Geodesic
Equations for probability distributions of local time on surface for diffusion processes and control problems
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 2, Tome 244 (1997), pp. 96-118
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The paper studies local time on surface for general $n$-dimensional diffusion process. Analogs of Kolmogorov–Fokker–Planck equations for the characteristic function and probability distributions of local time are derived and investigated for a wide class of $(n-1)$-dimensional surfaces. A general explicit formula which is an analog of the Tanaka formula is presented. Optimal control problems with functionals which depend on local time are investigated. 
@article{ZNSL_1997_244_a5,
     author = {N. G. Dokuchaev},
     title = {Equations for probability distributions of local time on surface for diffusion processes and control problems},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {96--118},
     year = {1997},
     volume = {244},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1997_244_a5/}
}
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N. G. Dokuchaev. Equations for probability distributions of local time on surface for diffusion processes and control problems. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 2, Tome 244 (1997), pp. 96-118. http://geodesic.mathdoc.fr/item/ZNSL_1997_244_a5/