Geodesic
Distributions of functionals of diffusions with non-standard switching
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 29, Tome 495 (2020), pp. 87-101

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The paper dials with the results that allow us to calculate the joint distributions of functionals of diffusions with switchings that occur in random time moments depending on the diffusion trajectory. Standard switching from one set of diffusion coefficients to another set occur in random moments of time corresponding to the moments of jumps of the Poisson process, independent of the initial diffusions. More general non-standard switching occur when the integral functional of the trajectory reaches the value equal to the exponentially distributed variable. With a unit integrand such switching become standard. 
@article{ZNSL_2020_495_a4,
     author = {A. N. Borodin},
     title = {Distributions of functionals of diffusions with non-standard switching},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {87--101},
     publisher = {mathdoc},
     volume = {495},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a4/}
}
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A. N. Borodin. Distributions of functionals of diffusions with non-standard switching. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 29, Tome 495 (2020), pp. 87-101. http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a4/