Paper solves investment and consumption problem with utility duality.
problem Continuous-time consumption and investment problem with incomplete markets and stochastic differential utility.
method Introduces a dual problem to study the problem, establishing duality for Epstein-Zin utility and identifying optimal strategies.
result Optimal strategies identified without technical conditions, with dual minimizer interpreted as least favorable market completion.
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 …
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.
This note introduces a duality principle for nonlinear equations.
problem Maximum principles at infinity for nonlinear equations.
method Ahlfors property and Khas'minskii potentials.
result Unified framework for various maximum principles.
This paper studies dynamic stochastic optimization problems parametrized by a random variable. Such problems arise in many applications in operations research and mathematical finance. We give sufficient conditions for the existence of solutions and the absence of a duality gap. Our proof uses extended dynamic programm…
Researchers develop a pricing method for contingent claims under partial information and short selling constraints.
problem Pricing contingent claims with partial information and short selling restrictions.
method Derive a dual problem using conjugate duality theory and conditions for strong duality.
result Characterization of contingent claim prices involving martingale and super-martingale conditions.
This paper optimizes portfolio management in incomplete markets with stochastic factors, considering periodic wealth evaluations.
problem Optimizing portfolio performance in an incomplete market model with stochastic factors and periodic wealth evaluations.
method Developed a martingale duality approach to find optimal portfolio processes and dual minimizers.
result Established the existence of optimal portfolio processes and identified dual minimizers as the 'least favorable' market completion.
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…
Unified approach solves Kyle model with dynamic information.
problem Solving a generalized Kyle model with dynamic information.
method Monge-Kantorovich duality and backward stochastic partial differential equations.
result Characterization of optimal strategies and pricing rules.
The thesis examines stochastic calculus in option pricing with logistic models and numerical methods.
problem Exploring the application of stochastic calculus in option pricing.
method Monte-Carlo Simulation and machine learning algorithms.
result Insights from Peter Carr and Lorenzo Torricelli's convex duality in continuous models.
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.
Paper proves game-theoretic and measure-theoretic expectations match for a specific financial scenario.
problem Proving equivalence between game-theoretic and measure-theoretic probability.
method New broad definition of game-theoretic probability; proving coincidence of expectations for lower semicontinuous positive functionals.
result Coincidence of game-theoretic and measure-theoretic expectations for specific financial scenario.
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.
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.
Paper tackles fast convergence for non-convex strongly-concave min-max problems.
problem Non-convex strongly-concave min-max problems in deep learning.
method Proximal stage-based method with PL condition for faster convergence.
result Established fast convergence in primal objective gap and duality gap.
This survey reviews portfolio choice in settings where investment opportunities are stochastic due to, e.g., stochastic volatility or return predictability. It is explained how to heuristically compute candidate optimal portfolios using tools from stochastic control, and how to rigorously verify their optimality by mea…
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.
We propose a randomized block-coordinate variant of the classic Frank-Wolfe algorithm for convex optimization with block-separable constraints. Despite its lower iteration cost, we show that it achieves a similar convergence rate in duality gap as the full Frank-Wolfe algorithm. We also show that, when applied to the d…
For a stochastic factor model we maximize the long-term growth rate of robust expected power utility with parameter λ∈(0,1). Using duality methods the problem is reformulated as an infinite time horizon, risk-sensitive control problem. Our results characterize the optimal growth rate, an optimal long-term trading s…
A new method estimates SDEs using occupation kernels.
problem Learning multivariate stochastic differential equations (SDEs).
method Two-step procedure: estimate drift, then diffusion. Occupation kernels used in RKHS.
result Validated on simulated and real-world data.
Kernel DRO uses RKHS to optimize under distributional uncertainty.
problem Optimizing under distributional uncertainty with limited knowledge.
method Kernel DRO using RKHS ambiguity sets and duality theory.
result Unified approach to robust and stochastic optimization.
Investors face constraints in Heston's model; optimal allocation differs from naive capped strategy.
problem Optimizing portfolio allocation with convex constraints in Heston's stochastic volatility model.
method Applied duality methods to derive a closed-form solution.
result The optimal constrained portfolio allocation differs from the naive capped portfolio, leading to different wealth outcomes.
Reinforcement learning for continuous-time risk-sensitive asset allocation
problem Continuous-time risk-sensitive asset allocation
method Free energy-entropy duality reformulation and q-learning actor-critic method result Optimal policy learning with high accuracy
Defines stochastic integrals for arbitrary collections of continuous semimartingales in finance.
problem Developing a framework for stochastic integration in complex financial models.
method Finite-dimensional approximation and operational characterization of stochastic integrals.
result Enriched class of wealth processes leads to exact analogues of optional decomposition and hedging duality.
The paper optimizes investment strategies with random endowments and transaction costs.
problem Maximizing utility with random endowments and transaction costs.
method Using consistent price system (CPS) and duality theory, the paper establishes optimal investment solutions.
result Existence and uniqueness of optimal solution for utility maximization problem.
This work explores duality between nonlinear potential theory and geometry.
problem Investigating properties of nonlinear equations on manifolds.
method Analyzing parabolicity and maximum principles at infinity for non-linear equations.
result Shows a unifying duality between properties and existence of Khas'minskii potentials.
