Abstract
Inspired by recent progress in quantum algorithms for ordinary and partial differential equations, we study quantum algorithms for stochastic differential equations (SDEs). Firstly we provide a quantum algorithm that gives a quadratic speed-up for multilevel Monte Carlo methods in a general setting. As applications, we apply it to compute expectation values determined by classical solutions of SDEs, with improved dependence on precision. We demonstrate the use of this algorithm in a variety of applications arising in mathematical finance, such as the Black-Scholes and Local Volatility models, and Greeks. We also provide a quantum algorithm based on sublinear binomial sampling for the binomial option pricing model with the same improvement.
Highlights
Differential equations are ubiquitous throughout mathematics, science, and engineering
For numerous systems arising in statistical physics, molecular dynamics, finance, and other real-world models, the dynamics is captured by a stochastic differential equation (SDE) [42, 50]
We have presented quantum-accelerated multilevel Monte Carlo methods for stochastic processes
Summary
Differential equations are ubiquitous throughout mathematics, science, and engineering. Given a typical SDE, a fundamental computational problem is to provide an expected value of a random variable Y , denoted E[Y ], which is a functional determined by the solution of the SDE. Such a computational problem has been widely studied in mathematical finance, where the quantity Y represents the payoff in option and derivative pricing. It is often computationally expensive to estimate E[Y ], since a scheme that approximates the SDE is necessarily run many times to average over the randomness. Monte Carlo (MC) methods are basic tools with a provable complexity analysis
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