Optimal hedging of an informed broker facing many traders

Authors: Philippe Bergault, Pierre Cardaliaguet, Wenbin Yan

Abstract

This paper investigates the optimal hedging strategies of an informed broker interacting with multiple traders in a financial market. We develop a theoretical framework in which the broker, possessing exclusive information about the drift of the asset’s price, engages with traders whose trading activities impact the market price. Using a mean-field game approach, we derive the equilibrium strategies for both the broker and the traders, illustrating the intricate dynamics of their interactions. The broker’s optimal strategy involves a Stackelberg equilibrium, where the broker leads and the traders follow. Our analysis also addresses the mean field limit of finite-player models and shows the convergence to the mean-field solution as the number of traders becomes large.

Complexity vs Empirical Score

Math Complexity: 9.5/10
Empirical Rigor: 2.0/10
Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper employs advanced mathematical tools such as mean-field games, Stackelberg equilibria, backward HJ equations, and convergence proofs, indicating very high mathematical complexity. However, it lacks any empirical backtesting, code, or real-world data implementation, with the excerpt focusing solely on theoretical model derivation and existence proofs.

Research Flowchart

  flowchart TD
  A["Research Goal: Optimal hedging for an informed broker with market impact"] --> B["Methodology: Mean-field game & Stackelberg equilibrium"]
  B --> C["Inputs: Broker's private info & trader interactions"]
  C --> D["Process: Finite-player model & mean-field limit"]
  D --> E["Computation: Deriving equilibrium strategies"]
  E --> F["Outcome: Optimal hedging strategy & convergence proof"]

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Applied stochastic control of jump diffusions (2007)
Modeling and pricing precipitation derivatives under weather forecasts (2016)
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A class of backward stochastic differential equations with discontinuous coefficients (2008)
Viscosity solutions of path-dependent integro-differential equations (2016)
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Existence of the solutions of backward–forward SDE ’s with continuous monotone coefficients (2006)
Theory and applications of decoupling fields for forward–backward stochastic differential equations (2015)
Dynamic risk measure for BSVIE with jumps and semimartingale issues (2019)
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Backward stochastic Volterra integral equations in Hilbert spaces (2006)
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Time-inconsistent stochastic linear–quadratic control (2012)
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Measuring marginal utility by reactions to risk (1945)
The methods of ethics (1874)
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Monotonicity of optimal contracts without the first-order approach (2018)
The informativeness principle without the first–order approach (2019)
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Moral hazard and the spanning condition without the first-order approach (2017)
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On the consistency of the first-order–approach to principal–agent problems (2012)
A note on time–dependent additive functionals (2017)
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Path–dependent martingale problems and additive functionals (2019)
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The behavior of high-frequency traders under different market stress scenarios (2017)
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Mean–variance hedging for partially observed drift processes (2001)
Optimal portfolio in partially observed stochastic volatility models (2001)
Optimal dealer pricing under transactions and return uncertainty (1981)
Liquidity and market structure (1988)
High-frequency trading in a limit order book (2008)
Dealing with the inventory risk: a solution to the market making problem (2013)
Algorithmic and high-frequency trading (2015)
Algorithmic trading with model uncertainty (2017)
Enhancing trading strategies with order book signals (2018)
Deep reinforcement learning for market making in corporate bonds: beating the curse of dimensionality (2019)
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Mean-field games with differing beliefs for algorithmic trading (2020)
Optimal make take fees in a multi market maker environment (2021)
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Optimal high-frequency trading with limit and market orders (2013)
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The generalized W eierstrass approximation theorem (1948)
Applications of the theory of B oolean rings to general topology (1937)
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Counterexamples in analysis (2003)
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Brownian motion, $``$diverse and undulating’’ (2006)
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Fundamentals and advanced techniques in derivatives hedging (2016)
Stochastic calculus for finance I : the binomial asset pricing model (2004)
The axiom of choice (1973)
Sur les suites des fonctions mesurables (1911)
