1,720,980 research outputs found
A Hamilton-Jacobi-Bellman Approach for the Numerical Computation of Probabilistic State Constrained Reachable Sets
Full text linkAim of this work is to characterise and compute the set of initial conditions for a system of controlled diffusion processes which allow to reach a terminal target satisfying pointwise state constraints with a given probability of success. Defining a suitable auxiliary optimal control problem, the characterization of this set is related to the solution of a particular Hamilton-Jacobi-Bellman equation. A semi-Lagrangian numerical scheme is defined and its convergence to the unique viscosity solution of the equation is proved. The validity of the proposed approach is then tested on some numerical examples
Zubov's method for controlled diffusions with state constraints
Full text linkWe consider a controlled stochastic system in presence of state-constraints. Under the assumption of exponential stabilizability of the system near a target set, we aim to characterize the set of points which can be asymptotically driven by an admissible control to the target with positive probability. We show that this set can be characterized as a level set of the optimal value function of a suitable unconstrained optimal control problem which in turn is the unique viscosity solution of a second order PDE which can thus be interpreted as a generalized Zubov equation
Quantization of stochastic volatility models: Numerical tests and an open source implementation
No full textThe aim of this paper is to discuss the implementation of the recursive marginal quantization algorithm of [FPS2018] to several stochastic volatility models. After recalling the theoretical framework and the main features of the method, we evaluate the performance of the algorithm for the pricing of derivatives. We also discuss an open source implementation of the algorithm. For most models we consider, with the exception of the Stein and Stein model, recursive marginal quantization provides a viable alternative to Monte Carlo simulations
Probabilistic error analysis for some approximation schemes to optimal control problems
Full text linkWe introduce a class of numerical schemes for optimal stochastic control problems based on a novel Markov chain approximation, which uses, in turn, a piecewise constant policy approximation, Euler– Maruyama time stepping, and a Gauß-Hermite approximation of the Gaußian increments. We provide lower error bounds of order arbitrarily close to 1/2 in time and 1/3 in space for Lipschitz viscosity solutions, coupling probabilistic arguments with regularization techniques as introduced by Krylov. The corresponding order of the upper bounds is 1/4 in time and 1/5 in space. For sufficiently regular solutions, the order is 1 in both time and space for both bounds. Finally, we propose techniques for further improving the accuracy of the individual components of the approximation
Optimal management of pumped hydroelectric production with state constrained optimal control
Full text linkWe present a novel technique to solve the problem of managing optimally a pumped hydroelectric storage system. This technique relies on representing the system as a stochastic optimal control problem with state constraints, these latter corresponding to the finite volume of the reservoirs. Following the recent level-set approach presented in O. Bokanowski, A. Picarelli, H. Zidani, "State-constrained stochastic optimal control problems via reachability approach", SIAM J. Control and Optim. 54 (5) (2016) , we transform the original constrained problem in an auxiliary unconstrained one in augmented state and control spaces, obtained by introducing an exact penalization of the original state constraints. The latter problem is fully treatable by classical dynamic programming arguments
A Level-Set Approach for Stochastic Optimal Control Problems under Controlled-Loss Constraints
Full text linkWe study a family of optimal control problems under a set of controlled- loss constraints holding at different deterministic dates. The characterization of the associated value function by a Hamilton-Jacobi-Bellman equation usually calls for strong assumptions on the dynamics of the processes involved and the set of constraints. To treat this problem in absence of those assumptions, we first convert it into a state-constrained stochastic target problem and then solve the latter by a level-set approach. With this approach, state constraints are managed through an exact penalization technique
A deep solver for BSDEs with jumps
Full text linkThe aim of this work is to propose an extension of the Deep BSDE solver by Han, E, Jentzen (2017) to the case of FBSDEs with jumps. As in the aforementioned solver, starting from a discretized version of the BSDE and parametrizing the (high dimensional) control processes by means of a family of artificial neural networks (ANNs), the BSDE is viewed as model-based reinforcement learning problem and the ANN parameters are fitted so as to minimize a prescribed loss function. We take into account both finite and infinite jump activity by introducing, in the latter case, an approximation with finitely many jumps of the forward process
Deep Quadratic Hedging
Full text linkWe propose a novel computational procedure for quadratic hedging in high-dimensional incomplete markets, covering mean-variance hedging and local risk minimization. Starting from the observation that both quadratic approaches can be treated from the point of view of backward stochastic differential equations (BSDEs), we (recursively) apply a deep learning-based BSDE solver to compute the entire optimal hedging strategies paths. This allows us to overcome the curse of dimensionality, extending the scope of applicability of quadratic hedging in high dimension. We test our approach with a classic Heston model and with a multiasset and multifactor generalization thereof, showing that this leads to high levels of accuracy.Accepted version. Final edited version available at https://doi.org/10.1287/moor.2023.021
Deep xVA solver - A neural network based counterparty credit risk management framework
Full text linkIn this paper, we present a novel computational framework for portfolio-wide risk management problems wherethe presence of a potentially large number of risk factors makes traditional numerical techniques ineffective.The new method utilises a coupled system of BSDEs for the valuation adjustments (xVA) and solves these by a recursive application of a neural network based BSDE solver.This not only makes the computation of xVA for high-dimensional problems feasible, but also produces hedge ratios and dynamic risk measures for xVA, and allows simulations of the collateral account
Improved order 1/4 convergence for piecewise constant policy approximation of stochastic control problems
Full text linkIn N. V. Krylov, "Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies", Electron. J. Probab., 4(2), 1999, it is proved under standard assumptions that the value functions of controlled diffusion processes can be approximated with order 1/6 error by those with controls which are constant on uniform time intervals. In this note we refine the proof and show that the provable rate can be improved to 1/4, which is optimal in our setting. Moreover, we demonstrate the improvements this implies for error estimates derived by similar techniques for approximation schemes, bringing these in line with the best available results from the PDE literature
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