# Dylan PossamaïETH Zurich | ETH Zürich · Department of Mathematics

Dylan Possamaï

PhD

## About

76

Publications

10,647

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Introduction

Additional affiliations

August 2020 - present

September 2017 - July 2020

September 2015 - August 2016

Education

October 2009 - December 2011

September 2008 - August 2009

September 2005 - August 2008

## Publications

Publications (76)

Motivated by the recent studies on the green bond market, we build a Principal–Agent model in which an investor trades on a portfolio of green and conventional bonds, both issued by the same governmental entity. The government provides incentives to the bondholder in order to increase the amount invested in green bonds. These incentives are, optima...

In this work, we provide a general mathematical formalism to study the optimal control of an epidemic, such as the COVID-19 pandemic, via incentives to lockdown and testing. In particular, we model the interplay between the government and the population as a principal–agent problem with moral hazard, à la Cvitanić et al. (Finance Stoch 22(1):1–37,...

We design a market-making model \`a la Avellaneda-Stoikov in which the market-takers act strategically, in the sense that they design their trading strategy based on an exogenous trading signal. The market-maker chooses her quotes based on the average market-takers' behaviour, modelled through a mean-field interaction. We derive, up to the resoluti...

Demand response programs in retail electricity markets are very popular. However, despite their success in reducing average consumption, the random responsiveness of consumers to price events makes their efficiency questionable to achieve the flexibility needed for electric systems with a large share of renewable energy. This paper aims at designin...

We study a McKean–Vlasov optimal control problem with common noise in order to establish the corresponding limit theory as well as the equivalence between different formulations, including strong, weak, and relaxed formulations. In contrast to the strong formulation, in which the problem is formulated on a fixed probability space equipped with two...

In this paper we study a pollution regulation problem in an electricity market with a network structure. The market is ruled by an independent system operator (ISO for short) who has the goal of reducing the pollutant emissions of the providers in the network, by encouraging the use of cleaner technologies. The problem of the ISO formulates as a co...

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 resu...

This work is mainly concerned with the so-called limit theory for mean-field games. Adopting the weak formulation paradigm put forward by \citeauthor*{carmona2015probabilistic} \cite{carmona2015probabilistic}, we consider a fully non-Markovian setting allowing for drift control, and interactions through the joint distribution of players' states and...

We study risk-sharing economies where heterogeneous agents trade subject to quadratic transaction costs. The corresponding equilibrium asset prices and trading strategies are characterised by a system of nonlinear, fully coupled forward–backward stochastic differential equations. We show that a unique solution exists provided that the agents’ prefe...

Motivated by the recent studies on the green bond market, we build a model in which an investor trades on a portfolio of green and conventional bonds, both issued by the same governmental entity. The government provides incentives to the bondholder in order to increase the amount invested in green bonds. These incentives are, optimally, indexed on...

We study the problem of demand response contracts in electricity markets by quantifying the impact of considering a continuum of consumers with mean–field interaction, whose consumption is impacted by a common noise. We formulate the problem as a Principal–Agent problem with moral hazard in which the Principal—she—is an electricity producer who obs...

We consider the control of the COVID-19 pandemic via incentives, through either stochastic SIS or SIR compartmental models. When the epidemic is ongoing, the population can reduce interactions between individuals in order to decrease the rate of transmission of the disease, and thus limit the epidemic. However, this effort comes at a cost for the p...

We study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite dimensional system of standard backward SDEs and establish its well-posedness, and we show that it is equivalent to that of a type-I backward stochastic Volterra integral equation. We also establish a...

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 of 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 opt...

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 th...

In this paper, we extend the optimal securitization model of Pag\`es [41] and Possama\"i and Pag\`es [42] between an investor and a bank to a setting allowing both moral hazard and adverse selection. Following the recent approach to these problems of Cvitani\'c, Wan and Yang [12], we characterize explicitly and rigorously the so-called credible set...

