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 expection 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.

翻译：在普通和部分差异方程式量子算法最近进展的启发下,我们研究随机差异方程式量子算法(SDEs) 。 首先,我们提供量子算法(SDEs),在一般环境下为多层次的蒙特卡洛方法提供二次加速。作为应用,我们应用它来计算由SDEs的经典解决方案决定的预期值,并更加依赖精确性。我们展示了在数学融资中产生的各种应用中,例如黑雪素和地方波动模型,以及希腊人,使用这种算法。我们还提供了基于亚线性二元论抽样的量子算法,以同样的改进来计算二元式选择定价模型。