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On the variation of the difference of singular components in the Skorokhod problem and on stochastic differential systems in a half-space. (English) Zbl 0648.60055
Let $$Y=(Y_ t)_{t\geq 0}$$ is a real-valued, continuous adapted process with $$Y_ 0\geq 0$$ a.s. and $$K_ t=\sup_{s\leq t}\max (0,-Y_ s)$$, $$Z_ t=Y_ t+K_ t$$. The pair (K,Z) is the unique solution of the so- called Skorokhod problem for Y, see for example N. El Karoui and M. Chaleyat-Maurel [Temps locaux. Exposés du Séminaire J. Azema-M. Yor (1976-1977), Asterisque 52-53, 117-144 (1978; Zbl 0385.60063)]. The author proved the following theorem:
Let $$Y^ 1$$, $$Y^ 2$$ be two adapted continuous processes with $$Y^ i_ 0\geq 0$$, $$i=1,2$$, a.s. and suppose that $$Y=Y^ 1-Y^ 2$$ is a semimartingale. Then the following representation holds for all $$t\geq 0$$ a.s. $\int^{t}_{0}| d(K^ 1_ s-K^ 2_ s)| =| Y_ 0| -| Z_ t| -\int^{t}_{0}u_ s dY_ s+\int^{t}_{0}I_{(u_ s=1)} dL^ 0_ s(Z)+\int^{t}_{0}I_{(u_ s=-1)} dL^ 0_ s(-Z),$ where $$Z=Z^ 1-Z^ 2$$, $$Z^ i$$, $$i=1,2$$, is the solution of the Skorokhod problem for $$Y^ i$$, u is some predictable process taking the values $$+1$$, -1 and $$L^ 0_ s(Z)$$, $$L^ 0_ s(-Z)$$ are local times at zero of the semimartingales Z, -Z.
From the theorem a representation for $$\int^{t}_{0}| d(X^ 1_ s-X^ 2_ s)|$$, $$X^ i_ t=\sup_{s\leq t}Y^ i_ s$$, $$i=1,2$$, is obtained. These results are applied to the investigation of the existence and uniqueness of a strong solution to a stochastic differential system in a half space and to the estimation of the distance in variation between the local times of two reflected (at zero) stochastic processes with the same diffusion and drift coefficients.
Reviewer: R.Manthey

##### MSC:
 60G48 Generalizations of martingales 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J55 Local time and additive functionals
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##### References:
  Chaleyat Maurel M., Asterisque 52 pp 117– (1978)  Chitashvili R. J., Stochastics 5 pp 255– (1981) · Zbl 0479.60062  Kinkladze G. N., Some construction methods for strong and weak solutions of boundary problems in multi-dimensional stochastic differential equations (in russian) (1985)  DOI: 10.1214/aop/1176994567 · Zbl 0448.60043  DOI: 10.1007/BFb0087205
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