Sulle successioni di funzioni ortogonali (1910)
Asset pricing with heterogeneous beliefs and illiquidity (2020)
Dynamic trading: price inertia and front–running (2016)
A two-dimensional control problem arising from dynamic contracting theory (2019)
Facelifting in utility maximization (2016)
Valuation of exotic options under shortselling constraints (2002)
Consistent price systems and face–lifting pricing under transaction costs (2008)
When terminal facelift enforces D elta constraints (2015)
Optimal replication of contingent claims under portfolio constraints (1998)
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Principal–agent problems revisited part II : dynamic models (2006)
Principal–agent problems revisited part I : static models (2006)
Principal–agent analysis in continuous-time (2006)
The continuous-time principal–agent problem with moral hazard and recursive preferences (2010)
The role of boundary solutions in principal–agent problems of the H olmstr "o m– M ilgrom type (2007)
Strategies in the principal–agent model (2013)
Principal–agent problems with exit options (2008)
Optimal compensation and pay–performance sensitivity in a continuous-time principal–agent model (2012)
A dynamic principal–agent problem as a feedback S tackelberg differential game (2010)
Overconfidence in the continuous-time principal–agent problem (2003)
Leadership, trust, and power: dynamic moral hazard in high office (2008)
Sticky incentives and dynamic agency (2011)
Reward for luck in a dynamic agency model (2010)
Optimal dynamic contracts with moral hazard and costly monitoring (2016)
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Optimal dynamic contracts with hidden actions in continuous time (2006)
A solvable continuous time dynamic principal–agent model (2015)
Stochastic control, and applications to finance (2018)
Moral hazard for time–inconsistent agents (2020)
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Continuous-time principal–agent problem in degenerate systems (2019) — link
Principal–agent problem with multiple principals (2019) — link
Regulation of renewable resource exploitation (2020)
Market making and incentives design in the presence of a dark pool: a deep reinforcement learning approach (2019) — link
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Privacy-preserving contact tracing of covid-19 patients (2020)
Contact tracing mobile apps for COVID –19: privacy considerations and related trade–offs (2020) — link
A simple stochastic epidemic (1950)
COVID –19: forecasting short term hospital needs in F rance (2020)
Optimal quarantine strategies for COVID –19 control models (2020) — link
Beyond just `` flattening the curve " : optimal control of epidemics with purely non–pharmaceutical interventions (2020) — link
Optimal control of an SIR epidemic through finite–time non–pharmaceutical intervention (2020) — link
The effect of stay–at–home orders on COVID –19 infections in the U nited S tates (2020) — link
Susceptible–infected–recovered ( SIR ) dynamics of COVID –19 and economic impact (2020) — link
COVID –19: extending or relaxing distancing control measures (2020)
The benefits and costs of using social distancing to flatten the curve for COVID –19 (2020)
Trauma does not quarantine: violence during the COVID –19 pandemic (2020)
Report 9: Impact of non–pharmaceutical interventions ( NPI s) to reduce COVID –19 mortality and healthcare demand (2020)
The optimal COVID –19 quarantine and testing policies (2020)
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A transformation method to study the solvability of fully coupled FBSDE s (2019)
Existence of optimal stochastic control laws (1971)
Neighbors, knowledge, and nuggets: two natural field experiments on the role of incentives on energy conservation (2015)
Random horizon principal–agent problem (2020) — link
Regular variation (1987)
Regularly varying functions (1976)
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Mechanism design for general screening problems with moral hazard (1992)
Risk independence and multiattributed utility functions (1973)
Risk sharing and incentives in the principal and agent relationship (1979)
Optimal incentive contracts with imperfect information (1979)
The economic incentive provided by sampling inspection (1960)
The pure theory of taxation (1897)
Overinsurance and public provision of insurance: the roles of moral hazard and adverse selection (1974)
The economics of moral hazard: further comment (1968)
The economics of moral hazard: comment (1968)
Taxing price makers (1978)
Accident and health insurance (1948)
Health insurance (1959)
Uncertainty and the welfare economics of medical care (1963)
Research in management controls: a critical synthesis (1963)
The functions of the executive (1938)
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An inquiry into the nature and causes of the wealth of nations (1776)
Distributions for the first-order approach to principal–agent problems (2003)
Two new conditions supporting the first-order approach to multisignal principal–agent problems (2009)
Time–consistent feedback strategies with V olterra processes (2019) — link
Moment constrained optimal dividends: precommitment $&$ consistent planning (2019) — link
Time-inconsistent stopping, myopic adjustment & equilibrium stability: with a mean–variance application (2019) — link
