Study solves HJB equations for time-inconsistent control problems.
problem Time-inconsistent deterministic linear quadratic control problems.
method Characterized solutions using Riccati equations with integral terms, proving uniqueness.
result Uniqueness of solutions to equilibrium HJB equations proved.
We consider a semilinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity which is related to a stochastic control problem with fuel constraint. The fuel constraint translates into a singular initial condition for the HJB equation. We first propose a transformation based on a change of vari…
Study uses FEM for HJB in option pricing with borrowing fees, improving accuracy and efficiency.
problem Optimal control problems in financial markets with frictions.
method Finite element method with non-uniform mesh, theta-scheme time integration, Newton-type algorithm.
result Efficient and accurate solution to HJB equation for option pricing with borrowing fees.
In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if admissible controls satisfy an integrability condition or if it is contin…
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.
In this paper, we present a novel penalty approach for the numerical solution of continuously controlled HJB equations and HJB obstacle problems. Our results include estimates of the penalisation error for a class of penalty terms, and we show that variations of Newton's method can be used to obtain globally convergent…
Paper explores solving HJB equations using neural networks.
problem Solving high-dimensional time-dependent HJB equations.
method Neural Galerkin methods with nonlinearly parametrized trial functions.
result Closed-form solutions for trial functions.
Solves pair trading problem using consumption-investment theory.
problem Pair trading consumption-investment problem
method Reduces HJB equation to a linear parabolic equation solvable explicitly
result Simple solution to pair trading problem
Proposes a new uncertain volatility model with worst-case scenario analysis.
problem Modeling and pricing options under uncertain volatility.
method Connection between G-HJB equations and 2BSDEs for option pricing.
result Derives a limit model for worst-case price scenario.
We consider the value function originating from an expected utility maximization problem with finite fuel constraint and show its close relation to a nonlinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity. On one hand, we give a so-called verification argument based on the dynamic progr…
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.
Paper solves a complex stopping problem using regularization and HJB equations.
problem Time-inconsistent mean-variance optimal stopping problem
method Vanishing regularization method to derive HJB equations and prove existence of solutions
result Formally recovers variational inequalities for original problem
We introduce a dynamic credit portfolio framework where optimal investment strategies are robust against misspecifications of the reference credit model. The risk-averse investor models his fear of credit risk misspecification by considering a set of plausible alternatives whose expected log likelihood ratios are penal…
Study optimal stopping in random exploration, deriving HJB and designing a reinforcement learning algorithm.
problem Optimal stopping problem in continuous time with random exploration.
method Transformed optimal stopping to optimal control problem, derived HJB equation, designed reinforcement learning algorithm.
result Convergence rate of policy iteration and comparison to classical optimal stopping.
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 controlled contagion with state-dependent killing, proving a comparison principle.
problem Analyzing controlled McKean--Vlasov contagion with state-dependent killing.
method Proof of a comparison principle using Wasserstein smooth-gauge comparison and killing-jump absorption estimates.
result Established a comparison principle for the two-population killed-particle HJB.
A new method solves complex financial equations efficiently.
problem Solving worst-case and best-case prices for two-factor uncertain volatility models.
method Decompose and integrate, then optimize; piecewise constant control; closed-form Green's functions; 2D convolution integrals; monotone numerical integration; Fast Fourier Transforms.
result The method efficiently computes the value function and optimal control, converging to the viscosity solution of the HJB equation.
Develops a new method for pricing GMWBs with jumps and stochastic interest rates.
problem Pricing guaranteed minimum withdrawal benefits (GMWBs) with jumps and stochastic interest rates.
method Combines semi-Lagrangian method with Fourier pricing and Green's function.
result Mathematically demonstrates convergence to the viscosity solution of the HJB-QVI.
Paper tackles time inconsistency in portfolio management with stochastic volatility and power utility.
problem Time inconsistency in portfolio management with stochastic volatility and power utility.
method Extended Hamilton Jacobi Bellman (HJB) equation, fixed point iteration, and linear parabolic PDE.
result Subgame perfect strategies are characterized and solved through numerical experiments.
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.
We extend the stochastic Perron method to analyze the framework of stochastic target games, in which one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the other player. Within this framework, our method produces a viscosity sub-solut…
Unified framework for growth models with environmental risk and pollution-dependent disasters.
problem Analyzing how rare but catastrophic shocks interact with capital accumulation and pollution in stochastic growth models.
method General Poisson point process formulation leading to non-local HJB equations with closed-form solutions.
result Unified framework captures how environmental degradation amplifies macroeconomic vulnerability and strengthens incentives for abatement.
This paper concerns the continuous time mean-variance portfolio selection problem with a special nonlinear wealth equation. This nonlinear wealth equation has a nonsmooth coefficient and the dual method developed in [6] does not work. We invoke the HJB equation of this problem and give an explicit viscosity solution of…
Model stock price dynamics using semi-Markov processes.
problem Model stock price dynamics through a semi-Markov process.
method Use semi-Markov process with Poisson random measure, establish existence and uniqueness of solution, derive HJB equation.
result Obtain expressions for optimal controls and value function using HJB equation.
