One model to solve them all: 2BSDE families via neural operators

Authors: Takashi Furuya, Anastasis Kratsios, Dylan Possamaï, Bogdan Raonić

Abstract

We introduce a mild generative variant of the classical neural operator model, which leverages Kolmogorov–Arnold networks to solve infinite families of second-order backward stochastic differential equations ($2$BSDEs) on regular bounded Euclidean domains with random terminal time. Our first main result shows that the solution operator associated with a broad range of $2$BSDE families is approximable by appropriate neural operator models. We then identify a structured subclass of (infinite) families of $2$BSDEs whose neural operator approximation requires only a polynomial number of parameters in the reciprocal approximation rate, as opposed to the exponential requirement in general worst-case neural operator guarantees.

Complexity vs Empirical Score

Math Complexity: 9.2/10
Empirical Rigor: 1.5/10
Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is highly mathematically complex, featuring dense advanced mathematics including stochastic differential equations, elliptic PDEs, Sobolev spaces, and neural operator theory with polynomial/exponential approximation rates. It shows very low empirical rigor as it presents theoretical approximation theorems without any code, backtests, datasets, or implementation-heavy details; it focuses on pure mathematical theory rather than practical application.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Solve infinite families of 2BSDEs<br>using Neural Operators"] --> B["Methodology:<br>Neural Operator +<br>Kolmogorov-Arnold Networks KANs"]
  B --> C["Data/Input:<br>Regular bounded Euclidean domains<br>with random terminal time"]
  C --> D["Computational Process:<br>Approximation of the<br>solution operator for 2BSDEs"]
  D --> E["Key Finding 1:<br>Solution operator is approximable<br>by Neural Operators"]
  D --> F["Key Finding 2:<br>Structured subclass requires only<br>Polynomial parameters for convergence"]
  E --> G["Outcome:<br>One model to solve<br>infinite families of 2BSDEs"]
  F --> G

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Extinction and quasi-stationarity for discrete-time, endemic SIS and SIR models (2021)
Permanence and extinction for the stochastic SIR epidemic model (2020)
Classification of asymptotic behavior in a stochastic SIR model (2016)
The law of mass action in epidemiology (1945)
Equilibria of an epidemic game with piecewise linear social distancing cost (2013)
Provisioning of public health can be designed to anticipate public policy responses (2017)
Game theory of social distancing in response to an epidemic (2010)
Optimal tax/subsidy combinations for the flu season (2004)
An examination of the R eed– F rost theory of epidemics (1952)
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Stochastic epidemic modeling (2009)
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Liquidity in a binomial market (2012)
Second-order stochastic target problems with generalized market impact (2019)
Hedging of covered options with linear market impact and gamma constraint (2017)
Testing statistical hypotheses (2005)
Decomposition formula for rough V olterra stochastic volatility models (2021)
A rough SABR formula (2022)
Weak error rates for option pricing under linear rough volatility (2022)
Multilevel M onte C arlo simulation for VIX options in the rough B ergomi model (2022)
Pathwise large deviations for the rough B ergomi model: corrigendum (2021)
Pathwise large deviations for the rough B ergomi model (2018)
ADOL — M arkovian approximation of rough lognormal model (2019)
Hierarchical adaptive sparse grids and quasi– M onte C arlo for option pricing under the rough B ergomi model (2020)
Strong convergence rates for M arkovian representations of fractional processes (2021)
Donsker-type theorems for correlated geometric fractional B rownian motions and related processes (2017)
Hybrid simulation scheme for volatility modulated moving average fields (2019)
Hybrid scheme for B rownian semistationary processes (2017)
Long vs short time scales: the rough dilemma and beyond (2022)
Affine representations of fractional processes with applications in mathematical finance (2019)
The L aplace transform of the integrated V olterra W ishart process (2022)
Dispersion-constrained martingale S chr "o dinger problems and the exact joint S&P $500$/ VIX smile calibration puzzle (2024)
Joint modelling and calibration of SPX and VIX by optimal transport (2022)
Calibration of local–stochastic volatility models by optimal transport (2022)
A regime-switching H eston model for VIX and S&P $500$ implied volatilities (2014)
Smiling twice: the H eston++ model (2018)
Joint pricing of VIX and SPX options with stochastic volatility and jump models (2015)
The VIX future in B ergomi models: fast approximation formulas and joint calibration with S & P $500$ skew (2022)
Inversion of convex ordering in the VIX market (2020)
Regime-switching stochastic volatility model: estimation and calibration to VIX options (2017)
Heston stochastic vol-of-vol model for joint calibration of VIX and S&P $500$ options (2018)
A consistent pricing model for index options and volatility derivatives (2013)
Consistent modelling of VIX and equity derivatives using a $3/2$ plus jumps model (2014)
Inversion of convex ordering: local volatility does not maximize the price of VIX futures (2020)
Forward variance dynamics: B ergomi’s model revisited (2014)
The extended SSVI volatility surface (2019)
The joint S&P $500$/ VIX smile calibration puzzle solved (2020)
Volatility markets: consistent modeling, hedging and practical implementation (2006)
Fast N inomiya– V ictoir calibration of the double–mean-reverting model (2013)
Inferring volatility dynamics and risk premia from the S&P $500$ and VIX markets (2019)
Variance curve models: finite dimensional realizations and beyond (2016)
Fast hybrid schemes for fractional R iccati equations (rough is not so tough) (2021)
