consider a path-dependent version of the planning problem introduced by P.L. Lions: given a pair of marginal distributions $(μ_0, μ_1)$, find a specification of the game problem starting from the initial distribution $μ_0$, and inducing the target distribution $μ_1$ at the mean field game equilibrium. Our main result reduces the path-dependent planning problem into an embedding problem, that is, constructing a McKean-Vlasov dynamics with given marginals $(μ_0,μ_1)$. Some sufficient conditions on $(μ_0,μ_1)$ are provided to guarantee the existence of solutions. We also characterize, up to integrability, the minimum entropy solution of the planning problem. In particular, as uniqueness does not hold anymore in our path-dependent setting, one can naturally introduce an optimal planning problem which would be reduced to an optimal transport problem along with controlled McKean-Vlasov dynamics.

math.OC↗
arXiv2023

Mean field game of mutual holding with defaultable agents, and systemic risk

We introduce the possibility of default in the mean field game of mutual holding of Djete and Touzi [11]. This is modeled by introducing absorption at the origin of the equity process. We provide an explicit solution of this mean field game. Moreover, we provide a particle system approximation, and we derive an autonomous equation for the time evolution of the default probability, or equivalently the law of the hitting time of the origin by the equity process. The systemic risk is thus described by the evolution of the default probability.

math.PR↗
arXiv2023

Dynamic programming equation for the mean field optimal stopping problem

We study the optimal stopping problem of McKean-Vlasov diffusions when the criterion is a function of the law of the stopped process. A remarkable new feature in this setting is that the stopping time also impacts the dynamics of the stopped process through the dependence of the coefficients on the law. The mean field stopping problem is introduced in weak formulation in terms of the joint marginal law of the stopped underlying process and the survival process. This specification satisfies a dynamic programming principle. The corresponding dynamic programming equation is an obstacle problem on the Wasserstein space, and is obtained by means of a general Itô formula for flows of marginal laws of càdlàg semimartingales. Our verification result characterizes the nature of optimal stopping policies, highlighting the crucial need to randomized stopping. The effectiveness of our dynamic programming equation is illustrated by various examples including the mean-variance optimal stopping problem.

math.PR↗
arXiv2022

Viscosity solutions for obstacle problems on Wasserstein space

This paper is a continuation of our accompanying paper [Talbi, Touzi and Zhang (2021)], where we characterized the mean field optimal stopping problem by an obstacle equation on the Wasserstein space of probability measures, provided that the value function is smooth. Our purpose here is to establish this characterization under weaker regularity requirements. We shall define a notion of viscosity solutions for such equation, and prove existence, stability, and comparison principle.

math.PR↗
arXiv2022

Is there a Golden Parachute in Sannikov's principal-agent problem?

This paper provides a complete review of the continuous-time optimal contracting problem introduced by Sannikov, in the extended context allowing for possibly different discount rates for both parties. The agent's problem is to seek for optimal effort, given the compensation scheme proposed by the principal over a random horizon. Then, given the optimal agent's response, the principal determines the best compensation scheme in terms of running payment, retirement, and lump-sum payment at retirement. A Golden Parachute is a situation where the agent ceases any effort at some positive stopping time, and receives a payment afterwards, possibly under the form of a lump sum payment, or of a continuous stream of payments. We show that a Golden Parachute only exists in certain specific circumstances. This is in contrast with the results claimed by Sannikov, where the only requirement is a positive agent's marginal cost of effort at zero. Namely, we show that there is no Golden Parachute if this parameter is too large. Similarly, in the context of a concave marginal utility, there is no Golden Parachute if the agent's utility function has a too negative curvature at zero. In the general case, we prove that an agent with positive reservation utility is either never retired by the principal, or retired above some given threshold (as in Sannikov's solution). We show that different discount factors induce a face-lifted utility function, which allows to reduce the analysis to a setting similar to the equal discount rates one. Finally, we also confirm that an agent with small reservation utility may have an informational rent, meaning that the principal optimally offers him a contract with strictly higher utility than his participation value.

econ.TH↗
arXiv2022

Mean Field Game of Mutual Holding

We introduce a mean field model for optimal holding of a representative agent of her peers as a natural expected scaling limit from the corresponding $N-$agent model. The induced mean field dynamics appear naturally in a form which is not covered by standard McKean-Vlasov stochastic differential equations. We study the corresponding mean field game of mutual holding in the absence of common noise. Our first main result provides an explicit equilibrium of this mean field game, defined by a bang--bang control consisting in holding those competitors with positive drift coefficient of their dynamic value. We next use this mean field game equilibrium to construct (approximate) Nash equilibria for the corresponding $N$--player game. We also provide some numerical illustrations of our mean field game equilibrium which highlight some unexpected effects induced by our results.