In this paper, we study the dividend strategies for a shareholder with non-constant discount rate in a diffusion risk model. We assume that the dividends can only be paid at a bounded rate and restrict ourselves to the Markov strategies. This is a time inconsistent control problem. The extended HJB equation is given an…
Deep learning method proves convergence for high-dimensional PDEs.
problem Solving high-dimensional nonlinear PDEs for mean field control problems.
method Deep Galerkin method (DGM) for Hamilton-Jacobi-Bellman (HJB) equations.
result DGM converges to the true value function of mean field control problems.
We consider a time-consistent mean-variance portfolio selection problem of an insurer and allow for the incorporation of basis (mortality) risk. The optimal solution is identified with a Nash subgame perfect equilibrium. We characterize an optimal strategy as solution of a system of partial integro-differential equatio…
Study bond market making with hit-ratio target using optimal control and HJB equations.
problem Optimizing bond market making with hit-ratio target in OTC markets.
method Stochastic optimal control approach, dualizing hit-ratio target, HJB equation, Riccati equation, linearization.
result Explicit quote decompositions into riskless spread, inventory-risk correction, and hit-ratio correction.
This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet …
Study optimal consumption for loss-averse agents considering past spending peaks.
problem Optimal consumption for loss-averse agents with reference to past spending maximum.
method Adopted S-shaped utility, concave envelope, HJB variational inequality, dual transform, and smooth-fit conditions.
result Obtained piecewise closed-form solutions for optimal consumption and investment control.
Study portfolio selection with exogenous and endogenous transaction costs using deep learning.
problem Portfolio selection with both exogenous and endogenous transaction costs.
method Deep learning-driven policy iteration scheme for high-dimensional HJB equations.
result Proposes a scheme to address the curse of dimensionality and adapt to high-dimensional control spaces.
In this article we extend earlier work on the jump-diffusion risk-sensitive asset management problem [SIAM J. Fin. Math. (2011) 22-54] by allowing jumps in both the factor process and the asset prices, as well as stochastic volatility and investment constraints. In this case, the HJB equation is a partial integro-diffe…
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.
This paper first describes a class of uncertain stochastic control systems with Markovian switching, and derives an Itô-Liu formula for Markov-modulated processes. And we characterize an optimal control law, which satisfies the generalized Hamilton-Jacobi-Bellman (HJB) equation with Markovian switching. Then, by using …
Optimal credit and consumption strategies in a switching market with default contagion.
problem Optimal portfolio and consumption decisions in a credit market with default contagion.
method Cobb-Douglas utility, recursive ODE system, backward solution from all-default state.
result Existence and uniqueness of optimal feedback controls, verification theorem.
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.
Study optimizes dividend payout strategies under fluctuating interest rates.
problem Maximizing dividends under stochastic interest rates with negative values.
method Analytical HJB approach and backward SDEs for analysis.
result Explicit optimal strategies found for both time-dependent and strategy-independent stopping times.
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.
Study optimal consumption with drawdown limits over a fixed time frame.
problem Maximizing utility with consumption limits during a fixed period.
method Extended utility maximization problem with drawdown constraint, using PDE arguments and dual transform.
result Existence and uniqueness of classical solution to HJB variational inequality, with explicit free boundaries.
Paper tackles DOCTR-L with SciPhy RL, solving neural PDEs from data.
problem High-dimensional optimal control with stochastic policies.
method Soft HJB equation, Neural PDEs, Physics-Informed Neural Networks.
result Reduces DOCTR-L to solving neural PDEs from data.
Investor selects portfolios based on news attention in a hidden Markov model.
problem Mean-variance portfolio selection in a dynamic attention context.
method Closed-loop equilibrium strategies via extended HJB equation and Markov chain approximation.
result Equilibrium strategies found through iterative algorithm and numerical examples.
The paper studies scaling limits of hedging prices in financial models.
problem Scaling limits of exponential utility indifference prices in financial models.
method Formulated dual problem as stochastic control, solved HJB equation for upper bound, used duality result for lower bound.
result Represented scaling limit in terms of specific relative entropy and constructed asymptotic optimal hedging strategies.
Protocol diagnoses neural HJB-PIDE solvers for Lévy jumps, revealing a missing factor in their importance-proposal density.
problem Neural PDE solvers can match scalar diagnostics but miscompute operators, leading to systematic errors.
method Five-step diagnostic protocol decomposes neural solve into components, compares them with independent reference solutions.
result Corrected a missing 1/2-mixture factor in the neural method's importance-proposal density, improving control accuracy.
We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [9] for representing fully nonlinear HJB equations. In particular, this allows us to numerically solve stochastic control problems with controlled volatility,…
We solve a complex Bayesian control problem with novel methods.
problem Optimizing control of a hidden signal's influence on noisy observations.
method Measure-valued HJB perspective, viscosity theory, approximation arguments.
result Equivalence to HJB equation and continuous viscosity solution.
Market-maker optimizes quotes based on strategic market-takers' behavior.
problem Optimizing market-making strategies in a strategic trading environment.
method Mean-field game approach to model strategic market-takers and derive optimal controls.
result Derivation of optimal controls for market-maker and market-taker.
We introduce Taylor expansions that do not require the differentiability. We also provide new solutions to partial differential equations. We apply our methods to finance.
Optimizes investment and reinsurance strategies with unknown parameters.
problem Optimal investment and reinsurance with partially unknown model parameters.
method Robust approach using generalized HJB equation and Clarke gradients.
result Explicit optimal investment strategy and bounds for optimal reinsurance.