Stochastic volatility and fractional B rownian motion (2004)
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Properly discounted asset prices are semimartingales (2020)
Large financial markets, discounting, and no asymptotic arbitrage (2020)
Towards a discounting-invariant no-arbitrage theory (2020)
Making no-arbitrage discounting-invariant: a new FTAP beyond NFLVR and NUPBR (2022)
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Weak and strong no-arbitrage conditions for continuous financial markets (2015)
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No arbitrage and the growth optimal portfolio (2007)
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American options in the V olterra H eston model (2022)
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Markovian structure of the V olterra H eston model (2019)
Markowitz portfolio selection for multivariate affine and quadratic V olterra models (2021)
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Markovian lifts of positive semidefinite affine V olterra-type processes (2019)
Affine V olterra processes (2019)
Inhomogeneous affine V olterra processes (2022)
Approximation formula for option prices under rough H eston model and short-time implied volatility behavior (2020)
Small-time VIX smile for the rough H eston model (2021)
Series expansion and fourth-order global P ad 'e approximation for a rough H eston solution (2020)
Rough H eston with jumps—joint calibration to SPX/VIX level and skew as $ T \longrightarrow0$, and issues with the quadratic rough H eston model (2021)
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Regime switching rough H eston model (2019)
The Z umbach effect under rough H eston (2020)
Rational approximation of the rough H eston solution (2019)
The quadratic rough H eston model and the joint S & P 500/ VIX smile calibration problem (2020)
Lifting the H eston model (2019)
Roughening H eston (2019)
Asymptotic behavior of the fractional H eston model (2018)
The characteristic function of G aussian stochastic volatility models: an analytic expression (2022)
A comparison principle between rough and non-rough H eston models—with applications to the volatility surface (2020)
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A fast algorithm for simulation of rough volatility models (2022)
Rough fractional diffusions as scaling limits of nearly unstable heavy tailed H awkes processes (2016)
The R iemann– L iouville field and its GMC as $ H \longrightarrow0$, and skew flattening for the rough B ergomi model (2022)
Asymptotic behaviour of randomised fractional volatility models (2019)
Robustness and sensitivity analyses for rough V olterra stochastic volatility models (2023)
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Deep curve-dependent PDE s for affine rough volatility (2023)
Asymptotics for volatility derivatives in multi-factor rough volatility models (2021)
Perfect hedging in rough H eston models (2018)
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Option pricing under fast-varying and rough stochastic volatility (2018)
The order barrier for strong approximation of rough volatility models (2016)
Pricing options under rough volatility with backward SPDE s (2022)
Log-modulated rough stochastic volatility models (2021)
On VIX futures in the rough B ergomi model (2018)
Rough volatility, CGMY jumps with a finite history and the rough H eston model—small-time asymptotics in the $k\sqrt t $ regime (2021)
Turbocharging M onte C arlo pricing for the rough B ergomi model (2018)
Hedging under rough volatility (2021) — link
From quadratic H awkes processes to super- H eston rough volatility models with Z umbach effect (2021)
Fractional stochastic volatility model (2021)
From microscopic price dynamics to multidimensional rough volatility models (2021)
The characteristic function of rough H eston models (2019)
Short-time near-the-money skew in rough fractional volatility models (2019)
Deep hedging under rough volatility (2021)
No-arbitrage implies power-law market impact and rough volatility (2020)
Functional central limit theorems for rough volatility (2017)
Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models (2021)
Volatility of volatility is (also) rough (2019)
Calibrating rough volatility models: a convolutional neural network approach (2020)
Volatility has to be rough (2021)
Asymptotics for rough stochastic volatility models (2017)
The microstructural foundations of leverage effect and rough volatility (2018)
On deep calibration of (rough) stochastic volatility models (2019)
Short-time at-the-money skew and rough fractional volatility (2017)
Volatility options in rough volatility models (2020)
A regularity structure for rough volatility (2020)
Multifactor approximation of rough volatility models (2019)
Pricing under rough volatility (2016)
Consistent estimation for fractional stochastic volatility model under high-frequency asymptotics (2022)
Rough volatility: evidence from option prices (2018)
Volatility is rough (2018)
Decoupling the short- and long-term behavior of stochastic volatility (2022)
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Intraday periodicity and volatility persistence in financial markets (1997)
A unified stochastic volatility–stochastic correlation model (2020)
Correlation skew via stochastic correlation and jumps (2015)
Advanced equity derivatives: volatility and correlation (2014)
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A pairwise local correlation model (2019)
Calibration of local correlation models to basket smiles (2017)
A new class of local correlation models (2013)
Being particular about calibration (2013)
A mean-reverting SDE on correlation matrices (2013)
Breaking correlation breaks (2010)
A dynamic model for correlation (2010)
Monte C arlo pricing scheme for a stochastic–local volatility model (2014)
Local stochastic volatility with jumps: analytical approximations (2013)
