Study solves optimal portfolio selection using HJB equation.
problem Optimal portfolio selection problem.
method Maximal monotone operator method, Banach fixed-point theorem, Fourier transform, monotone operators technique.
result Existence and uniqueness of solution to HJB equation.
We study utility maximization for power utility random fields with and without intermediate consumption in a general semimartingale model with closed portfolio constraints. We show that any optimal strategy leads to a solution of the corresponding Bellman equation. The optimal strategies are described pointwise in term…
Optimizes portfolios with costs, showing existence of optimal strategies.
problem Risk-sensitive portfolio optimization with transaction costs.
method Log-return i.i.d. framework, Bellman equation analysis.
result Existence of optimal strategies for risk-averse and risk-seeking cases.
Deep neural nets approximate high-dimensional HJB equations efficiently.
problem Approximating solutions to high-dimensional HJB equations.
method Deep neural networks for approximating solutions.
result Deep neural networks can approximate solutions without the curse of dimensionality.
New RL algorithm minimizes distributional learning error.
problem Improving distributional reinforcement learning for better error minimization.
method Proposes a new model-based algorithm with theoretical minimax optimality.
result Proves minimax optimality for approximating return distributions.
Paper solves investment strategy optimization with deep learning.
problem Maximizing investor utility with optimal asset allocation.
method Solves PDEs with Deep Galerkin method.
result Deep learning algorithm outperforms finite difference method.
Optimal contracts are found for agents with quadratic effort costs.
problem Finding optimal contracts in principal-agent problems with quadratic effort costs.
method Modeling the problem using Hamilton-Jacobi-Bellman (HJB) equations and proving the existence of classical solutions.
result Existence of optimal contracts for agents with quadratic effort costs is proven.
The aim of this paper is to construct and analyze solutions to a class of Hamilton-Jacobi-Bellman equations with range bounds on the optimal response variable. Using the Riccati transformation we derive and analyze a fully nonlinear parabolic partial differential equation for the optimal response function. We construct…
The problem of determining the European-style option price in the incomplete market has been examined within the framework of stochastic optimization. An analytic method based on the discrete dynamic programming equation (Bellman equation) has been developed that gives the general formalism for determining the option p…
In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a quasi-linear paraboli…
The paper solves a complex financial optimization problem using a novel mathematical technique.
problem Optimizing portfolio selection in financial markets.
method Maximal monotone operator method and Riccati transformation.
result Existence and uniqueness of a solution to the transformed parabolic equation in a Sobolev space.
The paper explores solutions to the distributional Bellman equation in reinforcement learning.
problem Distributional reinforcement learning considers complete return distributions, not just expected returns.
method Study existence and uniqueness of solutions to general distributional Bellman equations, linking them to multivariate affine equations.
result Any solution to a distributional Bellman equation can be derived from a multivariate affine distributional equation.
We study the structure of a simple dynamic optimization problem consisting of one state and one control variable, from a physicist's point of view. By using an analogy to a physical model, we study this system in the classical and quantum frameworks. Classically, the dynamic optimization problem is equivalent to a clas…
The Bellman error is a poor proxy for value function accuracy, even with all state-action pairs.
problem The Bellman error is a poor proxy for the accuracy of the value function.
method Study of the Bellman equation as a surrogate objective for value prediction accuracy.
result The magnitude of the Bellman error is only weakly related to the distance to the true value function, even with all state-action pairs.
Optimizes control of infectious disease spread using stochastic methods.
problem Optimizing control of highly infectious diseases like COVID-19.
method Reformulated Hamilton-Jacobi-Bellman equation as stochastic minimum principle, leading to forward-backward stochastic differential equations.
result Numerous numerical solutions presented under various scenarios.
Improved risk-sensitive RL with exponential Bellman equation and better regret bounds.
problem Exponential gap between upper and lower bounds in risk-sensitive RL.
method Identified and addressed deficiencies in existing algorithms and analysis; developed novel analysis and exploration mechanism.
result Improved regret upper bounds over existing ones.
Deep Bellman Hedging uses reinforcement learning to optimize financial portfolio hedging.
problem Optimizing financial portfolio hedging with derivatives and trading frictions.
method Actor-critic reinforcement learning algorithm with continuous state and action spaces.
result Trained model provides optimal hedge for any initial portfolio and market state.
A neural network approach solves optimal decumulation problems for pension plans.
problem Optimal asset allocation and withdrawal strategies for DC pension holders.
method Data-driven neural network optimization with customized activation functions.
result The neural network approach learns near-optimal solutions comparable to HJB PDE methods.
