The paper simplifies utility maximization problems using dynamic convex duality.
problem Constrained utility maximization problems.
method Formulating primal and dual problems, using FBSDEs, and characterizing optimal controls.
result Explicit dynamic characterization of optimal controls and wealth processes.
Develops a method for near-optimal asset allocation with trading constraints.
problem Optimizing investment strategies in financial markets with trading constraints.
method Dual-control method using convex duality to generate bounds on optimal value function.
result Derives near-optimal asset allocation explicitly and demonstrates its accuracy in a real financial market.
Convex duality for two two different super--replication problems in a continuous time financial market with proportional transaction cost is proved. In this market, static hedging in a finite number of options, in addition to usual dynamic hedging with the underlying stock, are allowed. The first one the problems consi…
We consider an investor with constant absolute risk aversion who trades a risky asset with general Ito dynamics, in the presence of small proportional transaction costs. Kallsen and Muhle-Karbe (2012) formally derived the leading-order optimal trading policy and the associated welfare impact of transaction costs. In th…
The paper solves a finance problem using stochastic equations.
problem Risk minimization with portfolio constraints in financial markets.
method Uses Forward and Backward Stochastic Differential Equations (FBSDEs) to model and solve the problem.
result Explicit representations of solutions to quadratic risk minimization problems with constraints are derived.
The paper explores how duality applies to reinforcement learning.
problem Exploring the use of duality in reinforcement learning.
method Demonstrates duality in various RL methods, including value iteration and approximations.
result Lagrangian duality is found to be prevalent in reinforcement learning.
We study a robust maximization problem from terminal wealth and consumption under a convex constraints on the portfolio. We state the existence and the uniqueness of the consumption-investment strategy by studying the associated quadratic backward stochastic differential equation (BSDE in short). We characterize the op…
The paper analyzes portfolio selection with non-linear wealth dynamics and random coefficients.
problem Mean-variance portfolio selection with non-linear wealth dynamics and random coefficients.
method Solves an auxiliary stochastic control problem to construct a candidate portfolio, verifies optimality using convex duality, and provides the efficient frontier.
result Obtains the efficient frontier in closed form, showing people prefer riskless assets over classical linear markets.
Study optimal portfolio management with periodic evaluations in stochastic models, considering convex constraints.
problem Optimal portfolio management under ratio-type periodic evaluations in stochastic factor models with convex trading constraints.
method Transformed infinite horizon optimal control problem into an auxiliary terminal wealth optimization problem. Introduced an auxiliary unconstrained optimization problem in a modified market model. Used martingale duality approach to establish dual minimizer and optimal unconstrained wealth process.
result Derived and verified the optimal constrained portfolio process for the original problem over an infinite horizon.
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
Unified treatment of reinforcement learning via convex duality.
problem Applying Fenchel-Rockafellar duality to reinforcement learning.
method Unified derivation of various RL settings using convex duality.
result Ability to perform policy evaluation and on-policy policy gradient with offline data.
Introduces new info-geometric structure for dynamics on graphs and hypergraphs.
problem Modeling dynamics on discrete structures like graphs and hypergraphs.
method Introduces two dually flat structures: one on vertex space and another on edge space.
result Extends gradient flows to include nonequilibrium dynamics.
This paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration. The framework unifies many well-known duality frameworks from operations research and mathematical finance. The unification allows the extension …
We study the convex duality method for robust utility maximization in the presence of a random endowment. When the underlying price process is a locally bounded semimartingale, we show that the fundamental duality relation holds true for a wide class of utility functions on the whole real line and unbounded random endo…
Optimizes risk management in incomplete markets using convex risk measures.
problem Risk management in financial markets with incomplete information.
method Dynamic optimization problem split into static and representation problems; convex duality methods used.
result Optimal strategy involves superhedging a modified claim with a randomized test.
The paper analyzes the mean field Langevin dynamics and its convergence rate.
problem The convergence property of the mean field Langevin dynamics in the context of neural networks.
method The analysis uses a proximal Gibbs distribution and techniques from convex optimization.
result A concise convergence rate analysis of the mean field Langevin dynamics in both continuous and discrete time settings.
Solves signal recovery from few linear measurements using convex duality.
problem Recovering signals from limited linear measurements in various applications.
method Develops a convex-concave min-max reformulation for linear inverse problems.
result Simple ascent-descent algorithms for solving linear inverse problems.
