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Proof of a Tanaka-like formula stated by J. Rosen in Séminaire XXXVIII

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Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 1934)

Abstract

Let B t be a one dimensional Brownian motion, and let α′ denote the derivative of the intersection local time of B t as defined by J. Rosen in [2]. The object of this paper is to prove the following formula

$$\frac{1}{2}\alpha _t^\prime (x) + \frac{1}{2}\operatorname{sgn} (x)t = \int_0^t {L_s^{B_s - x} dB_s - \frac{1}{2}\int_0^t {\operatorname{sgn} (B_t - B_u - x)du} }$$
((1))

which was given as a formal identity in [2] without proof.

Keywords

  • Brownian Motion
  • Local Time
  • Formal Identity
  • Eral Form
  • Dominate Convergence Theorem

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References

  1. Revuz, D., Yor, M. (1986) Continuous Martingales and Brownian Motion Springer, Berlin.

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  2. Rosen, J. (2005). Derivatives of self-intersection local times, Séminaire de Probabilités, XXXVIII, Springer-Verlag, New York, LNM 1857, 171–184.

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Markowsky, G. (2008). Proof of a Tanaka-like formula stated by J. Rosen in Séminaire XXXVIII. In: Donati-Martin, C., Émery, M., Rouault, A., Stricker, C. (eds) Séminaire de Probabilités XLI. Lecture Notes in Mathematics, vol 1934. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77913-1_9

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