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.
New method uses neural networks to solve complex PDEs from optimal control theory.
problem Solving high-dimensional Hamilton-Jacobi-Bellman PDEs.
method Iterative diffusion optimization techniques, focusing on path measures and divergences.
result Favourable properties of log-variance divergence for Monte Carlo estimators.
New approach uses PDE learning for faster RL fine-tuning.
problem Learning optimal control policy for diffusion process.
method Solves variational inequality based on HJB equations.
result Shows fine-tuning can be done via supervised regression.
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.
The paper proves well-posedness of nonlocal PDEs related to stochastic control problems.
problem Characterizing equilibrium strategies and value functions for time-inconsistent stochastic control problems.
method Method of continuity and Banach's fixed point arguments, with Schauder prior estimates.
result Global well-posedness of nonlocal fully nonlinear PDEs with sharp a-priori estimates.
We find a simple strategy approximating optimal portfolio for short time horizons.
problem Optimizing portfolios in incomplete markets with general utility functions.
method Closed-form formula derived from HJB PDE, approximated by sub- and super-solutions.
result Approximation formula for optimal trading strategy is accurate for small time horizons.
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.
Method solves high-dimensional nonlinear PDEs using neural networks.
problem Solving high-dimensional fully nonlinear PDEs.
method Backward induction with multi-layer neural networks to estimate solution and its gradient, with Hessian approximated by automatic differentiation.
result Method extends previous work on semi-linear PDEs to fully nonlinear cases, demonstrating accuracy on various examples.
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…
Study optimal dividend policies for firms with random profitability.
problem Firms face a trade-off between bankruptcy and profit extraction.
method General cash flow drifts (Ornstein-Uhlenbeck, CIR) considered; rigorous proofs, numerical scheme provided.
result Optimal strategy includes barrier and band strategies, voluntary liquidation.
Extends DGM to solve PDEs and HJB equations in optimal control.
problem Solving PDEs and HJB equations in optimal control problems.
method Reparameterization and neural networks for positivity and normalization. Novel importance sampling for integral terms. Alternating stochastic gradient descent for simultaneous optimization.
result Solves PDEs and HJB equations in their primal form.
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.
This paper optimizes dividend payout rates with a drawdown constraint in a stochastic model.
problem Optimizing dividend payout rates while avoiding drawdowns in a stochastic model.
method Solving a path-dependent stochastic control problem using Hamilton-Jacobi-Bellman equations and PDE methods.
result Explicit characterization of an optimal feedback control strategy, including two free boundaries and the running maximum surplus process.
Neural networks solve high-dimensional HJB PDEs with asymptotic guarantees.
problem Solving high-dimensional Hamilton-Jacobi-Bellman PDEs in stochastic control theory.
method Actor-critic machine learning algorithm with a structured critic and biased gradient actor.
result The training dynamics converge to an ODE, ensuring solutions to the original problem.
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.
The paper calibrates LSV models using optimal transport and convex optimisation.
problem Calibrating Local-Stochastic Volatility (LSV) models with European option prices.
method Optimal transport problem, convex optimisation, PDE formulation, Hamilton-Jacobi-Bellman equation.
result Numerical solution of dual problem yields calibrated LSV model parameters.
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.
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.
Deep learning solves high-dimensional PDEs efficiently.
problem High-dimensional PDEs are computationally challenging.
method Approximate PDE solutions with a deep neural network trained to satisfy PDE conditions.
result Solves PDEs in up to 200 dimensions accurately.
Efficiently samples complex distributions using tensor train format.
problem Sampling from high-dimensional complex probability densities efficiently.
method Integrates tensor train format with backward stochastic differential equations (BSDEs) for fast, robust, and accurate sampling.
result Improved efficiency in sampling from challenging target distributions.
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.
Quantum machine learning solves high-dimensional PDEs with lower variance and improved accuracy.
problem Approximating solutions to high-dimensional parabolic PDEs.
method Pure Variational Quantum Circuit (VQC) for BSDE approximation, using temporal discretization and Monte Carlo simulation.
result VQC achieves lower variance and improved accuracy in most cases, particularly in highly nonlinear regimes.
New algorithm solves high-dimensional PDEs and BSDEs using neural networks.
problem Solving high-dimensional PDEs and BSDEs efficiently and accurately.
method Analogy with reinforcement learning, neural network approximation of policy function.
result Efficiency and accuracy demonstrated in solving 100-dimensional equations.
New method solves high-dimensional PDEs fast using physics-informed neural networks.
problem High computational cost in solving high-dimensional PDEs.
method Stochastic Dimension Gradient Descent (SDGD) for physics-informed neural networks (PINNs).
result Solves many high-dimensional PDEs including HJB and Schrödinger equations in 100,000 dimensions in 12 hours.
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.
This is the first in a series of papers in which we study an efficient approximation scheme for solving the Hamilton-Jacobi-Bellman equation for multi-dimensional problems in stochastic control theory. The method is a combination of a WKB style asymptotic expansion of the value function, which reduces the second order …
New method solves high-dimensional PDEs and 2BSDEs efficiently.
problem High-dimensional fully nonlinear PDEs and 2BSDEs in financial models.
method Connection between PDEs and 2BSDEs, merged formulation, temporal discretization, spatial approximation via neural nets, stochastic gradient descent.
result Efficient and accurate solution for high-dimensional nonlinear expectations.