We develop a theory for continuous-time non-Markovian stochastic control problems which are inherently time-inconsistent. Their distinguishing feature is that the classical Bellman optimality principle no longer holds. Our formulation is cast within the framework of a controlled non-Markovian forward stochastic differential equation, and a general...

We study a McKean-Vlasov optimal control problem with common noise, in order to establish the corresponding limit theory, as well as the equivalence between different formulations, including the strong, weak and relaxed formulation. In contrast to the strong formulation, where the problem is formulated on a fixed probability space equipped with two...

Following the recent literature on make take fees policies, we consider an exchange wishing to set a suitable contract with several market makers in order to improve trading quality on its platform. To do so, we use a principal-agent approach, where the agents (the market makers) optimise their quotes in a Nash equilibrium fashion, providing best r...

We study the McKean-Vlasov optimal control problem with common noise in various formulations, namely the strong and weak formulation, as well as the Markovian and non-Markovian formulations, and allowing for the law of the control process to appear in the state dynamics. By interpreting the controls as probability measures on an appropriate canonic...

In this article, we propose a wellposedness theory for a class of second
order backward doubly stochastic differential equation (2BDSDE). We prove
existence and uniqueness of the solution under a Lipschitz type assumption on
the generator, and we investigate the links between our 2BDSDEs and a class of
parabolic Fully nonLinear Stochastic PDes. Pre...

In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales, each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated...

We study the problem of demand response contracts in electricity markets by quantifying the impact of considering a mean-field of consumers, whose consumption is impacted by a common noise. We formulate the problem as a Principal-Agent problem with moral hazard in which the Principal - she - is an electricity producer who observes continuously the...

We study a risk-sharing equilibrium where heterogenous agents trade subject to quadratic transaction costs. The corresponding equilibrium asset prices and trading strategies are characterised by a system of nonlinear, fully-coupled forward-backward stochastic differential equations. We show that a unique solution generally exists provided that the...

We study a risk-sharing equilibrium where heterogenous agents trade subject to quadratic transaction costs. The corresponding equilibrium asset prices and trading strategies are characterised by a system of nonlinear, fully-coupled forward-backward stochastic differential equations. We show that a unique solution generally exists provided that the...

In this paper, we extend the Holmstr\"{o}m and Milgrom problem [30] 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 sec...

Despite the success of demand response programs in retail electricity markets in reducing average consumption, the literature shows failure to reduce the variance of consumers' responses. This paper aims at designing demand response contracts which allow to act on both the average consumption and its variance. The interaction between the producer a...

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 th...

In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated...

We consider a general formulation of the Principal-Agent problem from
Contract Theory, on a finite horizon. We show how to reduce the problem to a
stochastic control problem which may be analyzed by the standard tools of
control theory. In particular, Agent's value function appears naturally as a
controlled state variable for the Principal's proble...

Despite the success of demand response programs in retail electricity markets in reducing average consumption, the literature shows failure to reduce the variance of consumers' responses. This paper aims at designing demand response contracts which allow to act on both the average consumption and its variance. The interaction between the producer a...

The aim of this short note is to fill in a gap in our earlier paper [7] on 2BSDEs with reflections, and to explain how to correct the subsequent results in the second paper [6]. We also provide more insight on the properties of 2RBSDEs, in the light of the recent contributions [5, 13] in the so-called $G-$framework.

We study the optimal design of electricity contracts among a population of consumers with different needs. This question is tackled within the framework of Principal-Agent problem in presence of adverse selection. The particular features of electricity induce an unusual structure on the production cost, with no decreasing return to scale. We are ne...

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...

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 FBSDE, we are able to rewrite the Principal's problem as a control...

We provide a unified approach to a priori estimates for supersolutions of BSDEs in general filtrations, which may not be quasi left-continuous. As an example of application, we prove that reflected BSDEs are well-posed in a general framework.

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, pos...

In a framework close to the one developed by Holmstr\"om 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 eco...

