On the variation of the difference of singular components in the Skorokhod problem and on stochastic differential systems in a half-space.

*(English)*Zbl 0648.60055Let \(Y=(Y_ t)_{t\geq 0}\) 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.

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 |

##### Keywords:

Skorokhod problem; representation; predictable process; local times; existence and uniqueness of a strong solution
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\textit{M. A. Shashiashvili}, Stochastics 24, No. 3, 151--169 (1988; Zbl 0648.60055)

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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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