Investment strategy optimized for markets and credit risks.
problem Optimal investment/consumption problem in models with market and credit risk dependencies.
method Martingale approach and analysis of nonlinear Hamilton-Jacobi-Bellman equations transformed into semi-linear PDEs.
result Explicit representations for optimal strategy, consumption path, and wealth process.
This paper considers a portfolio optimization problem in which asset prices are represented by SDEs driven by Brownian motion and a Poisson random measure, with drifts that are functions of an auxiliary diffusion factor process. The criterion, following earlier work by Bielecki, Pliska, Nagai and others, is risk-sensit…
Optimal investment strategies in a contagious distress model using dynamic optimization.
problem Analyzing optimal portfolio allocations in a model influenced by hidden Markov chain and economic distress.
method Dynamic optimization framework, recursive HJB PDEs, uniform bounds, convergence to Sobolev solution.
result Optimal investment strategies depend on the gradient of value functions and distress states.
New sampling method uses stochastic interpolants and FBSDEs.
problem Sampling from high-dimensional distributions with unnormalized densities.
method Stochastic interpolants and FBSDEs to define and solve diffusion process.
result Effective sampling from challenging distributions.
We derive a closed form portfolio optimization rule for an investor who is diffident about mean return and volatility estimates, and has a CRRA utility. The novelty is that confidence is here represented using ellipsoidal uncertainty sets for the drift, given a volatility realization. This specification affords a simpl…
Study indifference pricing for insurance policies in a regime-switching market model.
problem Indifference pricing of pure endowment policies in a stochastic-factor model with different economic regimes.
method Stochastic control approach based on Hamilton-Jacobi-Bellman equation, Feynman-Kac formula, and sensitivity analysis.
result Characterization of indifference price as a solution to a linear PDE and a backward PDE.
We study the portfolio problem of maximizing the outperformance probability over a random benchmark through dynamic trading with a fixed initial capital. Under a general incomplete market framework, this stochastic control problem can be formulated as a composite pure hypothesis testing problem. We analyze the connecti…
Turnpike property applies to optimal control of PDEs and ResNets.
problem Optimal control of PDEs and ResNets.
method Mathematical formalization and controllability analysis.
result Optimal controls and states are nearly constant over most of the time.
We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic coefficients driven by a diffusion process. We assume that an agent makes consumption and investment decisions based on CRRA utility functions. The dynamical programming approach leads to an investigation of t…
This paper considers a portfolio optimization problem in which asset prices are represented by SDEs driven by Brownian motion and a Poisson random measure, with drifts that are functions of an auxiliary diffusion 'factor' process. The criterion, following earlier work by Bielecki, Pliska, Nagai and others, is risk-sens…
SGMs are robust to practical errors via uncertainty quantification.
problem Robustness of SGMs to practical implementation errors.
method Wasserstein uncertainty propagation (WUP) theorem and Bernstein estimates.
result SGMs are provably robust to multiple sources of error.
Optimal investment strategy with price impact model.
problem Maximizing expected utility from liquidation wealth with price impact.
method Price impact model accounting for market depth, liquidity costs, and convexity. Singular optimal stochastic control problem reduced to deterministic optimal tracking problem.
result Explicit solution constructed, free boundaries described, optimal trading strategy identified.
Improves SGM convergence bounds in W2-distance without strict assumptions.
problem Convergence bounds for SGMs in W2-distance require stringent assumptions.
method Novel framework using the OU process and PDE analysis.
result Log-concavity evolves from weak to strong over time.
Paper tackles stochastic control with mean and higher-order moments, finding Nash equilibria.
problem Time-inconsistent stochastic control problems with mean and higher-order moments.
method Developed closed-loop and open-loop Nash equilibrium controls using PDEs and maximum principles.
result Identical closed-loop and open-loop Nash equilibria controls, independent of state value and random path.
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.
This paper tackles collision avoidance for many UAVs using MFG and ML.
problem Collision avoidance for many UAVs in real-time missions.
method Mean-field game (MFG) theory combined with machine learning (ML) to reduce computation and communication energy.
result The proposed MFG learning control method achieves collision avoidance with low communication and acceptable computation energy.
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.
Optimal dividend strategy with ratcheting and capital injection under Cramér-Lundberg model.
problem Optimal dividend payout for an insurance company with ratcheting constraints and capital injections.
method Systematic probabilistic and PDE-based approach to solve HJB equation, constructing strong solution and optimal strategy.
result Existence and uniqueness of strong solution, explicit optimal feedback control strategy.
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.
Market makers optimize trading with a new implicit scheme for complex inequalities.
problem Optimizing trading in a limit order book with stochastic and impulse control.
method Implicit numerical scheme coupled with policy iteration algorithm.
result Convergence to the unique viscosity solution of the HJBQVI.
Two deep learning algorithms solve utility maximisation problems in finance.
problem Solving utility maximisation problems in finance with deep learning.
method Two algorithms: one for Markovian problems via HJB equation and 2BSDE, the other for non-Markovian problems via adjoint BSDE.
result Highly accurate results with low computational cost, solving problems with power, log, and non-HARA utilities in various models.