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Efficient computation of the stochastic behavior of partial sum processes


In this paper the computational aspects of probability calculations for dynamical partial sum expressions are discussed. Such dynamical partial sum expressions have many important applications, and examples are provided in the fields of reliability, product quality assessment, and stochastic control. While these probability calculations are ostensibly of a high dimension, and consequently intractable in general, it is shown how a recursive integration methodology can be implemented to obtain exact calculations as a series of two-dimensional calculations. The computational aspects of the implementation of this methodology, with the adoption of Fast Fourier Transforms, are discussed.

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  • Aitsahlia F, Lai TL (1997) Valuation of discrete barrier and hindsight options. J Financ Eng 6(2):169–177

    Google Scholar 

  • Andricopoulos AD, Widdicks M, Duck PW, Newton DP (2003) Universal option valuation using quadrature methods. J Financ Econ 67:447–471

    Article  Google Scholar 

  • Bélisle CJP, Romeijn HE, Smith RL (1993) Hit-and-run algorithm for generating multivariate distribution. Math Oper Res 18:255–266

    Article  MathSciNet  Google Scholar 

  • Carverhill AP, Clewlow LJ (1990) Flexible convolution. Risk 3:25–29

    Google Scholar 

  • Dunnett CW, Sobel M (1955) Approximations to the probability integral and certain percentage points of a multivariate analogue of Student’s \(t\)-distribution. Biometrika 42:258–260

    Article  MathSciNet  Google Scholar 

  • Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer, New York

    Book  Google Scholar 

  • Fusai G, Meucci A (2008) Pricing discretely monitored asian options under levy processes. J Bank Finance 32:2076–2088

    Article  Google Scholar 

  • Fusai G, Recchioni MC (2007) Analysis of quadrature methods for pricing discrete barrier options. J Econ Dyn Control 31(3):826–860

    Article  MathSciNet  Google Scholar 

  • Hayter AJ (2006) Recursive integration methodologies with statistical applications. J Stat Plan Inference 136:2284–2296

    Article  MathSciNet  Google Scholar 

  • Hayter AJ (2014) Recursive formulas for multinomial probabilities with applications. Comput Stat 29(5):1207–1219

    Article  MathSciNet  Google Scholar 

  • Jones E, Oliphant E, Peterson P, et al (2001) Scipy: open source scientific tools for python.

  • Kiatsupaibul S, Hayter AJ (2015) Recursive confidence band construction for an unknown distribution function. Biom J 57(1):39–51

    Article  MathSciNet  Google Scholar 

  • Kiatsupaibul S, Hayter AJ, Liu W (2017) Rank constrained distribution and moment computations. Comput Stat Data Anal 105:229–242

    Article  MathSciNet  Google Scholar 

  • Kiatsupaibul S, Smith RL, Zabinsky ZB (2011) An analysis of a variation of hit-and-run for uniform sampling from general regions. ACM Trans Model Comput Simul 21(3), Article number 16

    Article  Google Scholar 

  • Lovász L (1999) Hit-and-run mixes fast. Math Program 86:443–461

    Article  MathSciNet  Google Scholar 

  • Øksendal B (2014) Stochastic differential equations, 6th edn. Springer, Heidelberg

    MATH  Google Scholar 

  • Smith RL (1984) Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions. Oper Res 32:1296–1308

    Article  MathSciNet  Google Scholar 

  • Smith JO (2007) Mathematics of the discrete Fourier transform (DFT), with audio applications, 2nd edn. W3K Publishing, Seattle

    Google Scholar 

  • Soong WC, Hsu JC (1997) Using complex integration to compute multivariate normal probabilities. J Comput Gr Stat 6(4):397–415

    MathSciNet  Google Scholar 

  • Sullivan MA (2000) Pricing discretely monitored barrier options. J Comput Finance 3(4):35–52

    Article  Google Scholar 

Download references

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Correspondence to Seksan Kiatsupaibul.

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Saengkyongam, S., Hayter, A., Kiatsupaibul, S. et al. Efficient computation of the stochastic behavior of partial sum processes. Comput Stat 35, 343–358 (2020).

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