In this paper, we study the existence of densities (with respect to the Lebesgue measure) for marginal laws of the solution (Y,Z) to a quadratic growth BSDE. Using the (by now) well-established connection between these equations and their associated semi-linear PDEs, together with the Nourdin-Viens formula, we provide estimates on these densities.

We consider a stochastic control problem for a class of nonlinear kernels.
More precisely, our problem of interest consists in the optimization, over a
set of possibly non-dominated probability measures, of solutions of backward
stochastic differential equations (BSDEs). Since BSDEs are non-linear
generalizations of the traditional (linear) expecta...

We provide a unified approach to a priori estimates for supersolutions of BSDEs in general filtrations, which may not be quasi left-continuous. Unlike the previous related approaches in simpler settings, our results do not only rely on a simple application of Itô's formula and classical estimates, but use crucially appropriate generalizations of de...

In this article, we prove the existence of bounded solutions of quadratic backward SDEs with jumps, that is to say for which the generator has quadratic growth in the variables (z, u). From a technical point of view, we use a direct fixed point approach as in Tevzadze [38], which allows us to obtain existence and uniqueness of a solution when the t...

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,...

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 nu...

In this paper, we provide a strong formulation of the stochastic G{\^a}teaux
differentiability in order to study the sharpness of a new characterization,
introduced in [6], of the Malliavin-Sobolev spaces. We also give a new internal
structure of these spaces in the sense of sets inclusion.

The aim of this paper is twofold. First, we extend the results of Matoussi et al. (2012) concerning the existence and uniqueness of second-order reflected 2BSDEs to the case of two obstacles. Under some regularity assumptions on one of the barriers, similar to the ones in Crépey and Matoussi (2008), and when the two barriers are completely separate...

We consider a contracting problem in which a principal hires an agent to
manage a risky project. When the agent chooses volatility components of the
output process and the principal observes the output continuously, the
principal can compute the quadratic variation of the output, but not the
individual components. This leads to moral hazard with re...

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\^ateaux derivative in the
directions of the Cameron-Martin space. Incidentally, we provide a new
formulation for the characterization of the Malliavin-Sob...

In this article, we follow the study of quadratic backward SDEs with
jumps,that is to say for which the generator has quadratic growth in the
variables (z; u), started in our accompanying paper [15]. Relying on the
existence and uniqueness result of [15], we define the corresponding
g-expectations and study some of their properties. We obtain in pa...

In this paper, we study the existence of densities (with respect to the
Lebesgue measure) for marginal laws of the solution $(Y,Z)$ to a quadratic
growth BSDE. Using the (by now) well-established connection between these
equations and their associated semi-linear PDEs, together with the
Nourdin-Viens formula, we provide estimates on these densities...

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...

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 M\'emin
[7], and tightness of solutions to discrete tim...

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 nondominated. We propose studying this problem in the framework of seco...

The problem of robust hedging requires to solve the problem of superhedging
under a nondominated family of singular measures. Recent progress was achieved
by [9,11]. We show that the dual formulation of this problem is valid in a
context suitable for martingale optimal transportation or, more generally, for
optimal transportation under controlled s...

In the context of the multi-dimensional infinite horizon optimal
consumption-investment problem with proportional transaction costs, we provide
the first order expansion in small transact costs. Similar to the
one-dimensional derivation in our accompanying paper [42], the asymptotic
expansion is expressed in terms of a singular ergodic control prob...

The aim of this paper is twofold. First, we extend the results of [32]
concerning the existence and uniqueness of second-order reflected 2BSDEs to the
case of upper obstacles. Then, under some regularity assumptions on one of the
barriers, similar to the ones in [9], and when the two barriers are completely
separated, we provide a complete wellpose...

We prove the existence of bounded solutions of quadratic backward SDEs with
jumps, using a direct fixed point approach as in Tevzadze [36]. Under an
additional standard assumption, we prove a uniqueness result, thanks to a
comparison theorem. Then we study the properties of the corresponding
$g$-expectations, we obtain in particular a non linear Do...