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Time-inconsistent stochastic optimal control problems and backward stochastic V olterra integral equations (2019) — link
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Conditional optimal stopping: a time–inconsistent optimization (2020)
A course in game theory (1994)
Stochastic equations in infinite dimensions (2014)
Time-inconsistent optimal control problems and the equilibrium HJB equation (2012)
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Time-inconsistent recursive stochastic optimal control problems (2017)
Contracting with a na " i ve time-inconsistent agent: to exploit or not to exploit? (2015) — link
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Psychology and economics (1998)
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Incentives for procrastinators (1999)
Doing it now or later (1999)
The equity premium: a puzzle (1985)
Approximating B ochner integrals by R iemann sums (2002)
A solvable time-inconsistent principal–agent problem (2018)
Equilibrium strategies for the mean–variance investment problem over a random horizon (2018)
Inconsistent investment and consumption problems (2015)
Behavioral contract theory (2014)
Behavioral portfolio selection in continuous time (2008)
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Exploiting na " i vet'e about self-control in the credit market (2010)
Optimal equilibria for time-inconsistent stopping problems in continuous time (2020)
On the equilibrium strategies for time-inconsistent problems in continuous time (2019)
Optimal exit time from casino gambling: strategies of precommitted and naive gamblers (2019)
Le comportement de l’homme rationnel devant le risque : critique des postulats et axiomes de l’ 'e cole am 'e ricaine (1953)
Ramsey meets L aibson in the neoclassical growth model (1999)
Dynamic mean–variance asset allocation (2010)
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Mean–variance portfolio optimization with state-dependent risk aversion (2014)
Psychology and economics: evidence from the field (2009)
Contracting with diversely naive agents (2006)
Risk, ambiguity and the S avage axioms (1961)
The existence of regular conditional probabilities: necessary and sufficient conditions (1985)
Convex viscosity solutions and state constraints (1997)
The utility analysis of choices involving risk (1948)
Consistent plans (1980)
On quadratic $g$-evaluations/expectations and related analysis (2010)
Optimal stochastic control, stochastic target problems, and backward SDE (2013)
Continuous exponential martingales and BMO (1994)
Why has CEO pay increased so much? (2008)
Corporate liquidity management under moral hazard (2019)
Moral hazard with bounded payments (2008)
Renegotiating moral hazard contracts under limited liability and monotonicity (2001)
A theory of sales quotas with limited liability and rent sharing (2000)
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On certain questions in the theory of optimal control (1962)
Viscosity solutions of fully nonlinear elliptic path dependent partial differential equations (2016)
Backward stochastic differential equations with no driving martingale, M arkov processes and associated pseudo partial differential equations (2017) — link
Martingale driven BSDE s, PDE s and other related deterministic problems (2017) — link
Forward backward SDE s in weak formulation (2018)
Weak existence and uniqueness for forward–backward SDE s (2006)
On weak solutions of backward stochastic differential equations (2005)
On the continuity of weak solutions of backward stochastic differential equations (2008)
Weak solutions of backward stochastic differential equations with continuous generator (2014)
Weak solutions of forward–backward SDE ’s (2003)
Well-posedness of backward stochastic differential equations with general filtration (2013)
Numerical F ourier method and second-order T aylor scheme for backward SDE s in finance (2016)
A F ourier cosine method for an efficient computation of solutions to BSDE s (2015)
Numerical approximation of general L ipschitz BSDE s with branching processes (2019)
Optimal and better transport plans (2009)
On a class of optimal transportation problems with infinitely many marginals (2013)
Grossissement initial, hypoth `e se $( H ^\prime)$ et th 'e or `e me de G irsanov (1985)
Sopra l’integrazione delle serie (1906)
Le \c c ons sur l’int 'e gration et la recherche des fonctions primitives (1904)
Causal optimal transport: theory and applications (2017)
Mean field games via controlled martingale problems: existence of M arkovian equilibria (2015)
Statistics of random processes (1977)
Martingale measure approximation, application to the control of diffusions (1992)
Extended mean field control problems: stochastic maximum principle and transport perspective (2019)
Wellposedness of mean field games with common noise under a weak monotonicity condition (2016)
Points d’ 'e quilibre dans les jeux stochastiques de somme non nulle (1994)
Nonzero sum linear–quadratic stochastic differential games and backward–forward equations (1999)
Backward–forward SDE ’s and stochastic differential games (1998)
Existence of N ash equilibrium points for M arkovian non–zero-sum stochastic differential games with unbounded coefficients (2015)
On a non-linear semi-group attached to stochastic optimal control (1976)
Path-dependent equations and viscosity solutions in infinite dimension (2018)
