Quantum Encoding and Analysis on Continuous Time Stochastic Process with Financial Applications

Xi-Ning Zhuang1,2, Zhao-Yun Chen3, Cheng Xue3, Yu-Chun Wu1,4,5,3, and Guo-Ping Guo1,4,5,3,2

1CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
2Origin Quantum Computing, Hefei, China
3Institute of Artificial Intelligence, Hefei Comprehensive National Science Center
4CAS Center for Excellence and Synergistic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, 230026, China
5Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China

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Abstract

Modeling stochastic phenomena in continuous time is an essential yet challenging problem. Analytic solutions are often unavailable, and numerical methods can be prohibitively time-consuming and computationally expensive. To address this issue, we propose an algorithmic framework tailored for quantum continuous time stochastic processes. This framework consists of two key procedures: data preparation and information extraction. The data preparation procedure is specifically designed to encode and compress information, resulting in a significant reduction in both space and time complexities. This reduction is exponential with respect to a crucial feature parameter of the stochastic process. Additionally, it can serve as a submodule for other quantum algorithms, mitigating the common data input bottleneck. The information extraction procedure is designed to decode and process compressed information with quadratic acceleration, extending the quantum-enhanced Monte Carlo method. The framework demonstrates versatility and flexibility, finding applications in statistics, physics, time series analysis and finance. Illustrative examples include option pricing in the Merton Jump Diffusion Model and ruin probability computing in the Collective Risk Model, showcasing the framework’s ability to capture extreme market events and incorporate history-dependent information. Overall, this quantum algorithmic framework provides a powerful tool for accurate analysis and enhanced understanding of stochastic phenomena.

In the realm of physics, tackling complex continuous-time stochastic processes has long been a challenge due to the lack of analytic solutions and the formidable computational consumption of numerical methods. However, this research proposes a new quantum algorithmic framework offering a game-changing solution. This framework consists of two crucial components: data preparation and information extraction. Data preparation reduces time and space complexity by statistics-inspired information compression. It can also be used in other quantum algorithms, addressing data input bottlenecks. Information extraction processes this compressed data with quadratic acceleration, expanding the quantum-enhanced Monte Carlo method. The impact is far-reaching, with applications in statistics, physics, time series analysis, and finance. Examples include option pricing and ruin probability calculation, showcasing its ability to handle extreme market events and history-dependent data. In essence, this quantum algorithmic framework provides a powerful tool for a more accurate analysis of stochastic phenomena.

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[1] Yewei Yuan, Chao Wang, Bei Wang, Zhao-Yun Chen, Meng-Han Dou, Yu-Chun Wu, and Guo-Ping Guo, "An improved QFT-based quantum comparator and extended modular arithmetic using one ancilla qubit", New Journal of Physics 25 10, 103011 (2023).

[2] Sascha Wilkens and Joe Moorhouse, "Quantum computing for financial risk measurement", Quantum Information Processing 22 1, 51 (2023).

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