Study utility maximization with costs under uncertain models.
problem Maximizing utility in a market with transaction costs and model uncertainty.
method Transformed semi-static utility maximization problem on an enlarged space using randomization techniques and dynamic programming.
result Existence of optimal strategy and convex duality theorem proved.
Deep neural networks with multiple branches are less non-convex, improving performance.
problem Improving neural network performance through multi-branch architectures.
method Quantitative measurement of duality gap for neural networks with multi-branches and various activation functions.
result The duality gap of multi-branch neural networks decreases as the number of branches increases, leading to less non-convex optimization problems.
We price and hedge American options robustly in continuous time.
problem Pricing and hedging American options in continuous time with model uncertainty.
method Assumes continuous semimartingale asset prices and closed convex constraints on volatility. Proves robust pricing-hedging duality and identifies American options as European options on an enlarged space.
result We prove robust pricing-hedging duality and show it holds against richer models with dynamic trading of European options.
The paper explores optimal investment and contingent claim valuation in illiquid markets using convex duality.
problem Optimal investment and contingent claim valuation in markets with nonlinear trading costs and portfolio constraints.
method Convex duality theory applied to markets with general conditions on utility functions and market models.
result Dual expressions decompose into terms for risk preferences, trading costs, and portfolio constraints.
We develop a general theory of convex duality for certain singular control problems, taking the abstract results by Kramkov and Schachermayer (1999) for optimal expected utility from nonnegative random variables to the level of optimal expected utility from increasing, adapted controls. The main contributions are the f…
A general duality proof for Wasserstein distributionally robust optimization.
problem Optimizing under uncertainty with Wasserstein distance.
method One-dimensional convex analysis and interchangeability principle.
result General duality result holds for various distributions and costs.
A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: (i) The optimal terminal wealth X∗(T):=Xφ∗(T) of the problem to maximize the expected U-utility of the terminal wealth Xφ(T) generated by admissible portfolios $\varp…
Develops a convex duality framework for analyzing GANs.
problem Analyzing how GANs behave under different discriminator constraints.
method Introduces a convex duality framework to interpret GANs under constrained discriminators.
result Shows that the GAN formulation can be interpreted as minimizing a divergence to penalized moments of the data distribution.
Novel framework shows exact recoverability of matrix completion and robust PCA with optimal sample complexity.
problem Efficiently recovering a hidden matrix from limited observations.
method Strong duality and novel analytical framework for non-convex matrix factorization problems.
result Exact recoverability and strong duality hold with nearly-optimal sample complexity guarantees for matrix completion and robust PCA.
Unified analysis of conjugate gradients and accelerated methods using duality gap.
problem Minimizing convex quadratic functions efficiently.
method Approximate Duality Gap Technique to unify conjugate gradients and accelerated methods.
result Unified and self-contained proof of conjugate gradients without relying on Chebyshev polynomials.
We consider the robust utility maximization using a static holding in derivatives and a dynamic holding in the stock. There is no fixed model for the price of the stock but we consider a set of probability measures (models) which are not necessarily dominated by a fixed probability measure. By assuming that the set of …
Paper improves data embeddings with Lagrange duality.
problem Near isometric orthogonal embeddings for data points.
method Formulated as non-convex optimization, used Lagrange duality for relaxation, and provided a polynomial time algorithm.
result Achieved better approximation guarantees and lower distortion compared to baselines.
The paper derives robust dualities for pricing and hedging in financial markets.
problem Tackles pricing and hedging of contingent claims in financial markets with various constraints.
method Derives dualities for super- and subhedging, considering strict and relaxed versions.
result Yields tighter price bounds and robust hedging strategies.
Optimizes portfolios with constraints and stochastic factors, deriving explicit solutions.
problem Optimizing expected utility in an incomplete market with stochastic factors and convex constraints.
method Fundamental duality results and HJB PDE, derived condition for exponential affine solutions.
result Explicit expressions for optimal allocations and Riccati ODE solutions in specific markets.
Paper studies convex risk measures linked to optimization.
problem Risk assessment in finance and insurance.
method Investigates a wide class of risk measures on Orlicz spaces.
result Characterizes the dual of risk measures and provides complementary representations.
