Abstract
The authors report on the construction of a new algorithm for the weak approximation of stochastic differential equations. In this algorithm, an ODE-valued random variable whose average approximates the solution of the given stochastic differential equation is constructed by using the notion of free Lie algebras. It is proved that the classical Runge–Kutta method for ODEs is directly applicable to the ODE drawn from the random variable. In a numerical experiment, this is applied to the problem of pricing Asian options under the Heston stochastic volatility model. Compared with some other methods, this algorithm is significantly faster.
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This research was partly supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (C), 15540110, 2003 and 18540113, 2006, the 21st century COE program at Graduate School of Mathematical Sciences, the University of Tokyo, and JSPS Core-to-Core Program 18005.
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Ninomiya, M., Ninomiya, S. A new higher-order weak approximation scheme for stochastic differential equations and the Runge–Kutta method. Finance Stoch 13, 415–443 (2009). https://doi.org/10.1007/s00780-009-0101-4
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DOI: https://doi.org/10.1007/s00780-009-0101-4
Keywords
- Free Lie algebra
- Mathematical finance
- Runge–Kutta method
- Stochastic differential equations
- Weak approximation