Stochastic Control Portfolio

Stochastic Control Portfolio is a high-performance computational engine designed to solve optimal decision-making problems in finance. By transforming a modular reaction-diffusion PDE framework into a Hamilton-Jacobi-Bellman (HJB) solver, this project provides a robust numerical approach to the Merton Portfolio Problem, determining the optimal allocation of wealth between risky and risk-free assets over a continuous-time horizon.

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📝 Repository Description

This repository implements a high-performance Merton Portfolio Problem solver developed for quantitative finance. Built from a modular PDE engine, this repo optimizes investment strategies under uncertainty by solving the Hamilton-Jacobi-Bellman (HJB) equation. Features include GPU-accelerated time-stepping and Neumann boundary support.

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🚀 Key Features

• Optimal Control Solver: Solves the non-linear HJB equation to determine the optimal portfolio weight ($\pi^*$) at every point on a wealth-time grid.
• Modular PDE Architecture: Decouples the financial "physics" (Merton dynamics) from the numerical integration engine (Implicit-Explicit time-stepping).
• High-Performance Computing: Utilizes vectorized operations and optional GPU acceleration (via Numba/CuPy) to handle dense spatial grids efficiently.
• Advanced Boundary Management: Implements Neumann (Reflective) boundary conditions to ensure utility conservation across the wealth domain.
• Automated Risk Monitoring: Built-in health checks to detect and prevent numerical instabilities during non-linear optimization steps.

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🧠 Mathematical Framework

The core of the repository is the Merton Portfolio Problem, which seeks to maximize the expected power utility of terminal wealth:

$$E\left[ \frac{W_T^\gamma}{\gamma} \right]$$

The solver iterates backward in time from the terminal utility state, solving the HJB equation: $$V_t + \sup_{\pi} \left[ (rW + \pi W(\mu - r))V_W + \frac{1}{2}(\pi W \sigma)^2 V_{WW} \right] = 0$$

At each step, the engine performs a local optimization to find the optimal control $\pi^*$, effectively transforming a global portfolio strategy into a series of local "reaction" terms within the PDE grid.

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📂 Repository Structure

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Getting Started

Prerequisites

• Python ≥ 3.11
• NumPy, SciPy, Matplotlib
• Numba (CPU acceleration)
• CuPy (optional — GPU acceleration)

Installation

git clone https://github.com/Diegotistical/stochastic-control-portfolio.git
cd stochastic-control-portfolio
pip install -r requirements.txt