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Stochastic Differential Equations with Transformed Anticipating Conditions |
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PP: 865-875 |
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doi:10.18576/jsap/120301
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Author(s) |
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John Abonongo,
Patrick Chidzalo,
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Abstract |
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| Stochastic differential equations can be specified with anticipatory initial value constraints (IVC) which is useful in many applications where future filtration is known. However, such specification leads to different results than those used in the usual Itoˆ’s calculus, and choice of transformations, at IVC, affects the orientation of equivalent isometry moments and related quantities. To solve this problem, the paper analyzes linear stochastic differential equations with anticipatory initial value constraints specified by an exponential transformation. The conditions for general solutions when such function is defined are derived from Itoˆ’s lemma, Taylor, and Fourier Series. Exact solutions are found using the derived conditions as well as those found by other research works. The article also derives numerical scheme for stochastic differential equations with anticipating initial conditions from first principles, Euler-Muruyama and Monte-Carlo methods. The results show that, the Euler-Muruyama method without a Monte-Carlo extension gives reliable numerical solutions for stochastic differential equations with anticipatory initial conditions. The Monte-Carlo extension introduces a slight smoothing effect on the estimated numerical solution.
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