We consider a financial market with liquidity cost as in \c{C}etin, Jarrow
and Protter [2004], where the supply function S{\epsilon}(s,{\nu}) depends on a
parameter {\epsilon}\geq0 with S0(s,{\nu})=s corresponding to the perfect
liquid situation. Using the PDE characterization of \c{C}etin, Soner and Touzi
[2010], of the super-hedging cost of an op...

In this paper, we define a notion of second order backward stochastic
differential equation with jumps (2BSDEJs for short), which generalizes the
continuous case considered by Soner, Touzi and Zhang [33]. However, on the
contrary to their formulation, where they can define pathwise the density of
quadratic variation of the canonical process, in our...

In this paper, we follow the study of second order BSDEs with jumps started
in our accompanying paper [17]. We prove existence of these equations by a
direct method, thus providing complete wellposedness for second order BSDEs.
These equations are the natural candidates for the probabilistic interpretation
of fully non-linear partial integro-differ...

In this paper, we pursue the study of second order BSDEs with jumps (2BSDEJs for short) started in our accompanying paper [9]. We prove existence of these equations by a direct method, thus providing complete wellposedness for 2BSDEJs. These equations are natural candidates for the probabilistic interpretation of some fully non-linear partial integ...

In this paper, we define a notion of second order backward stochastic differential equation with jumps, which generalizes the continuous case considered by Soner, Touzi and Zhang [37]. In order to correctly define these notions, we first have to solve an important issue corresponding to the possibility to aggregate, in the sense of [39] and [10], b...

In this paper, we take up the analysis of a principal/agent model with moral
hazard introduced in \cite{pages}, 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 marting...

In this article, we build upon the work of Soner, Touzi and Zhang [35] to
define a notion of a second order backward stochastic differential equation
reflected on a lower cadlag obstacle. We prove existence and uniqueness of the
solution under a Lipschitz type assumption on the generator, and we investigate
some links between our reflected 2BSDEs a...

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 sec...

In a recent paper, Soner, Touzi and Zhang [20] have introduced a notion of
second order backward stochastic differential equations (2BSDEs for short),
which are naturally linked to a class of fully non-linear PDEs. They proved
existence and uniqueness for a generator which is uniformly Lipschitz in the
variables $y$ and $z$. The aim of this paper i...

We extend the wellposedness results for second order backward stochastic
differential equations introduced by Soner, Touzi and Zhang \cite{stz} to the
case of a bounded terminal condition and a generator with quadratic growth in
the $z$ variable. More precisely, we obtain uniqueness through a representation
of the solution inspired by stochastic co...

This PhD dissertation presents two independent research topics dealing with contemporary issues in mathematical finance, the second one being divided into into two distinct problems. Throughout the first part of the dissertation, we study the notion of second order backward stochastic differential equations (2BSDE in the following), first introduce...

Cette thèse présente deux principaux sujets de recherche indépendants, le dernier étantdécliné sous la forme de deux problèmes distincts. Dans toute la première partie de la thèse, nous nous intéressons à la notion d’équations différentielles stochastiques rétrogrades du second ordre (dans la suite 2EDSR), introduite tout d’abordpar Cheridito, Sone...

Stochastic volatility models have replaced Black-Scholes model since they are able to generate a volatility smile. However, standard models fail to capture the smile slope and level movements. The Double-Heston model provides a more flexible approach to model the stochastic variance. In this paper, we focus on numerical implementation of this model...

Using probability change techniques introduced by Drimus for Heston model, we derive a n-th order expansion formula of Wishart option price in terms of Black-Scholes price and Black-Scholes Greeks. Numerical results are given for the second order case. Thanks to this new approximation, the smile implied by Wishart model can be better understood. Th...

In financial mathematics, Wishart processes have emerged as an efficient tool to model stochastic covariance structures. Their numerical simulation may be quite challenging since they involve matrix processes. In this article, we propose an extensive study of financial applications of Wishart processes. First, we derive closed-form formulas for opt...

## Projects

Projects (7)

This project regroups some works on principal-agent problems with application to energy sector.