arXiv subjects

Dylan Possamaï

Publications and source records attributed to Dylan Possamaï.

At least 55 records · Page 3Linked to original sources

arXiv2019

Optimal electricity demand response contracting with responsiveness incentives

Despite the success of demand response programs in retail electricity markets in reducing average consumption, the random responsiveness of consumers to price event makes their efficiency questionable to achieve the flexibility needed for electric systems with a large share of renewable energy. The variance of consumers' responses depreciates the value of these mechanisms and makes them weakly reliable. This paper aims at designing demand response contracts which allow to act on both the average consumption and its variance. The interaction between a risk--averse producer and a risk--averse consumer is modelled through a Principal--Agent problem, thus accounting for the moral hazard underlying demand response contracts. We provide closed--form solution for the optimal contract in the case of constant marginal costs of energy and volatility for the producer and constant marginal value of energy for the consumer. We show that the optimal contract has a rebate form where the initial condition of the consumption serves as a baseline. Further, the consumer cannot manipulate the baseline at his own advantage. The second--best price for energy and volatility are non--constant and non--increasing in time. The price for energy is lower (resp. higher) than the marginal cost of energy during peak--load (resp. off--peak) periods. We illustrate the potential benefit issued from the implementation of an incentive mechanism on the responsiveness of the consumer by calibrating our model with publicly available data. We predict a significant increase of responsiveness under our optimal contract and a significant increase of the producer satisfaction.

math.OC↗

arXiv2019

Bank monitoring incentives under moral hazard and adverse selection

In this paper, we extend the optimal securitisation model of Pagès [50] and Possamaï and Pagès [51] between an investor and a bank to a setting allowing both moral hazard and adverse selection. Following the recent approach to these problems of Cvitanić, Wan and Yang [14], we characterise explicitly and rigorously the so-called credible set of the continuation and temptation values of the bank, and obtain the value function of the investor as well as the optimal contracts through a recursive system of first-order variational inequalities with gradient constraints. We provide a detailed discussion of the properties of the optimal menu of contracts.

econ.GN↗

arXiv2018

Existence and uniqueness results for BSDEs with jumps: the whole nine yards

This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a filtration only assumed to satisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete--time approximations of general martingales.

math.PR↗

arXiv2018

Zero-sum path-dependent stochastic differential games in weak formulation

We consider zero-sum stochastic differential games with possibly path-dependent controlled state. Unlike the previous literature, we allow for weak solutions of the state equation so that the players' controls are automatically of feedback type. Under some restrictions, needed for the a priori regularity of the upper and lower value functions of the game, we show that the game value exists when both the appropriate path-dependent Isaacs condition, and the uniqueness of viscosity solutions of the corresponding path-dependent Isaacs-HJB equation hold. We also provide a general verification argument and a characterisation of saddle-points by means of an appropriate notion of second-order backward SDEs.

math.PR↗

arXiv2018

A tale of a Principal and many many Agents

In this paper, we investigate a moral hazard problem in finite time with lump$-$sum and continuous payments, involving infinitely many Agents with mean field type interactions, hired by one Principal. By reinterpreting the mean$-$field game faced by each Agent in terms of a mean field forward backward stochastic differential equation (FBSDE for short), we are able to rewrite the Principal's problem as a control problem of McKean$-$Vlasov SDEs. We review one general approache to tackle it, introduced recently in [1, 43, 44, 45, 46] using dynamic programming and Hamilton$-$Jacobi$-$Bellman (HJB for short) equations, and mention a second one based on the stochastic Pontryagin maximum principle, which follows [10]. We solve completely and explicitly the problem in special cases, going beyond the usual linear$-$quadratic framework. We finally show in our examples that the optimal contract in the $N-$players' model converges to the mean$-$field optimal contract when the number of agents goes to $+\infty$, thus illustrating in our specific setting the general results of [8].

