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Arash Fahim

Publications and source records attributed to Arash Fahim.

16 recordsLinked to original sources
arXiv2025

WoSNN: Stochastic Solver for PDEs with Machine Learning

Solving elliptic partial differential equations (PDEs) is a fundamental step in various scientific and engineering studies. As a classic stochastic solver, the Walk-on-Spheres (WoS) method is a well-established and efficient algorithm that provides accurate local estimates for PDEs. In this paper, by integrating machine learning techniques with WoS and space discretization approaches, we develop a novel stochastic solver, WoS-NN. This new method solves elliptic problems with Dirichlet boundary conditions, facilitating precise and rapid global solutions and gradient approximations. The method inherits excellent characteristics from the original WoS method, such as being meshless and robust to irregular regions. By integrating neural networks, WoS-NN also gives instant local predictions after training without re-sampling, which is especially suitable for intense requests on a static region. A typical experimental result demonstrates that the proposed WoS-NN method provides accurate field estimations, reducing errors by around $75\%$ while using only $8\%$ of path samples compared to the conventional WoS method, which saves abundant computational time and resource consumption.

math.NA↗
arXiv2025

A hierarchical resource-efficient deep policy gradient method for continuous-time optimal control problems

In this paper, we propose an efficient implementation of a deep policy gradient method (PGM) for optimal control problems in continuous time. For continuous-time problems that require a fine time discretization to achieve a desired accuracy, the proposed method improves time efficiency and performance by strategically allocating computational resources across scales, i.e., the number of trajectories, the granularity of time discretization, and the complexity of the neural network architecture. The main idea of the paper is to avoid committing to a fine time discretization. At first, we train a policy, modeled by a neural network, for a discretized optimal control problem in a coarse time scale. Then, we only discretize more in the time intervals where there are indications that the coarse scheme is not accurate enough and train a new policy in the finer scale. We then proceed to refine the time grid further to achieve better accuracy in the new smaller time intervals. Our theoretical result indicates how the new schedule for allocation of resources in different time scales can lead to efficiency. We conclude the paper by numerical experiments on a linear-quadratic stochastic optimal control problem and an optimal execution problem from quantitative finance.

math.OC↗
arXiv2024

Gradient ascent method for fully nonlinear parabolic differential equations with convex nonlinearity

We introduce a generic numerical schemes for fully nonlinear parabolic PDEs on the full domain, where the nonlinearity is convex on the Hessian of the solution. The main idea behind this paper is reduction of a fully nonlinear problem to a class of simpler semilinear ones parameterized by the diffusion term. The contribution of this paper is to provide a directional maximum principle with respect to the diffusion coefficient for semilinear problems, which specifies how to modify the diffusion coefficient to approach to the solution of the fully nonlinear problem. While the objects of the study, diffusion coefficient, is infinite dimensional, the maximum direction of increase can be found explicitly. This also provides a numerical gradient ascent method for the fully nonlinear problem. To establish a proof-of-concept, we test our method in a numerical experiment on the fully nonlinear Hamilton-Jacobi-Bellman equation for portfolio optimization under stochastic volatility model.

math.AP↗
arXiv2023

Impact of carbon market on production emissions

The aim of this paper is to address the effect of the carbon emission allowance market on the production policy of a large polluter production firm. We investigate this effect in two cases; when the large polluter cannot affect the risk premium of the allowance market, and when it can change the risk premium by its production. In this simple model, we ignore any possible investment of the firm in pollution reducing technologies. We formulate the problem of optimal production by a stochastic optimization problem. Then, we show that, as expected, the market reduces the optimal production policy in the first case if the firm is not given a generous initial cheap allowance package. However, when the large producer activities can change the market risk premium, the cut on the production and consequently pollution cannot be guaranteed. In fact, there are cases in this model when the optimal production is {\it always} larger than expected, and an increase in production, and thus pollution, can increase the profit of the firm. We conclude that some of the parameters of the market which contribute to this effect can be wisely controlled by the regulators in order to diminish this manipulative behavior of the firm.

math.OC↗
arXiv2023

Optimal Investment in a Dual Risk Model

Dual risk models are popular for modeling a venture capital or high tech company, for which the running cost is deterministic and the profits arrive stochastically over time. Most of the existing literature on dual risk models concentrated on the optimal dividend strategies. In this paper, we propose to study the optimal investment strategy on research and development for the dual risk models to minimize the ruin probability of the underlying company. We will also study the optimization problem when in addition the investment in a risky asset is allowed.

q-fin.RM↗
arXiv2022

Asymptotic Analysis for Optimal Dividends in a Dual Risk Model

The dual risk model is a popular model in finance and insurance, which is often used to model the wealth process of a venture capital or high tech company. Optimal dividends have been extensively studied in the literature for a dual risk model. It is well known that the value function of this optimal control problem does not yield closed-form solutions except in some special cases. In this paper, we study the asymptotics of the optimal dividend problem when the parameters of the model go to either zero or infinity. Our results provide insights to the optimal strategies and the optimal values when the parameters are extreme.

q-fin.RM↗
arXiv2019

Generalized Duality for Model-Free Superhedging given Marginals

In a discrete-time financial market, a generalized duality is established for model-free superhedging, given marginal distributions of the underlying asset. Contrary to prior studies, we do not require contingent claims to be upper semicontinuous, allowing for upper semi-analytic ones. The generalized duality stipulates an extended version of risk-neutral pricing. To compute the model-free superhedging price, one needs to find the supremum of expected values of a contingent claim, evaluated not directly under martingale (risk-neutral) measures, but along sequences of measures that converge, in an appropriate sense, to martingale ones. To derive the main result, we first establish a portfolio-constrained duality for upper semi-analytic contingent claims, relying on Choquet's capacitability theorem. As we gradually fade out the portfolio constraint, the generalized duality emerges through delicate probabilistic estimations.