Handbook of mathematical functions with formulas, graphs, and mathematical tables (1948)
Bang-bang–type N ash equilibrium point for M arkovian nonzero-sum stochastic differential game (2014)
Stochastic volatility extensions of the swap market model (2008)
Jumps and stochastic volatility: exchange rate processes implicit in D eutsche mark options (1996)
A closed-form solution for options with stochastic volatility with applications to bond and currency options (1993)
The term structure of simple forward rates with jump risk (2003)
A stochastic volatility swap market model (2006)
Markov representation of the H eath– J arrow– M orton model (2001)
Manufacturing and managing customer-driven derivatives (2016)
Interest rate models—theory and practice (2006)
Swap market model: theory and empirical evidence (2008)
Arbitrage-free discretization of lognormal forward LIBOR and swap rate models (2000)
Constructive theory of market models for interest-rate derivatives (2007)
Modelling and calibration of swap market models (2003)
A nonparametric local volatility model for swaptions smile (2018)
Non-parametric local volatility swaption formula revisited (2016)
Towards a general local volatility model for all asset classes (2019)
A LIBOR market model with stochastic basis (2010)
LIBOR local volatility model: a new interest rate smile model (2016)
Extended LIBOR market models with affine and quadratic volatility (2000)
LIBOR market models with stochastic basis (2010)
LIBOR market models versus swap market models for pricing interest rate derivatives: an empirical analysis (2001)
Non-parametric local volatility formula for interest rate swaptions (2016)
Interest rate models enhanced with local volatility (2016)
The swap market model with local stochastic volatility (2018)
A unified view of LIBOR models (2016)
Old and new approaches to LIBOR modeling (2010)
New and robust drift approximations for the LIBOR market model (2008)
Lognormal approximations to LIBOR market models (2002)
Strong T aylor approximation of SDE s and application to the L 'e vy LIBOR model (2010)
Weak and strong T aylor methods for numerical solutions of stochastic differential equations (2011)
The affine LIBOR models (2013)
LIBOR model with expiry-wise stochastic volatility and displacement (2013)
Multiple stochastic volatility extension of the LIBOR market model and its implementation (2009)
Valuation of floating range notes in L 'evy term-structure models (2006)
Exact pricing formulae for caps and swaptions in a L 'e vy term structure model (2006)
LIBOR market model with semimartingales (1999)
The L 'e vy LIBOR model (2005)
The term structure of interbank risk (2013)
A note on the swap market model (2005)
Generic market models (2006)
Single and multi-currency co-initial swap market models (1997)
The co-initial swap market model (2004)
Theory and calibration of swap market models (2007)
Pricing swap options using the forward swap market (1990)
Models of forward LIBOR and swap rates (1999)
Solving SABR in exact form and unifying it with LIBOR market model (2009)
LIBOR market model with SABR style stochastic volatility (2008)
LIBOR market model with stochastic volatility (2003)
LIBOR market model with stochastic volatility (2006)
LIBOR market model: from deterministic to stochastic volatility (2002)
A stochastic volatility forward LIBOR model with a term structure of volatility smiles (2003)
A stochastic-volatility, displaced-diffusion extension of the LIBOR market model (2003)
Extended LIBOR market models with stochastic volatility (2001)
Managing smile risk (2002)
Alternative asset-price dynamics and volatility smile (2003)
Analytical pricing of the smile in a forward LIBOR market model (2003)
Lognormal-mixture dynamics and calibration to market volatility smiles (2002)
Displaced and mixture diffusions for analytically-tractable smile models (2002)
A mixed-up smile (2000)
Volatility skews and extensions of the LIBOR market model (2000)
Stochastic calculus for finance II : continuous-time models (2004)
Dual swap and swaption formulae in the normal and lognormal models (1996)
Rank-$2$ swaption formulae (1997)
Forward rate volatilities, swap rate volatilities, and implementation of the LIBOR market model (2000)
On the pricing implications of the joint lognormal assumption for the swaption and cap markets (1999)
Long forward and zero-coupon rates can never fall (1996)
Equilibrium asset pricing with transaction costs (2021)
A necessary and sufficient condition for rationalizability in a quasi-linear context (1987)
The taxation principle and multi-time H amilton– J acobi equations (1985)
Multi-dimensional screening: a user’s guide (1999)
Monopoly pricing of capacity usage under asymmetric information (1993)
The direct approach to debt option pricing (1994)
A note on pricing interest rate derivatives when forward LIBOR rates are lognormal (1997)
Option pricing in the presence of natural boundaries and a quadratic diffusion term (1997)
On the fundamental theorem of asset pricing (2015)
A version of the fundamental theorem of asset pricing (1999)
"U ber das P roblem der W ohlordnung (1915)
Two classical surprises concerning the axiom of choice and the continuum hypothesis (2002)
Ueber unendliche, lineare P unktmannichfaltigkeiten (1883)
On the integration of discontinuous functions (1874)
Sul problema della misura dei gruppi di punti di una retta (1905)