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An adjoint method for the exact calibration of stochastic local volatility models (2018)
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The H eston stochastic–local volatility model: efficient M onte C arlo simulation (2014)
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Polynomial jump–diffusions on the unit simplex (2018)
Markov cubature rules for polynomial processes (2020)
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Polynomial diffusions on compact quadric sets (2017)
Stochastic volatility with an O rnstein– U hlenbeck process: an extension (1999)
Correlations and bounds for stochastic volatility models (2007)
Multivariate J acobi process with application to smooth transitions (2006)
Quadratic variance swap models (2016)
Asset pricing with matrix jump diffusions (2008)
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A multivariate O rnstein– U hlenbeck type stochastic volatility model (2009)
Positive-definite matrix processes of finite variation (2007)
Correlation risk and optimal portfolio choice (2010)
Correlation risk and the term structure of interest rates (2008)
Hedging (co) variance risk with variance swaps (2011)
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From implied to spot volatilities (2010)
A market model for stochastic implied volatility (1999)
Markovian models in the stochastic implied volatility framework (2002)
Market model of stochastic implied volatility with application to the BGM model (2001)
Stochastic volatility’s orderly smiles (2012)
The alpha- H eston stochastic volatility model (2021)
Polynomial jump–diffusion models (2020)
Polynomial diffusions and applications in finance (2016)
The J acobi stochastic volatility model (2018)
Polynomial processes and their applications to mathematical finance (2012)
The multivariate sup OU stochastic volatility model (2013)
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A multifactor volatility H eston model (2008)
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Wishart stochastic volatility: asymptotic smile and numerical framework (2008)
Exact and high-order discretization schemes for W ishart processes and their affine extensions (2013)
Wishart processes (1991)
Derivative pricing with multivariate stochastic volatility: application to credit risk (2004)
Wishart quadratic term structure models (2003)
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Option pricing when correlations are stochastic: an analytical framework (2007)
Affine processes and applications in finance (2003)
Affine processes on positive semidefinite matrices (2011)
Affine processes on symmetric cones (2016)
Affine processes are regular (2011)
Affine processes with compact state space (2018)
Affine processes beyond stochastic continuity (2019)
Regularity of affine processes on general state spaces (2013)
Smile dynamics II (2005)
Local–stochastic volatility: models and non-nodels (2017)
The smile in stochastic volatility models (2011)
Stochastic volatility modeling (2015)
Smile dynamics IV (2009)
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Smile dynamics (2004)
Calibrating and pricing with a stochastic–local volatility model (2015)
The calibration of stochastic local-volatility models: an inverse problem perspective (2019)
Nonparametric model calibration for derivatives (2017)
Nonlinear option pricing (2014)
Local volatility models enhanced with jumps (2016)
Existence of a calibrated regime switching local volatility model (2020)
The vol smile problem (2002)
Time changes for L 'e vy processes (2001)
Generalized hyperbolic diffusions with applications towards finance (1999)
A hyperbolic diffusion model for stock prices (1996)
Some recent developments in stochastic volatility modelling (2002)
Processes of normal inverse G aussian type (1997)
Normal inverse G aussian distributions and stochastic volatility modelling (1997)
Option pricing for asset returns described by generalized hyperbolic distributions: discrete versus continuous models (1996)
Stochastic volatility for L 'e vy processes (2003)
Stochastic volatility, jumps and hidden time changes (2002)
The fine structure of asset returns: an empirical investigation (2002)
New insights into smile, mispricing, and value at risk: the hyperbolic model (1998)
Hyperbolic distributions in finance (1995)
Option pricing with VG martingale components (1991)
The variance gamma process and option pricing (1998)
The variance gamma ( VG ) model for share market returns (1990)
L 'e vy processes in finance: pricing financial derivatives (2003)
[ GLP & MEMM ] pricing models and related problems (2006)
Pure jump L 'e vy processes for asset price modelling (2002)
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Retrieving L 'e vy processes from option prices: regularization of an ill-posed inverse problem (2006)
Financial modelling with jump processes (2004)
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Jump–diffusion processes: volatility smile fitting and numerical methods for option pricing (2000)
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Post-‘87 crash fears in S & P $500$ futures options (2000)
Option pricing when underlying stock returns are discontinuous (1976)
Option pricing under a double exponential jump diffusion model (2004)
A jump–diffusion model for option pricing (2002)
Stochastic volatility and jump–diffusion—implications on option pricing (1999)
Valuation of A merican options in a jump–diffusion model (1997)
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Option pricing for jump diffusions: approximations and their interpretation (1993)
Martingale analysis for assets with discontinuous returns (1995)
Pricing options on securities with discontinuous returns (1993)
Jump diffusion option valuation in discrete time (1993)
Discontinuous asset prices and non-attainable contingent claims (1993)
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A simplified jump process for common stock returns (1983)