q-fin.PR↗
arXiv2017

A Numerical Scheme for A Singular control problem: Investment-Consumption Under Proportional Transaction Costs

This paper concerns the numerical solution of a fully nonlinear parabolic double obstacle problem arising from a finite portfolio selection with proportional transaction costs. We consider the optimal allocation of wealth among multiple stocks and a bank account in order to maximize the finite horizon discounted utility of consumption. The problem is mainly governed by a time-dependent Hamilton-Jacobi-Bellman equation with gradient constraints. We propose a numerical method which is composed of Monte Carlo simulation to take advantage of the high-dimensional properties and finite difference method to approximate the gradients of the value function. Numerical results illustrate behaviors of the optimal trading strategies and also satisfy all qualitative properties proved in Dai et al. (2009) and Chen and Dai (2013).

q-fin.PM↗
arXiv2015

Model-independent Superhedging under Portfolio Constraints

In a discrete-time market, we study model-independent superhedging, while the semi-static superhedging portfolio consists of {\it three} parts: static positions in liquidly traded vanilla calls, static positions in other tradable, yet possibly less liquid, exotic options, and a dynamic trading strategy in risky assets under certain constraints. By considering the limit order book of each tradable exotic option and employing the Monge-Kantorovich theory of optimal transport, we establish a general superhedging duality, which admits a natural connection to convex risk measures. With the aid of this duality, we derive a model-independent version of the fundamental theorem of asset pricing. The notion "finite optimal arbitrage profit", weaker than no-arbitrage, is also introduced. It is worth noting that our method covers a large class of Delta constraints as well as Gamma constraint.

q-fin.PR↗
arXiv2013

Strong Convergence to the homogenized limit of parabolic equations with random coefficients II

This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients, and their convergence to solutions of a homogenized equation. In [11] rate of convergence results in homogenization and estimates on the difference between the averaged Green's function and the homogenized Green's function for random environments which satisfy a Poincaré inequality were obtained. Here these results are extended to certain environments in which correlations can have arbitrarily small power law decay. Similar results for discrete elliptic equations were obtained in [12].

math.AP↗
arXiv2013

Balancing small fixed and proportional transaction cost in trading strategies

Transaction costs appear in financial markets in more than one form. There are several results in the literature on small proportional transaction cost and not that many on fixed transaction cost. In the present work, we heuristically study the effect of both types of transaction cost by focusing on a portfolio optimization. Here we assume the presence of fixed transaction cost and that there is a balance between fixed and proportional transaction cost, such that none of them dominates the other, asymptotically. We find out that the deviation of value function, when the fixed transaction cost is $\varepsilon$, from the Merton value function, without transaction cost, is of order $\varepsilon^1/2$ which is different from the pure proportional cost of $\varepsilon^2/3$. Based on this, we propose an expansion for the value function in terms of powers of $\varepsilon^1/2$.

q-fin.PM↗
arXiv2012

Strong Convergence to the homogenized limit of elliptic equations with random coefficients II

Consider a discrete uniformly elliptic divergence form equation on the $d$ dimensional lattice $\Z^d$ with random coefficients. In [3] rate of convergence results in homogenization and estimates on the difference between the averaged Green's function and the homogenized Green's function for random environments which satisfy a Poincaré inequality were obtained. Here these results are extended to certain environments with long range correlations. These environments are simply related via a convolution to environments which do satisfy a Poincaré inequality.

math.AP↗
arXiv2012

Strong Convergence to the Homogenized Limit of Parabolic Equations with Random Coefficients

This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients, and their convergence to solutions of a homogenized equation. It has previously been shown that if the random environment is translational invariant and ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized parabolic PDE. In this paper point-wise estimates are obtained on the difference between the averaged solution to the random equation and the solution to the homogenized equation for certain random environments which are strongly mixing.

math.AP↗
arXiv2010

A Probabilistic Scheme for Fully Nonlinear Nonlocal Parabolic PDEs with singular L\'evy measures

We introduce a Monte Carlo scheme for fully nonlinear parabolic nonlocal PDE's whose nonlinearity in of Hamilton-Jacobi-Bellman-Isaacs (HJBI for short). We avoid the difficulties of infinite L\'evy measure by truncation of the L\'evy integral. The first result provides the convergence of the scheme for general parabolic nonlinearities. The second result provides bounds on the rate of convergence for concave (or equivalently convex) nonlinearities. For both results, it is crucial to choose truncation of the infinite L\'evy measure appropriately dependent on the time discretization. We also introduce a Monte Carlo Quadrature method to approximate the nonlocal term in the HJBI nonlinearity.

math.PR↗
arXiv2009

A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs

We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in \cite{cstv}, and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main result provides the convergence of the discrete-time approximation and derives a bound on the discretization error in terms of the time step. An explicit implementable scheme requires to approximate the conditional expectation operators involved in the discretization. This induces a further Monte Carlo error. Our second main result is to prove the convergence of the latter approximation scheme, and to derive an upper bound on the approximation error. Numerical experiments are performed for the approximation of the solution of the mean curvature flow equation in dimensions two and three, and for two and five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations arising in the theory of portfolio optimization in financial mathematics.

math.PR↗