Study on pricing and hedging for American options in dynamic and static markets.
problem Investigating pricing-hedging duality for American options in discrete time financial models.
method Abstract setting with universal enlargement and dynamic consistency, applied to robust framework examples.
result Recovery of pricing-hedging duality through dynamic consistency and market extensions.
We define a class of L-convex-concave subsets of RPn, where L is a projective subspace of dimension l in RPn. These are sets whose sections by any (l+1)-dimensional space L' containing L are convex and concavely depend on L'. We introduce an L-duality for these sets, and prove that the L-dual to an L-…
New space for valuations in non-Archimedean setting with duality properties.
problem Developing a non-Archimedean analogue of classical valuation spaces.
method Construction of a new space of valuations with similar structures to classical spaces.
result The new space satisfies Poincaré duality and hard Lefschetz theorem.
We analyze deep neural networks using convex duality to reveal hidden layer structures.
problem Understanding the structure of deep neural networks.
method Introducing a convex analytic framework to characterize hidden layer weights.
result Optimal hidden layer weights align with previous layers via duality.
Wasserstein GANs are shown to have hidden convexity, enabling exact solutions with convex optimization.
problem Non-convex and non-concave optimization in GANs.
method Convex duality analysis of Wasserstein GANs with two-layer neural network discriminators.
result Wasserstein GANs can be solved exactly with convex optimization under certain conditions.
The paper extends Merton's problem by adding benchmark tracking, finding optimal strategies.
problem Maximizing consumption utility with a trade-off against benchmark performance.
method Developed a convex duality theorem and derived optimal strategies for specific cases.
result Found optimal portfolio and consumption strategies for CRRA utility and geometric Brownian motion benchmarks.
Optimal risk sharing without convex preferences using aggregate convexity.
problem Risk sharing among non-convex preferences.
method Aggregate convexity principles and Lyapunov convexity, combined with approximation arguments for law invariant risk measures.
result Derivation of a computationally tractable formula for the conjugate of the value function.
In a discrete-time market, we study model-independent superhedging, while the semi-static superhedging portfolio consists of {\it three} parts: static positions in liquidly traded vanilla calls, static positions in other tradable, yet possibly less liquid, exotic options, and a dynamic trading strategy in risky assets …
Develops exact convex optimization formulations for neural networks.
problem Training two-layer neural networks with rectified linear units.
method Uses semi-infinite duality and minimum norm regularization to develop exact convex optimization formulations.
result Shows equivalence of ReLU networks trained with weight decay to block ℓ1 penalized convex models. Study utility maximization with transaction costs and random endowment using numéraire-based model.
problem Maximizing utility with transaction costs and random endowment.
method Numéraire-based model and convex duality.
result Established standard convex duality results under proportional transaction costs.
New approach to asset pricing without martingale measures.
problem No-arbitrage condition and martingale measures in financial asset pricing theory.
method Convex duality and Fenchel conjugate for super-replication cost estimation.
result Super-hedging problem leads to a new condition called Absence of Immediate Profit (AIP).
Proves existence of AdS manifold with prescribed metrics on boundary.
problem Prescribing metrics on the boundary of AdS 3-manifolds.
method Using duality between convex space-like surfaces in AdS₃, proves existence of AdS manifold with prescribed metrics.
result Existence of AdS manifold with prescribed metrics on boundary.
Study collective pricing and hedging with admissible risk exchanges forming a finitely generated convex cone.
problem Collective pricing and hedging with exchanges forming a finitely generated convex cone.
method Extend collective First Fundamental Theorem of Asset Pricing and pricing-hedging duality.
result No collective arbitrage implies the closedness of the aggregate feasibility cone.
New method calculates super-hedging prices with transaction costs.
problem Super-hedging European contingent claims under proportional transaction costs.
method Explicit recursive scheme based on convex duality and Legendre-Fenchel transform.
result Computes super-hedging price and optimal strategy without martingale arguments.
New method solves complex optimization problems efficiently.
problem Optimizing complex functions with inner expectations in machine learning.
method Combines variance reduction methods with duality-free techniques.
result Proves linear convergence for convex and non-convex cases.
DGKIP extends KIP for dataset distillation without bi-level optimization.
problem Efficiently distill datasets for various loss functions.
method Leverages duality theory to avoid bi-level optimization.
result DGKIP supports a wider range of loss functions.