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The volatility surface: a practitioner’s guide (2011)
A general asymptotic implied volatility for stochastic volatility models (2005)
Fine-tune your smile: correction to H agan et al. (2007)
Equivalent B lack volatilities (1999)
Efficient, almost exact simulation of the H eston stochastic volatility model (2010)
Simple and efficient simulation of the H eston stochastic volatility model (2008)
A simple option formula for general jump–diffusion and other exponential L 'e vy processes (2001)
The moment formula for implied volatility at extreme strikes (2004)
The structure of the implied volatility in the H eston model (2000)
Multiscale stochastic volatility asymptotics (2003)
Multiscale stochastic volatility for equity, interest rate, and credit derivatives (2011)
Stochastic volatility and correction to the heat equation (2004)
Asymptotics of a two-scale stochastic volatility model (1998)
Stochastic volatility correction to B lack– S choles (2000)
Stochastic volatility and epsilon–martingale decomposition (2001)
Financial modeling in a fast mean-reverting stochastic volatility environment (1999)
Derivatives in financial markets with stochastic volatility (2000)
Complications with stochastic volatility models (1998)
A new approach for option pricing under stochastic volatility (2007)
A simple new formula for options with stochastic volatility (1997)
Option valuation under stochastic volatility II (2016)
Stochastic volatility model with time-dependent skew (2005)
Option valuation under stochastic volatility (2000)
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The pricing of options on assets with stochastic volatilities (1987)
Volatile volatilities (2002)
Moment explosions in stochastic volatility models (2007)
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Implied and local volatilities under stochastic volatility (2001)
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Stochastic volatility, smile & asymptotics (1999)
A risk-neutral stochastic volatility model (1998)
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Market volatility and feedback effects from dynamic hedging (1997)
Pricing and hedging long-term options (2000)
Empirical performance of alternative option pricing models (1997)
Derivative asset analysis in models with level-dependent and stochastic volatility (1997)
Analysis of the term structure of implied volatilities (1994)
Stochastic volatility option pricing (1994)
Option pricing under incompleteness and stochastic volatility (1992)
Stock price distributions with stochastic volatility: an analytic approach (1991)
Option pricing when the variance is changing (1987)
Option values under stochastic volatility: theory and empirical estimates (1987)
Random-variance option pricing: empirical tests of the model and delta–sigma hedging (1991)
Option pricing when the variance changes randomly: theory, estimation, and an application (1987)
Pricing E uropean currency options: a comparison of the modified B lack– S choles model and a random variance model (1989)
Implied volatility functions: empirical tests (1998)
A unified theory of volatility (2004)
A theory of volatility (2002)
Stochastic implied trees: arbitrage pricing with stochastic term and strike structure of volatility (1998)
Pricing and hedging with smiles (1993)
Arbitrage pricing with stochastic volatility (1992)
Pricing and hedging with smiles (1997)
Model art (1993)
A note on a simple, accurate formula to compute implied standard deviations (1996)
A simple formula to compute the implied standard deviation (1988)
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Bond and bond option pricing based on the current term structure (1988)
Asymptotics and calibration of local volatility models (2002)
Regularization of incorrectly posed problems (1963)
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Implied binomial trees (1994)
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A forward equation for barrier options under the B runick & S hreve M arkovian projection (2016)
Forward evolution equations for knock-out options (2007)
Calibrating and pricing with embedded local volatility models (2007)
On the M arkovian projection in the B runick– S hreve mimicking result (2014)
Trading and hedging local volatility (1997)
A class of L 'e vy process models with almost exact calibration to both barrier and vanilla FX options (2010)
Option price when the stock is a semimartingale (2002)
Mimicking the marginal distributions of a semimartingale (2009)
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On I t ^o s formula for multidimensional B rownian motion (2000)
Quadratic covariation and an extension of I t ^o ’s formula (1995)
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Quantum mechanics and path integrals (1965)
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Stochastic flow approach to D upire’s formula (2007)
The complete guide to option pricing formulas (2007)
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Two generalizations of the S tieltjes integral (1921)
The integral and its generalizations (1921)
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Closed-loop convergence for mean field games with common noise (2023)
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Semimartingales and stochastic differential equations (1985)
Stochastic integration without tears (with apology to P .- A . M eyer) (1986)
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Stochastic calculus and stochastic models (1974)
Martingales et int 'e gration stochastique (1984)
Semi-martingales et int 'e grales stochastiques en temps continu (1983)
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The stochastic integral as a vector measure (1980)
Un survol de la th 'e orie de l’int 'e grale stochastique (1980)
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Monopolistic competition in electricity networks with resistance losses (2010)
