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Mimicking the one-dimensional marginal distributions of processes having an ito differential
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  • Published: December 1986

Mimicking the one-dimensional marginal distributions of processes having an ito differential

  • I. Gyöngy1 

Probability Theory and Related Fields volume 71, pages 501–516 (1986)Cite this article

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Summary

Let ξ(t) be a stochastic process starting from 0 with Ito differential

$$d\xi \left( t \right) = \delta \left( {t,\omega } \right)dW\left( t \right) + \beta \left( {t,\omega } \right)dt,$$

where\(\left( {W\left( t \right),\mathfrak{F}_t } \right)\) is a Wiener process, δ and β are bounded\(\mathfrak{F}\) processes such that δδT is uniformly positive definite. Then it is proved that there exists a stochastic differential equation

$$dx\left( t \right) = \sigma \left( {t,x\left( t \right)} \right)dW\left( t \right) + b\left( {t,x\left( t \right)} \right)dt$$

with non-random coefficients which admits a weak solutionx(t) having the same one-dimensional probability distribution as ξ(t) for everyt. The coefficients σ andb have a simple interpretation:

$$\sigma \left( {t,x} \right) = \left( {E\left( {\delta \delta ^T \left( t \right)|\xi \left( t \right) = x} \right)} \right)^{\tfrac{1}{2}} ,b\left( {t,x} \right) = E\left( {\beta \left( t \right)|\xi \left( t \right) = x} \right).$$

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References

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Author information

Authors and Affiliations

  1. Department of Probability Theory and Statistics, Eötvös University Budapest, Muzeum krt 6-8, 1088, Budapest, Hungary

    I. Gyöngy

Authors
  1. I. Gyöngy
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Additional information

This paper was written while the author was visiting the J.W. Goethe-University in Frankfurt as a fellow of the Alexander v. Humboldt Foundation

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Cite this article

Gyöngy, I. Mimicking the one-dimensional marginal distributions of processes having an ito differential. Probab. Th. Rel. Fields 71, 501–516 (1986). https://doi.org/10.1007/BF00699039

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  • Received: 16 January 1985

  • Issue Date: December 1986

  • DOI: https://doi.org/10.1007/BF00699039

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Keywords

  • Differential Equation
  • Probability Distribution
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
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