Policy Gradient Optimal Correlation Search for Variance Reduction in Monte Carlo simulation and Maximum Optimal Transport
This work addresses the problem of high variance in Monte Carlo simulations for practitioners in computational finance and related fields, offering a novel method that integrates optimal transport concepts.
The paper tackles variance reduction in Monte Carlo simulations for stochastic differential equations by introducing a new estimator using two correlated processes with the same marginal law, and approximates the optimal correlation function with a deep neural network trained via policy gradient and reinforcement learning, achieving significant variance reduction.
We propose a new algorithm for variance reduction when estimating $f(X_T)$ where $X$ is the solution to some stochastic differential equation and $f$ is a test function. The new estimator is $(f(X^1_T) + f(X^2_T))/2$, where $X^1$ and $X^2$ have same marginal law as $X$ but are pathwise correlated so that to reduce the variance. The optimal correlation function $ρ$ is approximated by a deep neural network and is calibrated along the trajectories of $(X^1, X^2)$ by policy gradient and reinforcement learning techniques. Finding an optimal coupling given marginal laws has links with maximum optimal transport.