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$ N $-player games and mean-field games with absorption (2018)
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Probabilistic approach to finite state mean field games (2020)
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Simplified stochastic calculus with applications in economics and finance (2021)
Simplified calculus for semimartingales: multiplicative compensators and changes of measure (2023)
Radner equilibrium and systems of quadratic BSDE s with discontinuous generators (2022)
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Optimization of convex risk functions (2006)
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A nonlinear partial integro-differential equation from mathematical finance (2010)
Nonexponential Sanov and Schilder theorems on Wiener space: BSDEs, Schr "o dinger problems and control (2020)
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Optimal incentives to mitigate epidemics: a S tackelberg mean field game approach (2022)
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Financial economics without probabilistic prior assumptions (2015)
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Kantorovich duality for general transport costs and applications (2017)
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Weak transport for non-convex costs and model-independence in a fixed-income market (2021)
Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization (2020)
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Nash equilibrium point for one kind of stochastic nonzero-sum game problem and BSDE s (2009)
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Systemic risk and stochastic games with delay (2018)
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Analytical approximations of BSDE s with nonsmooth driver (2015)
$ \mathbf L _ 2 $-time regularity of BSDE s with irregular terminal functions (2010)
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Time discretization of FBSDE with polynomial growth drivers and reaction–diffusion PDE s (2015)
Convergence and qualitative properties of modified explicit schemes for BSDE s with polynomial growth (2018)
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Approximation of the solution of the backward stochastic differential equation. S mall noise, large sample and high frequency cases (2014)
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Path regularity and explicit convergence rate for BSDE with truncated quadratic growth (2010)
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Malliavin calculus for backward stochastic differential equations and application to numerical solutions (2011)
Adaptive importance sampling in least-squares M onte C arlo algorithms for backward stochastic differential equations (2017)
Quasi-regression M onte- C arlo scheme for semi-linear PDE s and BSDE s with large scale parallelization on GPU s (2020)
Solving backward stochastic differential equations with quadratic-growth drivers by connecting the short-term expansions (2019)
Approximation scheme for solutions of backward stochastic differential equations via the representation theorem (2006)
The steepest descent method for forward–backward SDE s (2005)
Numerical stability analysis of the Euler scheme for BSDE s (2015)
Cubature method to solve BSDE s: error expansion and complexity control (2020)
A primal–dual algorithm for BSDE s (2017)
Stability of solutions of BSDE s with random terminal time (2006)
Improved error bounds for quantization based numerical schemes for BSDE and nonlinear filtering (2018)
Discretization of forward–backward stochastic differential equations and related quasi-linear parabolic equations (2007)
Numerical approximation of backward stochastic differential equations with jumps (2014)
Multilevel approximation of backward stochastic differential equations (2014)
Two algorithms for the discrete time approximation of M arkovian backward stochastic differential equations under local conditions (2015)
Stratified regression M onte- C arlo scheme for semilinear PDE s and BSDE s with large scale parallelization on GPU s (2016)
A sequential M onte C arlo algorithm for solving BSDE s (2007)
Random walk approximation of BSDE s with H "o lder continuous terminal condition (2020)
Simulation of BSDE s with jumps by W iener chaos expansion (2016)
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Robust hedging of the lookback option (1998)
Backward stochastic differential equations driven by $ G $– B rownian motion with uniformly continuous generators (2021)
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Reflected solutions of backward stochastic differential equations driven by $ G $– B rownian motion (2018)
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Optimal high-frequency trading with limit and market orders (2013)
Incorporating order-flow into optimal execution (2016)
Optimal control and machine learning in finance: contributions to the literature on optimal execution, market making, and exotic options (2019)
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The generalized W eierstrass approximation theorem (1948)
Applications of the theory of B oolean rings to general topology (1937)
"U ber die analytische D arstellbarkeit sogenannter willk "u rlicher F unktionen einer reellen V er "a nderlichen—erste M itteilung (1885)
Correction to `` T he measurability of hitting times$ " $ (2011)
The measurability of hitting times (2010)
Probability and potentials (1966)
Counterexamples in analysis (2003)
Measure theory (1950)
Notes on measure theory (2004)
Brownian motion, $ `` $diverse and undulating$ " $ (2006)
Brownian motion (2010)
Almost sure—a random mathematical blog (2021)
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On utility maximization in discrete-time financial market models (2005)
Stochastic finance: an introduction in discrete time (2004)
Approximation pricing and the variance-optimal martingale measure (1996)
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)
