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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:
[1] Chaleyat Maurel M., Asterisque 52 pp 117– (1978)
[2] Chitashvili R. J., Stochastics 5 pp 255– (1981) · Zbl 0479.60062
[3] Kinkladze G. N., Some construction methods for strong and weak solutions of boundary problems in multi-dimensional stochastic differential equations (in russian) (1985)
[4] DOI: 10.1214/aop/1176994567 · Zbl 0448.60043
[5] DOI: 10.1007/BFb0087205
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