The paper extends collective arbitrage concepts to multi-agent markets with cooperation.
problem Understanding collective market completeness and pricing in multi-agent systems.
method Develops new techniques and theorems to establish collective pricing-hedging duality and collective replication.
result Established a Second Fundamental Theorem of Asset Pricing in cooperative multi-agent settings.
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.
Paper proves linear convergence of R-FDM and RC-FDM under weak strong convexity.
problem Optimizing SVM dual problem and LASSO problem.
method Randomized feasible descent method (R-FDM) and coordinate-wise random feasible descent method (RC-FDM).
result Both R-FDM and RC-FDM converge linearly under weak strong convexity assumption.
Dual IHT algorithm solves NP-hard non-convex sparse minimization problems.
problem Non-convex sparse minimization with ℓ2-regularized loss function. method Developed a dual IHT algorithm for maximizing the non-smooth dual objective.
result Sparse recovery performance is invariant to RIP, superior to primal IHT algorithms.
Solves risk-sensitive investment via duality, entropic regularization, and RL.
problem Risk-sensitive portfolio management in a factor-based setting.
method Free energy-entropy duality, Kuroda-Nagai change-of-measure, RL algorithm.
result Direct analytical solution, explicit controls, two interpretations of optimal allocation.
New framework for DP-SMO with near-optimal privacy-loss trade-off.
problem Optimal trade-off between privacy and population loss in DP-SMO.
method General framework using Phased-ERM method and black-box optimization.
result Near-linear time algorithms with near-optimal guarantees.
Paper analyzes complexity of PSGLA for sampling log-concave distributions.
problem Sampling from log-concave distributions with composite potentials.
method Uses primal-dual interpretation and duality gap to analyze PSGLA complexity.
result Complexity of PSGLA is O(1/ε2) for strongly convex potentials. New method uses optimal transport to determine best γ for almost stochastic dominance.
problem Tackles determining the best γ for almost stochastic dominance. method Generalizes optimal transport problem to determine γ for various test functions. result Derives dual characterization of order relations in terms of expectation comparisons.
We reveal an interesting convex duality relationship between two problems: (a) minimizing the probability of lifetime ruin when the rate of consumption is stochastic and when the individual can invest in a Black-Scholes financial market; (b) a controller-and-stopper problem, in which the controller controls the drift a…
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.
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…
Epoch-GDA achieves optimal convergence rate for SCSC min-max problems.
problem Solving stochastic min-max problems with strong convexity and strong concavity.
method Epoch-wise stochastic gradient descent ascent method (Epoch-GDA) without additional assumptions.
result Achieves the optimal rate of O(1/T) for the duality gap of general SCSC min-max problems. Survey of mathematical foundations for reinforcement learning.
problem Design and analysis of modern reinforcement learning algorithms.
method Organizes mathematical structures from probability, optimization, and operator theory.
result Unified mathematical entry point for researchers in various fields.
Study proves existence, uniqueness, and stability for specific stochastic Volterra equations.
problem Analyzing existence, uniqueness, and stability of affine stochastic Volterra equations with L1-kernels. method Approximations with L2-kernels, stability result, duality argument, deterministic Riccati--Volterra integral equation. result Established weak uniqueness for the equations using Fourier--Laplace transform and a deterministic Riccati--Volterra integral equation.
The important application of semi-static hedging in financial markets naturally leads to the notion of quasi self-dual processes which is, for continuous semimartingales, related to symmetry properties of both their ordinary as well as their stochastic logarithms. We provide a structure result for continuous quasi self…
We study the stochastic control problem of maximizing expected utility from terminal wealth under a non-bankruptcy constraint. The wealth process is subject to shocks produced by a general marked point process. The problem of the agent is to derive the optimal insurance strategy which allows "lowering" the level of the…
New duality found linking neural network weights and activities for better generalization.
problem Understanding and improving neural network generalization.
method Activity-weight duality mapping between neural network layers.
result Generalization loss can be decomposed into geometric factors of sharpness and weight standard deviation.
New bounds for VIX derivatives pricing using LS Monte Carlo.
problem Pricing VIX derivatives due to the square root of expected realised variance.
method Least Squares Monte Carlo with stochastic duality and adjustments.
result Effective upper and lower bounds for VIX derivatives pricing.
Paper introduces a new method for risk-sensitive investment management using RL.
problem Risk-sensitive portfolio management with unknown model parameters.
method Combines RL and risk-sensitive stochastic control with Gaussian perturbations for exploration.
result Endogenous relative-entropy regularization and optimal investment strategy derived.
Solves multi-objective risk-averse portfolio optimization with convex risk measures.
problem Portfolio optimization under risk and uncertainty.
method Convex vector optimization, Benson's algorithm, Lagrangian duality, scenario-wise decomposition.
result Developed methods to solve complex portfolio optimization problems.
Proves Poincaré duality for Hopf algebroids with bijective antipode.
problem Proving Poincaré duality for Hopf algebroids.
method Using twisted Poincaré duality and bijective antipode properties.
result Recovering and extending known Poincaré dualities for Hopf algebroids.
New proof of chain duality for simplicial complexes.
problem Proving the existence of chain duality for chain complexes over simplicial complexes.
method Geometric and conceptual treatment of chain duality.
result Fundamental for Ranicki's surgery exact sequence.