Optimal stopping contract for public private partnerships under moral hazard (2022)
On S annikov’s continuous-time principal–agent problem (2014)
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)
On the smoothness of value functions and the existence of optimal strategies in diffusion models (2015)
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)
A principal–agent approach to study capacity remuneration mechanisms (2020)
Continuous-time principal–agent problem in degenerate systems (2019)
Principal–agent problem with multiple principals (2023)
Regulation of renewable resource exploitation (2020)
Market making and incentives design in the presence of a dark pool: a S tackelberg actor–critic approach (2023)
Information technology-based tracing strategy in response to COVID -19 in S outh K orea—privacy controversies (2020)
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)
Optimal control of an SIR epidemic through finite-time non-pharmaceutical intervention (2021)
The effect of stay-at-home orders on COVID -19 cases and fatalities in the U nited S tates (2020) — link
Susceptible–infected–recovered ( SIR ) dynamics of COVID -19 and economic impact (2020)
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)
Contact rate epidemic control of COVID -19: an equilibrium view (2020)
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Existence, uniqueness and regularity of decoupling fields to multidimensional fully coupled FBSDE s (2013)
Solvability of coupled FBSDE s with diagonally quadratic generators (2017)
The method of decoupling fields generalized to higher spatial derivatives (2018)
A transformation method to study the solvability of fully coupled FBSDE s (2022)
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 (2022)
Regular variation (1987)
Regularly varying functions (1976)
Robust contracting in general contract spaces (2022)
On incentives and control in organizations (1977)
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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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-inconsistency with rough volatility (2021)
Moment constrained optimal dividends: precommitment & consistent planning (2022)
Time-inconsistent stopping, myopic adjustment and equilibrium stability: with a mean–variance application (2020)
Equilibrium concepts for time-inconsistent stopping problems in continuous time (2021)
Time-inconsistent stochastic optimal control problems and backward stochastic V olterra integral equations (2021)
General stopping behaviors of na "\i ve and noncommitted sophisticated agents, with application to probability distortion (2020)
A general theory of M arkovian time inconsistent stochastic control problems (2010)
Time-consistent conditional expectation under probability distortion (2021)
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)
Failure of smooth pasting principle and nonexistence of equilibrium stopping rules under time-inconsistency (2021)
The stochastic F ubini theorem revisited (2011)
A note on measurement of utility (1937)
Psychology and economics (1998)
On second-best national saving and game-equilibrium growth (1968)
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)
Forward rank-dependent performance criteria: time-consistent investment under probability distortion (2021)
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 (2021)
Optimal exit time from casino gambling: strategies of precommitted and naive gamblers (2019)
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Ramsey meets L aibson in the neoclassical growth model (1999)
Dynamic mean–variance asset allocation (2010)
Time inconsistent stochastic control in continuous time: theory and examples (2016)
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)
Superposition and mimicking theorems for conditional M c K ean– V lasov equations (2023)
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 and pseudo- PDEs (2022)
Martingale driven BSDE s, PDE s and other related deterministic problems (2021)
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 I (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)
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)
A re-examination of traditional hypotheses about the term structure of interest rates (1981)
A defense of traditional hypotheses about the term structure of interest rates (1986)
Uniqueness for linear–quadratic mean field games with common noise (2018)
Selection of equilibria in a linear quadratic mean-field game (2020)
Mean field game of controls and an application to trade crowding (2018)
Electricity price dynamics in the smart grid: a mean-field–type game perspective (2018)
Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability (2016)
Linear quadratic optimal control problems for mean-field backward stochastic differential equations (2019)
A linear–quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon (2015)
A weak martingale approach to linear–quadratic M c K ean– V lasov stochastic control problems (2019)
Linear quadratic mean field type control and mean field games with common noise, with application to production of an exhaustible resource (2016)
On mean field games with common noise and M c K ean– V lasov SPDE s (2019)
Translation invariant mean field games with common noise (2015)
Mc K ean– V lasov optimal control: the dynamic programming principle (2022)
Stability of M c K ean– V lasov stochastic differential equations and applications (2020)
Dynamic programming in complete separable spaces (1977)
Probability measures and the $ C $-sets of S elivanovskij (1978)
Resolution of measurability problems in discrete-time stochastic control (1979)
Dynamic programming for controlled M arkov families: abstractly and over martingale measures (2014)
Optimal control and zero-sum stochastic differential game problems of mean-field type (2020)
On the interpretation of the master equation (2017)
Quenched mass transport of particles towards a target (2020)