math.PR↗

arXiv2018

On a class of path-dependent singular stochastic control problems

This paper studies a class of non$-$Markovian singular stochastic control problems, for which we provide a novel probabilistic representation. The solution of such control problem is proved to identify with the solution of a $Z-$constrained BSDE, with dynamics associated to a non singular underlying forward process. Due to the non$-$Markovian environment, our main argumentation relies on the use of comparison arguments for path dependent PDEs. Our representation allows in particular to quantify the regularity of the solution to the singular stochastic control problem in terms of the space and time initial data. Our framework also extends to the consideration of degenerate diffusions, leading to the representation of the solution as the infimum of solutions to $Z-$constrained BSDEs. As an application, we study the utility maximisation problem with transaction costs for non$-$Markovian dynamics.

math.OC↗

arXiv2017

Stochastic control for a class of nonlinear kernels and applications

We consider a stochastic control problem for a class of nonlinear kernels. More precisely, our problem of interest consists in the optimisation, over a set of possibly non-dominated probability measures, of solutions of backward stochastic differential equations (BSDEs). Since BSDEs are nonlinear generalisations of the traditional (linear) expectations, this problem can be understood as stochastic control of a family of nonlinear expectations, or equivalently of nonlinear kernels. Our first main contribution is to prove a dynamic programming principle for this control problem in an abstract setting, which we then use to provide a semi-martingale characterisation of the value function. We next explore several applications of our results. We first obtain a wellposedness result for second order BSDEs (as introduced in [86]) which does not require any regularity assumption on the terminal condition and the generator. Then we prove a nonlinear optional decomposition in a robust setting, extending recent results of [71], which we then use to obtain a super-hedging duality in uncertain, incomplete and nonlinear financial markets. Finally, we relate, under additional regularity assumptions, the value function to a viscosity solution of an appropriate path-dependent partial differential equation (PPDE).

math.PR↗

arXiv2017

Dynamic programming approach to principal-agent problems

We consider a general formulation of the Principal-Agent problem with a lump-sum payment on a finite horizon, providing a systematic method for solving such problems. Our approach is the following: we first find the contract that is optimal among those for which the agent's value process allows a dynamic programming representation, for which the agent's optimal effort is straightforward to find. We then show that the optimization over the restricted family of contracts represents no loss of generality. As a consequence, we have reduced this non-zero sum stochastic differential game to a stochastic control problem which may be addressed by the standard tools of control theory. Our proofs rely on the backward stochastic differential equations approach to non-Markovian stochastic control, and more specifically, on the recent extensions to the second order case.

math.OC↗

arXiv2016

Moral hazard under ambiguity

In this paper, we extend the Holmstroöm and Milgrom problem [47] by adding uncertainty about the volatility of the output for both the Agent and the Principal. We study more precisely the impact of the "Nature" playing against the Agent and the Principal by choosing the worst possible volatility of the output. We solve the first--best and the second--best problems associated with this framework and we show that optimal contracts are in a class of contracts similar to [14, 15], linear with respect to the output and its quadratic variation. We compare our results with the classical problem in [47].

math.OC↗

arXiv2016

Contracting theory with competitive interacting agents

In a framework close to the one developed by Holmström and Milgrom [44], we study the optimal contracting scheme between a Principal and several Agents. Each hired Agent is in charge of one project, and can make efforts towards managing his own project, as well as impact (positively or negatively) the projects of the other Agents. Considering economic Agents in competition with relative performance concerns, we derive the optimal contracts in both first best and moral hazard settings. The enhanced resolution methodology relies heavily on the connection between Nash equilibria and multidimensional quadratic BSDEs. The optimal contracts are linear and each agent is paid a fixed proportion of the terminal value of all the projects of the firm. Besides, each Agent receives his reservation utility, and those with high competitive appetence are assigned less volatile projects, and shall even receive help from the other Agents. From the principal point of view, it is in the firm interest in our model to strongly diversify the competitive appetence of the Agents.

econ.GN↗

arXiv2015

Weak approximation of second-order BSDEs

We study the weak approximation of the second-order backward SDEs (2BSDEs), when the continuous driving martingales are approximated by discrete time martingales. We establish a convergence result for a class of 2BSDEs, using both robustness properties of BSDEs, as proved in Briand, Delyon and Mémin [Stochastic Process. Appl. 97 (2002) 229-253], and tightness of solutions to discrete time BSDEs. In particular, when the approximating martingales are given by some particular controlled Markov chains, we obtain several concrete numerical schemes for 2BSDEs, which we illustrate on specific examples.