Der B eweis des F undamentalsatzes der I ntegralrechnung (1880)
Alcune osservazioni sulle funzioni punteggiate discontinue (1881)
Introduction to option pricing theory (2000)
A new look at the fundamental theorem of asset pricing (1998)
What is a free lunch? (2004)
A note on the no arbitrage condition for international financial markets (1996)
The notion of arbitrage and free lunch in mathematical finance (2008)
Attainable claims with $p$‘th moments (1996)
Arbitrage possibilities in B essel processes and their relations to local martingales (1995)
The no-arbitrage property under a change of num 'e raire (1995)
The existence of absolutely continuous local martingale measures (1995)
The fundamental theorem of asset pricing for unbounded stochastic processes (1998)
A general version of the fundamental theorem of asset pricing (1994)
Non-arbitrage and the fundamental theorem of asset pricing: summary of main results (1999)
Non-arbitrage and the fundamental theorem of asset pricing (1993)
Almost sure characterization of martingales (1994)
The analysis of finite security markets using martingales (1987)
A stochastic calculus model of continuous trading: complete markets (1983)
Arbitrage and free lunch with bounded risk for unbounded continuous processes (1994)
On the existence of equivalent local martingale measures (1991)
Representing martingale measures when asset prices are continuous and bounded (1992)
Martingale measures for discrete-time processes with infinite horizon (1994)
Martingale measures for a class of right-continuous processes (1993)
A remark on arbitrage and martingale measure (1993)
Quelques remarques sur un th 'e or `e me de Y an (1990)
A H ilbert space proof of the fundamental theorem of asset pricing in finite discrete time (1992)
Equivalent martingale measures and no-arbitrage (1994)
A teacher’s note on no-arbitrage criteria (2001)
No-arbitrage and equivalent martingale measures: an elementary proof of the H arrison– P liska theorem (1995)
Local martingales and the fundamental asset pricing theorems in the discrete-time case (1998)
On the fundamental theorem of asset pricing with an infinite state space (1991)
Equivalent martingale measures and no-arbitrage in stochastic securities market models (1990)
Multiperiod security markets with differential information: martingales and resolution times (1986)
Caract 'e risation d’une classe d’ensembles convexes de $ L ^1$ ou $ H ^1$ (1980)
Arbitrage et lois de martingale (1990)
Arbitrage and equilibrium in economies with infinitely many commodities (1981)
Martingales and stochastic integrals in the theory of continuous trading (1981)
Martingales and arbitrage in multiperiod securities markets (1979)
The pricing of options and corporate liabilities (1973)
A family of humped volatility models (2001)
The market model of interest rate dynamics (1997)
Closed form term structure derivatives in a H eath– J arrow– M orton model with log-normal annually compounded interest rates (1994)
Lognormality of rates and term structure models (2000)
Different dynamical specifications of the term structure of interest rates and their implications (1993)
Closed form solutions for term structure derivatives with log-normal interest rates (1997)
LIBOR and swap market models and measures (1997)
Continuous-time term structure models: forward measure approach (1997)
Martingale methods in financial modeling (1997)
A multifactor G auss M arkov implementation of H eath, J arrow, and M orton (1994)
The pricing of commodity options with stochastic interest rates (1987)
Continuous-time limits in the generalized H o– L ee framework under the forward measure (1996)
The one-factor G aussian interest rate model: theory and implementation (1988)
Weak convergence of term structure movements and the connection of prices and interest rates (1993)
A review of the H o and L ee model (1992)
The structure of binomial lattice models for bonds (1998)
A simple approach to the pricing of interest rate options (1991)
Volatility structures of forward rates and the dynamics of the term structure (1995)
Term structure movements and pricing interest rate contingent claims (1986)
Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation (1992)
The stochastic properties of term structure movements (1987)
An arbitrage model of the term structure of interest rates (1978)
A multivariate model of the term structure (1980)
Savings bonds, retractable bonds and callable bonds (1977)
Factor models and the shape of the term structure (1998)
On short rate processes and their implications for term structure movements (1994)
Bond and option pricing when short rates are lognormal (1991)
The valuation of interest rate derivatives in a multi-factor term structure model with deterministic components (1995)
Positive interest rates and non-linear term structure models (1998)
Bond and bond option pricing based on the current term structure (1997)
On deterministic shift extensions of short-rate models (1998)
A deterministic-shift extension of analytically-tractable and time-homogeneous short-rate models (2001)
Which model for term-structure of interest rates should one use? (1995)
Solution of the extended CIR term structure and bond option valuation (1996)
A simple class of square-root interest-rate models (1995)
A note on the models of H ull and W hite for pricing options on the term structure (1995)