A mean-field stochastic control problem with partial observations (2017)
Mitteilungen zur L ehre vom T ransfiniten (1888)
On the relaxed mean-field stochastic control problem (2018)
Existence of optimal controls for systems governed by mean-field stochastic differential equations (2014)
Existence and optimality conditions for relaxed mean-field stochastic control problems (2017)
The relaxed optimal control problem for mean-field SDE s systems and application (2014)
Mc K ean– V lasov optimal control: limit theory and equivalence between different formulations (2022)
Mimicking an I t ^o process by a solution of a stochastic differential equation (2013)
Viscosity solutions to parabolic master equations and M c K ean– V lasov SDE s with closed-loop controls (2020)
Representation and approximation of martingale measures (1992)
Stochastic partial differential equations: six perspectives (1999)
Large deviation properties of weakly interacting processes via weak convergence methods (2012)
Continuous time mean–variance portfolio optimization through the mean field approach (2016)
On the M c K ean– V lasov limit for interacting diffusions (1988)
Stochastic particle approximations for generalized B oltzmann models and convergence estimates (1997)
Large systems of diffusions interacting through their ranks (2012)
Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation (2013)
Propagation of chaos and fluctuations for a moderate model with smooth initial data (1998)
A propagation of chaos result for a system of particles with moderate interaction (1987)
Asymptotic behaviour of some interacting particle systems; M c K ean– V lasov and B oltzmann models (1996)
A law of large numbers for moderately interacting diffusion processes (1985)
A martingale approach to the law of large numbers for weakly interacting stochastic processes (1984)
Backward stochastic V olterra integral equations—a brief survey (2013)
Recursive utility processes, dynamic risk measures and quadratic backward stochastic V olterra integral equations (2021)
On solutions of backward stochastic V olterra integral equations with jumps in Hilbert spaces (2010)
Path-dependent backward stochastic V olterra integral equations with jumps, differentiability and duality principle (2018)
Adapted solution of a backward stochastic nonlinear V olterra integral equation (2002)
Linear V olterra backward stochastic integral equations (2019)
On a class of backward stochastic V olterra integral equations (2013)
Backward stochastic V olterra integral equations with additive perturbations (2015)
Well-posedness and regularity of backward stochastic V olterra integral equations (2008)
A test for the rank of the volatility process: the random perturbation approach (2013)
Testing the maximal rank of the volatility process for continuous diffusions observed with noise (2017)
Estimation of the B rownian dimension of a continuous I t ^o process (2008)
On countervailing incentives (1995)
Countervailing incentives in agency problems (1989)
Participation constraints in adverse selection models (2000)
Calcul d’ I t ^o sans probabilit 'e s (1981)
Trading strategies generated by L yapunov functions (2017)
A martingale approach for fractional B rownian motions and related path dependent PDE s (2019)
Backward stochastic V olterra integral equations—representation of adapted solutions (2019)
Backward stochastic V olterra integral equations and some related problems (2006)
Zero-sum path-dependent stochastic differential games in weak formulation (2020)
Viscosity solutions of a system of nonlinear second-order elliptic PDE s arising in switching games (1991)
Viscosity solutions for monotone systems of second-order elliptic PDE s (1991)
Generalised solutions for fully nonlinear PDE systems and existence–uniqueness theorems (2017)
Principal–agent problem with common agency without communication (2018)
Mean-field moral hazard for optimal energy demand response management (2021)
Existence of optimal mechanisms in principal–agent problems (2017)
The existence of optimal contracts in the principal–agent model (1987)
Generalized equilibrium results for games with incomplete information (1988)
Stability results for martingale representations: the general case (2019)
Gesammelte A bhandlungen mathematischen und philosophischen I nhalts (1932)
"A hnliche (deutliche) A bbildung und "a hnliche S ysteme (1932)
Untersuchugen aus der M engenlehre (1901)
"U ber G . C antorsche S "a tze (1896)
Mitteilungen zur L ehre vom T ransfiniten (1887)
Beitr "a ge zur B egr "u ndung der transfiniten M engenlehre (1897)
Beitr "a ge zur B egr "u ndung der transfiniten M engenlehre (1895)
Alternative theoretical frameworks for finite horizon discrete-time stochastic optimal control (1978)
Universally measurable policies in dynamic programming (1979)
Dynamic programming in B orel spaces (1978)
Linear–quadratic optimal control problems for mean-field stochastic differential equations (2013)
A general stochastic maximum principle for SDE s of mean-field type (2011)
Foundations of kinetic theory (1956)
Propagation of chaos for a class of non-linear parabolic equations (1969)
Mimicking the one-dimensional marginal distributions of processes having an I t^o differential (1986)
Sur un type de convergence interm 'e diaire entre la convergence en loi et la convergence en probabilit 'e (1981)
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On the uniqueness of solutions of stochastic differential equations (1971)
Particle representations for a class of nonlinear SPDE s (1999)
On the approximation of measurable functions by continuous functions (1996)
Weak and strong solutions of general stochastic models (2014)
Repr 'e sentation des processus ponctuels multivari 'e s `a l’aide d’un processus de Poisson (1977)
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Portfolio choice with market closure and implications for liquidity premia (2016)
