The standard switching from one set of diffusion coefficients to another one occurs at random times corresponding to the moments of jumps of a Poisson process independent of the initial diffusion. The paper deals with the process of Brownian motion with variance taking one of two values by the switching depending on trajectories of the process. The most attractive from the computational point of view is the moment inverse to local time.
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A. N. Borodin, “Distributions of functionals of switching diffusions,” Zap. Nauchn. Semin. POMI, 454, 52–81 (2016); English transl. J. Math. Sci., 229, No. 6, 632–650 (2018).
M. Kac, “On distribution of certain Wiener functionals,” Trans. Amer. Math. Soc., 65, 1–13 (1949)
A. N. Borodin, Stochastic Processes, Birkhäuser, Cham (2017).
A. N. Borodin and P. Salminen, Handbook of Brownian Motion. Facts and Formulas, Birkhäuser, Basel, Heidelberg, New York, Dordrecht, London (2015).
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 486, 2019, pp. 35–43.
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Borodin, A.N. Distribution of Functionals of a Brownian Motion with Nonstandard Switching. J Math Sci 258, 758–763 (2021). https://doi.org/10.1007/s10958-021-05578-x