• Title, Summary, Keyword: Stratonovich integral

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A Wong-Zakai Type Approximation for the Multiple Ito-Wiener Integral

  • Lee, Kyu-Seok;Kim, Yoon-Tae;Jeon, Jong-Woo
    • Proceedings of the Korean Statistical Society Conference
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    • pp.55-60
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    • 2002
  • We present an extension of the Wong-Zakai type approximation theorem for a multiple stochastic integral. Using a piecewise linear approximation $W^{(n)}$ of a Wiener process W, we prove that the multiple integral processes {${\int}_{0}^{t}{\cdots}{\int}_{0}^{t}f(t_{1},{\cdots},t_{m})W^{(n)}(t_{1}){\cdots}W^{(n)}(t_{m}),t{\in}[0,T]$} where f is a given symmetric function in the space $C([0,T]^{m})$, converge to the multiple Stratonovich integral of f in the uniform $L^{2}$-sense.

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An Approximation Theorem for Two-Parameter Wiener Process

  • Kim, Yoon-Tae
    • Journal of the Korean Statistical Society
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    • v.26 no.1
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    • pp.75-88
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    • 1997
  • In this paper, a two-parameter version of Ikeda-Watanabe's mollifiers approximation of the Brownian motion is considered, and an approximation theorem corresponding to the one parameter case is proved. Using this approximation, we formulate Wong-Zakai type theorem is a Stochastic Differential Equation (SDE) driven by a two-parameter Wiener process.

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Lie Algebraic Solution of Stochastic Differential Equations

  • Kim, Yoon-Tae;Jeon, Jong-Woo
    • Proceedings of the Korean Statistical Society Conference
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    • pp.25-30
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    • 2003
  • We prove that the logarithm of the flow of stochastic differential equations is an element of the free Lie algebra generated by a finite set consisting of vector fields being coefficients of equations. As an application, we directly obtain a formula of the solution of stochastic differential equations given by Castell(1993) without appealing to an expansion for ordinary differential equations given by Strichartz (1987).

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HEAT EQUATION WITH A GEOMETRIC ROUGH PATH POTENTIAL IN ONE SPACE DIMENSION: EXISTENCE AND REGULARITY OF SOLUTION

  • Kim, Hyun-Jung;Lototsky, Sergey V.
    • Communications of the Korean Mathematical Society
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    • v.34 no.3
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    • pp.757-769
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    • 2019
  • A solution of the heat equation with a distribution-valued potential is constructed by regularization. When the potential is the generalized derivative of a $H{\ddot{o}}lder$ continuous function, regularity of the resulting solution is in line with the standard parabolic theory.

OPTION PRICING UNDER GENERAL GEOMETRIC RIEMANNIAN BROWNIAN MOTIONS

  • Zhang, Yong-Chao
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.5
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    • pp.1411-1425
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    • 2016
  • We provide a partial differential equation for European options on a stock whose price process follows a general geometric Riemannian Brownian motion. The existence and the uniqueness of solutions to the partial differential equation are investigated, and then an expression of the value for European options is obtained using the fundamental solution technique. Proper Riemannian metrics on the real number field can make the distribution of return rates of the stock induced by our model have the character of leptokurtosis and fat-tail; in addition, they can also explain option pricing bias and implied volatility smile (skew).