A note on the models of H ull and W hite for pricing options on the term structure: response (1995)
Numerical procedures for implementing term structure models II : two-factor models (1994)
Numerical procedures for implementing term structure models I : single-factor models (1994)
One-factor interest-rate models and the valuation of interest-rate derivative securities (1993)
The pricing of options on interest rate caps and floors using the H ull– W hite model (1993)
Bond option pricing based on a model for the evolution of bond prices (1993)
Pricing interest-rate–derivative securities (1990)
A nonlinear general equilibrium model of the term structure of interest rates (1989)
The valuation of options on coupon bonds (1993)
The pricing of options on default-free bonds (1982)
A term structure model and the pricing of interest rate derivatives (1993)
A term structure model and the pricing of interest rate options (1989)
A one-factor model of interest rates and its application to treasury bond options (1990)
Analyzing convertible bonds (1980)
A note on the stability of lognormal interest rate models and the pricing of Eurodollar futures (1997)
The lognormal interest rate model and eurodollar futures (1993)
Theory of rational option pricing (1973)
An exact bond option formula (1989)
On the term structure of interest rates (1978)
An equilibrium characterization of the term structure (1977)
A theory of the term structure of interest rates (1985)
An intertemporal general equilibrium model of asset prices (1985)
Pricing and hedging swaps (1991)
The term structure of interest rates. Volume 1 (1990)
A reexamination of traditional hypotheses about the term structure: a comment (1993)
Time varying term premia and traditional hypotheses about the term structure (1990)
Term premiums in bond returns (1984)
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Rebalancing with linear and quadratic costs (2017)
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Linear quadratic stochastic control problems with stochastic terminal constraint (2018)
Hedging with temporary price impact (2017)
Financial markets equilibrium with heterogeneous agents (2011)
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Hedging, arbitrage and optimality with superlinear frictions (2015)
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Investing with liquid and illiquid assets (2018)
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Continuous auctions and insider trading (1985)
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Portfolio choice with small temporary and transient price impact (2019)
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Rebalancing multiple assets with mutual price impact (2018)
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Scaling limits of processes with fast nonlinear mean reversion (2020)
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Portfolio choice with market closure and implications for liquidity premia (2016)
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Dynamic trading with predictable returns and transaction costs (2013)
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Asset prices and trading volume under fixed transaction costs (2004)
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Asset prices in general equilibrium with transactions costs and recursive utility (2015)
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An option to reduce transaction costs (2011)
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Competing mechanisms in a common value environment (2000)
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Analysis of H amilton– J acobi– B ellman equations arising in stochastic singular control (2013)
Investment under uncertainty (1994)
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Stochastic differential games and viscosity solutions of I saacs equations (1988)
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Backward stochastic differential equations—from linear to fully nonlinear theory (2017)
Uniqueness of the representation for $ G $-martingales with finite variation (2012)
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Limit theorems for BSDE with local time applications to non-linear PDE (2002)
Backward stochastic differential equation with local time (1999)
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Backward stochastic differential equations with stochastic monotone coefficients (2004)
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BSDE s with stochastic L ipschitz condition (2000)
A general comparison theorem for backward stochastic differential equations (2010)
Contracting theory with competitive interacting agents (2019)
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Multidimensional quadratic and subquadratic BSDE s with special structure (2015)
Splitting multidimensional BSDE s and finding local equilibria (2014)
Solvability of backward stochastic differential equations with quadratic growth (2008)
General existence results for reflected BSDE and BSDE (2011)
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BSDE s with a random terminal time driven by a monotone generator and their links with PDE s (2004)
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Social insurance and redistribution with moral hazard and adverse selection (2006)
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