math.OC↗
arXiv2022

Second order backward SDE with random terminal time

Backward stochastic differential equations extend the martingale representation theorem to the nonlinear setting. This can be seen as path-dependent counterpart of the extension from the heat equation to fully nonlinear parabolic equations in the Markov setting. This paper extends such a nonlinear representation to the context where the random variable of interest is measurable with respect to the information at a finite stopping time. We provide a complete wellposedness theory which covers the semilinear case (backward SDE), the semilinear case with obstacle (reflected backward SDE), and the fully nonlinear case (second order backward SDE).

math.PR↗
arXiv2022

Nonlinear predictable representation and $L^1$-solutions of backward SDEs and second-order backward SDEs

The theory of backward SDEs extends the predictable representation property of Brownian motion to the nonlinear framework, thus providing a path-dependent analog of fully nonlinear parabolic PDEs. In this paper, we consider backward SDEs, their reflected version, and their second-order extension, in the context where the final data and the generator satisfy $L^1$-type of integrability condition. Our main objective is to provide the corresponding existence and uniqueness results for general Lipschitz generators. The uniqueness holds in the so-called Doob class of processes, simultaneously under an appropriate class of measures. We emphasize that the previous literature only deals with backward SDEs, and requires either that the generator is separable in $(y,z)$, see Peng [Pen97], or strictly sublinear in the gradient variable $z$, see [BDHPS03], or that the final data satisfies an $L\ln L$-integrability condition, see [HT18]. We by-pass these conditions by defining $L^1$-integrability under the nonlinear expectation operator induced by the previously mentioned class of measures.

math.PR↗
arXiv2022

Random horizon principal-agent problems

We consider a general formulation of the random horizon Principal-Agent problem with a continuous payment and a lump-sum payment at termination. In the European version of the problem, the random horizon is chosen solely by the principal with no other possible action from the agent than exerting effort on the dynamics of the output process. We also consider the American version of the contract, which covers the seminal Sannikov's model, where the agent can also quit by optimally choosing the termination time of the contract. Our main result reduces such non-zero-sum stochastic differential games to appropriate stochastic control problems which may be solved by standard methods of stochastic control theory. This reduction is obtained by following Sannikov's approach, further developed by Cvitanic, Possamai, and Touzi. We first introduce an appropriate class of contracts for which the agent's optimal effort is immediately characterized by the standard verification argument in stochastic control theory. We then show that this class of contracts is dense in an appropriate sense so that the optimization over this restricted family of contracts represents no loss of generality. The result is obtained by using the recent well-posedness result of random horizon second-order backward SDE.

math.OC↗
arXiv2022

On path-dependent multidimensional forward-backward SDEs

This paper extends the results of Ma, Wu, Zhang, Zhang [11] to the context of path-dependent multidimensional forward-backward stochastic differential equations (FBSDE). By path-dependent we mean that the coefficients of the forward-backward SDE at time t can depend on the whole path of the forward process up to time t. Such a situation appears when solving path-dependent stochastic control problems by means of variational calculus. At the heart of our analysis is the construction of a decoupling random field on the path space. We first prove the existence and the uniqueness of decoupling field on small time interval. Then by introducing the characteristic BSDE, we show that a global decoupling field can be constructed by patching local solutions together as long as the solution of the characteristic BSDE remains bounded. Finally, we provide a stability result for path-dependent forward-backward SDEs.

math.PR↗
arXiv2019

On the Root solution to the Skorokhod embedding problem given full marginals

This paper examines the Root solution of the Skorohod embedding problem given full marginals on some compact time interval. Our results are obtained by limiting arguments based on finitely-many marginals Root solution of Cox, Oblój and Touzi. Our main result provides a characterization of the corresponding potential function by means of a convenient parabolic PDE.

math.OC↗
arXiv2019

Optimal make-take fees for market making regulation

We consider an exchange who wishes to set suitable make-take fees to attract liquidity on its platform. Using a principal-agent approach, we are able to describe in quasi-explicit form the optimal contract to propose to a market maker. This contract depends essentially on the market maker inventory trajectory and on the volatility of the asset. We also provide the optimal quotes that should be displayed by the market maker. The simplicity of our formulas allows us to analyze in details the effects of optimal contracting with an exchange, compared to a situation without contract. We show in particular that it leads to higher quality liquidity and lower trading costs for investors.

q-fin.TR↗
arXiv2019

Continuous-Time Principal-Agent Problem in Degenerate Systems

In this paper we present a variational calculus approach to Principal-Agent problem with a lump-sum payment on finite horizon in degenerate stochastic systems, such as filtered partially observed linear systems. Our work extends the existing methodologies in the Principal-Agent literature using dynamic programming and BSDE representation of the contracts in the non-degenerate controlled stochastic systems. We first solve the Principal's problem in an enlarged set of contracts defined by a forward-backward SDE system given by the first order condition of the Agent's problem using variational calculus. Then we use the sufficient condition of the Agent's problem to verify that the optimal contract that we obtain by solving the Principal's problem is indeed implementable (i.e. belonging to the admissible contract set). Importantly we consider the control problem in a weak formulation. Finally, we give explicit solution of the Principal-Agent problem in partially observed linear systems and extend our results to some mean field interacting Agents case.

math.PR↗