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Some two-dimensional extensions of Bougerol’s identity in law for the exponential functional of linear Brownian motion. (English) Zbl 1303.60073

The main result is a two-dimensional identity in law. Let \((B_t,L_t)\) and \((\beta_t, \lambda_t)\) be two independent pairs of a linear Brownian motion with its local time at 0. Let \(A_t=\int_0^t \exp(2B_s)\, ds\). Then, for fixed \(t\), the pair \((\sinh(B_t), \sinh(L_t))\) has the same law as \((\beta(A_t), \exp(-B_t)\lambda(A_t))\), and also as \((\exp(-B_t)\beta(A_t),\lambda(A_t))\). This result is an extension of an identity in distribution due to P. Bougerol [Ann. Inst. Henri Poincaré, Sect. B 19, 369–391 (1983; Zbl 0533.60010)] that concerned the first components of each pair. Some other related identities are also considered.

MSC:

60J65 Brownian motion
60J60 Diffusion processes
60J55 Local time and additive functionals

Citations:

Zbl 0533.60010
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References:

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