A model optimizes carbon emission reduction and allowance purchasing for companies.
problem Optimizing carbon emissions and allowance purchasing for companies.
method Established an optimal control model involving two stochastic processes with two control variables, converted into an HJB equation, proved existence and uniqueness of solution.
result Proved the existence and uniqueness of the solution to the HJB equation.
The Noether theorem is extended to stochastic control problems using contact symmetries.
problem Stochastic optimal control problems.
method Exploiting jet bundles and contact geometry, the authors prove the existence of conserved quantities.
result Optimal control problems admit infinitely many conserved quantities in the form of local martingales.
Corrects gaps in a method for optimizing high-frequency trading strategies.
problem Optimizing bid and ask limit order strategies in high-frequency trading.
method Uses an approximation method based on Avellaneda and Stoikov's 2008 article, correcting gaps found in it.
result The main answer in Avellaneda and Stoikov's article remains unchanged despite corrections.
The paper tackles non-cumulative objectives in reinforcement learning and proposes modifications to existing algorithms.
problem Optimizing objectives that are not naturally expressed as summations of rewards in various fields.
method The paper modifies the Bellman optimality equation to handle non-cumulative objectives by replacing summation with a generalized operation.
result The modified Bellman updates can converge to the globally optimal solution under certain conditions.
Deep nets solve MDPs without high dimensions.
problem Solving Bellman equations for MDPs in high dimensions.
method Deep neural networks with ReLU activation approximating payoff and transition functions.
result Deep nets can approximate Q-functions in polynomially bounded parameters. In this paper we investigate a dynamic stochastic portfolio optimization problem involving both the expected terminal utility and intertemporal utility maximization. We solve the problem by means of a solution to a fully nonlinear evolutionary Hamilton-Jacobi-Bellman (HJB) equation. We propose the so-called Riccati met…
The main purpose of this paper is to analyze solutions to a fully nonlinear parabolic equation arising from the problem of optimal portfolio construction. We show how the problem of optimal stock to bond proportion in the management of pension fund portfolio can be formulated in terms of the solution to the Hamilton-Ja…
A new macroscopic market making model connects market making and optimal execution.
problem Connecting market making and optimal execution problems.
method Using continuous processes for orders, the model bridges the gap between market making and optimal execution.
result Demonstrates the model's effectiveness through various noise and intensity function scenarios.
Deep learning for HJB PDEs using synthetic data and residual minimization.
problem Solving Hamilton-Jacobi-Bellman PDEs for optimal control problems.
method Gradient-augmented synthetic dataset for supervised learning, residual minimization.
result Improves accuracy and efficiency of deep learning for HJB PDEs.
Study optimal futures trading strategies for assets with multiscale central tendency price model.
problem Optimal dynamic trading of futures with multiscale central tendency price model.
method Derive no-arbitrage futures prices, solve HJB equations for optimal strategies.
result Optimal trading strategies depend on asset parameters and futures risk premia.
Study optimal investment strategies for an insurer in two currency markets.
problem Maximizing expected exponential utility of terminal wealth for an insurer in two currency markets.
method Dynamic programming method applied to solve Hamilton-Jacobi-Bellman equations.
result Optimal investment strategies and value functions are derived.
We solve continuous-time reinforcement learning using distributional Hamilton-Jacobi-Bellman equations.
problem Predicting the distribution of returns in continuous-time, stochastic environments.
method We derive a distributional Hamilton-Jacobi-Bellman equation for Itô diffusions and Feller-Dynkin processes, and propose an algorithm based on a JKO scheme.
result We propose an online control algorithm that can be used to approximately solve the distributional HJB equation.
The paper calibrates SPX and VIX options using optimal transport.
problem Joint calibration of SPX and VIX options or futures.
method Semimartingale optimal transport problem with PDE formulation and dual formulation.
result The model accurately calibrates SPX, VIX options, and futures simultaneously.
This paper optimizes DC pension plan investments using O-U process and loan.
problem Optimizing investment strategy for DC pension plans under specific market conditions.
method Dynamic programming and Hamilton-Jacobi-Bellman equation to derive optimal investment strategy.
result Explicit expression for optimal investment strategy derived.
Study optimal consumption and investment strategies with leverage constraints using Epstein-Zin utility.
problem Optimal portfolio choice under leverage constraints and Epstein-Zin utility.
method Established viscosity solution to HJB equation, demonstrated smoothness, characterized optimal strategies, derived explicit solutions.
result Explicit solutions for optimal consumption and investment strategies under leverage constraints.
