Séminaire de Probabilités XLI pp 199-202 | Cite as
Proof of a Tanaka-like formula stated by J. Rosen in Séminaire XXXVIII
Chapter
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 which was given as a formal identity in [2] without proof.
$$\frac{1}{2}\alpha _t^\prime (x) + \frac{1}{2}\operatorname{sgn} (x)t = \int_0^t {L_s^{B_s - x} dB_s - \frac{1}{2}\int_0^t {\operatorname{sgn} (B_t - B_u - x)du} }$$
(1)
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.Google Scholar
- 2.Rosen, J. (2005). Derivatives of self-intersection local times, Séminaire de Probabilités, XXXVIII, Springer-Verlag, New York, LNM 1857, 171–184.Google Scholar
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© Springer-Verlag Berlin Heidelberg 2008