math.PR↗

arXiv2015

Second-order BSDEs with jumps: Formulation and uniqueness

In this paper, we define a notion of second-order backward stochastic differential equations with jumps (2BSDEJs for short), which generalizes the continuous case considered by Soner, Touzi and Zhang [Probab. Theory Related Fields 153 (2012) 149-190]. However, on the contrary to their formulation, where they can define pathwise the density of quadratic variation of the canonical process, in our setting, the compensator of the jump measure associated to the jumps of the canonical process, which is the counterpart of the density in the continuous case, depends on the underlying probability measures. Then in our formulation of 2BSDEJs, the generator of the 2BSDEJs depends also on the underlying probability measures through the compensator. But the solution to the 2BSDEJs can still be defined universally. Moreover, we obtain a representation of the $Y$ component of a solution of a 2BSDEJ as a supremum of solutions of standard backward SDEs with jumps, which ensures the uniqueness of the solution.

math.PR↗

arXiv2015

On the Malliavin differentiability of BSDEs

In this paper we provide new conditions for the Malliavin differentiability of solutions of Lipschitz or quadratic BSDEs. Our results rely on the interpretation of the Malliavin derivative as a G{â}teaux derivative in the directions of the Cameron-Martin space. Incidentally, we provide a new formulation for the characterization of the Malliavin-Sobolev type spaces $D^{1,p}$ .

math.PR↗

arXiv2015

A general Doob-Meyer-Mertens decomposition for $g$-supermartingale systems

We provide a general Doob-Meyer decomposition for $g$-supermartingale systems, which does not require any right-continuity on the system. In particular, it generalizes the Doob-Meyer decomposition of Mertens (1972) for classical supermartingales, as well as Peng's (1999) version for right-continuous $g$-supermartingales. As examples of application, we prove an optional decomposition theorem for $g$-supermartingale systems, and also obtain a general version of the well-known dual formation for BSDEs with constraint on the gains-process, using very simple arguments.

math.PR↗

arXiv2015

Utility maximization with random horizon: a BSDE approach

In this paper we study a utility maximization problem with random horizon and reduce it to the analysis of a specific BSDE, which we call BSDE with singular coefficients, when the support of the default time is assumed to be bounded. We prove existence and uniqueness of the solution for the equation under interest. Our results are illustrated by numerical simulations.

math.PR↗

arXiv2015

Robust utility maximization in non-dominated models with 2BSDEs

The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is non-dominated. We propose studying this problem in the framework of second-order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models of [2].

math.PR↗

arXiv2015

A mathematical treatment of bank monitoring incentives

In this paper, we take up the analysis of a principal/agent model with moral hazard introduced in [17], with optimal contracting between competitive investors and an impatient bank monitoring a pool of long-term loans subject to Markovian contagion. We provide here a comprehensive mathematical formulation of the model and show using martingale arguments in the spirit of Sannikov [18] how the maximization problem with implicit constraints faced by investors can be reduced to a classical stochastic control problem. The approach has the advantage of avoiding the more general techniques based on forward-backward stochastic differential equations described in [6] and leads to a simple recursive system of Hamilton-Jacobi-Bellman equations. We provide a solution to our problem by a verification argument and give an explicit description of both the value function and the optimal contract. Finally, we study the limit case where the bank is no longer impatient.

math.PR↗

arXiv2015

General indifference pricing with small transaction costs

We study the utility indifference price of a European option in the context of small transaction costs. Considering the general setup allowing consumption and a general utility function at final time T, we obtain an asymptotic expansion of the utility indifference price as a function of the asymptotic expansions of the utility maximization problems with and without the European contingent claim. We use the tools developed in [54] and [48] based on homogenization and viscosity solutions to characterize these expansions. Finally we study more precisely the example of exponential utilities, in particular recovering under weaker assumptions the results of [6].

math.OC↗