We consider the exploration/exploitation problem in reinforcement learning. For exploitation, it is well known that the Bellman equation connects the value at any time-step to the expected value at subsequent time-steps. In this paper we consider a similar \textit{uncertainty} Bellman equation (UBE), which connects the…
In this paper, we extend the jump-diffusion model proposed by Davis and Lleo to include jumps in asset prices as well as valuation factors. The criterion, following earlier work by Bielecki, Pliska, Nagai and others, is risk-sensitive optimization (equivalent to maximizing the expected growth rate subject to a constrai…
New method uses TT approximations to solve HJB equations for efficient sampling.
problem Efficiently sampling from complex probability densities.
method Direct time integration of HJB equations using Tensor Train compression.
result Sample-free, dimensionality-avoiding integration method.
The paper studies the First Order BSPDEs (Backward Stochastic Partial Differential Equations) suggested earlier for a case of multidimensional state domain with a boundary. These equations represent analogs of Hamilton-Jacobi-Bellman equations and allow to construct the value function for stochastic optimal control pro…
The paper analyzes optimal consumption with past spending maximum as a reference.
problem Optimal consumption with past spending maximum as a reference.
method Path-dependent exponential utility, Hamilton-Jacobi-Bellman (HJB) equation, dual transform, smooth-fit principle.
result Closed-form solutions for optimal investment and consumption strategies in each region.
New RL formulation for maximizing maximum reward in molecule generation.
problem Traditional RL frameworks do not fit real-world applications like drug discovery.
method Formulated a new objective function to maximize maximum reward, derived Bellman equation, introduced operators, and proved convergence.
result Achieved state-of-the-art results in molecule generation.
Investigates optimal insurance and reinsurance strategies with incomplete market information.
problem Optimal investment-reinsurance problem for insurance companies with unknown market risk.
method Converted the original problem into a filtered observation problem, applied stochastic control theory, and used Hamilton-Jacobi-Bellman equations.
result Explicit formulas for value function and optimal strategy provided.
Model quantifies uncertainty's impact on European option prices.
problem Uncertainty in market volatility risk affects option pricing.
method Hamilton-Jacobi-Bellman framework and finite element method.
result Dependence of Delta on uncertainty is nonlinear and varied.
The paper studies risk-sensitive MDPs with recursive risk measures.
problem Risk-sensitive decision-making in MDPs with unbounded costs.
method Recursive application of static risk measures, Bellman equation derivation, existence of optimal policies.
result Existence of Markovian optimal policies for infinite planning horizons, contractive model for stationary optimal policy.
Optimal dividend payout strategy found for Brownian risk model with ratcheting constraint.
problem Optimal dividend payout from a surplus process governed by Brownian motion with drift under ratcheting constraint.
method Solved a two-dimensional optimal control problem using viscosity solutions of Hamilton-Jacobi-Bellman equations.
result Threshold and curve strategies identified as optimal for different dividend rate sets.
Paper solves Bayesian bandit problem with continuous-time limit and approximate policy.
problem Finding optimal policy in Bayesian bandit problems with large horizons.
method Reformulates Bayesian bandit problem as continuous Hamilton-Jacobi-Bellman (HJB) equation and proposes approximate Bayes-optimal policy.
result Approximate Bayes-optimal policy for large horizons with constant computational cost.
From the Hamilton-Jacobi-Bellman equation for the value function we derive a non-linear partial differential equation for the optimal portfolio strategy (the dynamic control). The equation is general in the sense that it does not depend on the terminal utility and provides additional analytical insight for some optimal…
We consider a utility maximization problem for an investment-consumption portfolio when the current utility depends also on the wealth process. Such kind of problems arise, e.g., in portfolio optimization with random horizon or with random trading times. To overcome the difficulties of the problem we use the dual appro…
Study optimal investment strategies with entropy regularization in volatile markets.
problem Optimal portfolio selection under stochastic volatility with constraints.
method Entropy-regularized relaxed controls, dynamic programming, nonlinear PDEs.
result Existence of classical solutions to nonlinear HJB equation for value function.
Optimal control theory connects diffusion models to generative modeling.
problem Sampling from unnormalized densities in statistics and computational sciences.
method Deriving a Hamilton-Jacobi-Bellman equation and applying control theory to minimize Kullback-Leibler divergence.
result Time-reversed diffusion sampler (DIS) outperforms other diffusion-based sampling methods.