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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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Value in mixed strategies for zero-sum stochastic differential games without I saacs condition (2014)
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Uniqueness of the representation for $ G $-martingales with finite variation (2012)
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Stochastic differential games with asymmetric information (2009)
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One dimensional BSDE s with logarithmic growth application to PDE s (2017)
$ L ^1$ solutions of non-reflected BSDE s and reflected BSDE s with one and two continuous barriers under general assumptions (2019)
Existence of solution to scalar BSDE s with $ L \exp\bigg(\sqrt \frac 2 \lambda \log(1+ L ) \bigg)$-integrable terminal values (2018)
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)
$\mathbb L ^p$ solutions of backward stochastic differential equations with jumps (2017)
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Second order backward SDE with random terminal time (2020)
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BSDE associated with L 'e vy processes and application to PDIE (2003)
BSDE s with stochastic L ipschitz condition (2000)
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Contracting theory with competitive interacting agents (2019)
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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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Dealing with moral hazard and adverse selection simultaneously (1985)
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An exploration in the theory of optimum income taxation (1971)
The investment banking contract for new issues under asymmetric information: delegation and the incentive problem (1980)
Regulating a monopolist with unknown costs (1982)
Non linear pricing (1993)
Monopoly with incomplete information (1984)
Welfare considerations on nonlinear pricing (1979)
Optimal capital structure in agency relationships (1995)
Monitoring of performance in organizational contracting: the case of defense procurement (1988)
Optimal contract mechanisms for principal–agent problems with moral hazard and adverse selection (1991)
Social insurance and redistribution with moral hazard and adverse selection (2006)
Pareto optima and competitive equilibria with adverse selection and moral hazard (1984)
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Noisy observation in adverse selection models (1992)
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Monitoring, moral hazard, asymmetric information, and risk sharing in procurement contracting (1987)
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Ambiguity, risk, and asset returns in continuous time (2002)
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A verification theorem for optimal stopping problems with expectation constraints (2019)
Adding constraints to BSDE s with jumps: an alternative to multidimensional reflections (2014)
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BSDE formulation of combined regular and singular stochastic control problems (2018)
Moral hazard in welfare economics: on the advantage of planner’s advices to manage employees’ actions (2017)
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Existence and uniqueness theorems for solutions of M c K ean– V lasov stochastic equations (2020)
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Backward stochastic differential equations with jumps are stable (2017)
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Contribution aux 'e quations diff 'e rentielles stochastiques r 'e trogrades r 'e fl 'e chies du second ordre (2017)
Generalized second order reflected BSDE s (2020)
'E quations diff 'e rentielles stochastiques sous $ G $-esp 'e rance et applications (2017)
Nonlinear expectation theory and stochastic calculus under K nightian uncertainty (2013)
Do residential customers respond to hourly prices? E vidence from a dynamic pricing experiment (2011)
Dynamic salience with intermittent billing: evidence from smart electricity meters (2014)
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Split incentives in residential energy consumption (2012)
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Partage des co ^u ts et tarification des infrastructures (2006)
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BSDE s with diffusion constraint and viscous H amilton– J acobi equations with unbounded data (2017)
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Partial hedging and cash requirements in discrete time (2016)
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Moral hazard in dynamic risk management (2017)
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Asymptotics for fixed transaction costs (2015)
Multi-asset portfolio optimization with transaction cost (2004)
Option pricing with transaction costs and a nonlinear Black–Scholes equation (1998)
Pricing a contingent claim liability with transaction costs using asymptotic analysis for optimal investment (2014)
Asymptotic analysis for optimal investment in finite time with transaction costs (2012)
Utility maximization trading two futures with transaction costs (2013)
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No-arbitrage in discrete-time markets with proportional transaction costs and general information structure (2006)
Hedging under an expected loss constraint with small transaction costs (2016)
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Dynamic contracting: accidents lead to nonlinear contracts (2015)
Equilibrium in a reinsurance market (1962)
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Backward stochastic differential equations with jumps and related non-linear expectations (2006)
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A stochastic control approach to a robust utility maximization problem (2007)
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