Geodesic
On reflection of continuous functions and random processes having local times
Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 3, pp. 665-669

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Assuming that the local time $\alpha(t,u)$, $t\in[0,\infty)$, $u\in\mathbf R$, of a real-valued continuous function $X(s)$, $s\in[0,\infty)$, is continuous in the time parameter, we show that $$ -\min_{0\le s\le t}\min(X(s),0)=\int_{-\infty}^0\mathbf{1}(\alpha(t,v)>0)\,dv, $$ where the function $\int_{-\infty}^01(\alpha(t,v)>0)\,dv$ is the local time for $\xi(s)=\alpha(s,X(s))$. We apply this result to random processes.
Keywords: local time, reflection problem, Brownian motion. 
@article{TVP_1995_40_3_a16,
     author = {F. S. Nasyrov},
     title = {On reflection of continuous functions and random processes having local times},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {665--669},
     publisher = {mathdoc},
     volume = {40},
     number = {3},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1995_40_3_a16/}
}
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F. S. Nasyrov. On reflection of continuous functions and random processes having local times. Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 3, pp. 665-669. http://geodesic.mathdoc.fr/item/TVP_1995_40_3_a16/