The Arc-Sine Laws for the Skew Brownian Motion and Their Interpretation

We consider the skew Brownian motion as a solution of some stochastic differential equation. We prove for the skew Brownian motion the analogues of the arc-sine laws for Wiener process. Unlike of existing results, we are forced to consider a stochastic differential equation with discontinuous diffusion coefficient. Possible interpretations of obtained results are suggested.


Introduction
Let { } ( ) A I x be the indicator function of set A. Levy in [1] proved such result:  be the first zero of Wiener process after the instant Among other researches that contain results of arc-sine law type for different functionals we should mention paper [4] where the result such as Theorem 1 for a process obtained by pasting together two Brownian motion processes was obtained.
In the article [5], besides another results, the analogue of Theorem 1 for the Brownian motion with linear drift was proved.
In ([6], remark 1) the result similar to Theorem 1 was proved for solution of stochastic equation if there exist such finite limits The article [8] contains results analogous to Theorems 1 and 2 for process ( ) ξ -Brownian bridge of length u.In [9] author investigated joint distribution of functionals from Theorems 1, 2, 3.
In the paper [10] is considered asymptotic behavior of probability The work [11] contains results such as Theorem 3 for the instant of hitting by solution of homogeneous stochastic equation The article [12] is devoted to proof of results such as Theorem 1 for the telegraph process.

I. H. Krykun
Papers [7] and [13] contain summary of further results which generalized results of Levy and others.
In this paper we consider a skew Brownian motion.It was defined by Ito & McKean [14] and later was investigated by many other authors.We mention works such scientists as Harrison & Shepp [15] and Le Gall [16] where skew Brownian motion was connected with solution of stochastic process involving the local time.Moreover, in papers [16] [17] [18] were obtained formulae which connect solutions of stochastic equations involving the local time with solutions of Ito's stochastic equations.Two recent entry points into the literature of occupation times and related functionals of skew Brownian motion are [19] and [20].
In this paper for the skew Brownian motion arc-sine laws analogous Theorems 1 -4 for Brownian motion are proved.We consider Ito's stochastic equation with discontinuous diffusion coefficient unlike existing results.
This paper is organized as follows: the second section contains denotation and definition of considered functionals and main results-Theorems 5 -8.In the third section Theorems 5 -8 and auxiliary Lemma are proved.The fourth section contains some interpretation of obtained results.

Main Results
Let us consider the skew Brownian motion as a solution of stochastic equation involving the local time If 1 β ≤ then there are the unique strong solution of Equation (1) [15], i.e.  ( ) exists almost surely and Equation (1) fulfils almost surely.We denote ( ) The main results of this article are four following theorems.
Theorem 5. Let ξ be the skew Brownian motion, defined by (1) and con- stant 1 will tend to 0 for 0 1 x < < , and for Obviously that the same result follows from formula (2).

Let us define
inf , here set inf 1.
and for Theorem 6.Let ξ be the skew Brownian motion, defined by (1) and con- stant 1 Let us denote ( ) Theorem 7. Let ξ be the skew Brownian motion, defined by (1) and cons- tant 1 x T ≤ = Theorem 8. Let ξ be the skew Brownian motion, defined by (1) and cons- tant 1

Proofs of Theorems 5 -8
We denote 1) is closely connected with solution of some Ito's stochastic equation.Namely let's define the function Now we consider such Ito's stochastic equation The diffusion coefficient of this process is a discontinuous function of bounded variation so from ( [21], Theorem) it follows the existence of strong solution of Equation (7).
It is well known [17] that ( ) ( ) ( ) Proof of Theorem 5.For some constant 1 β < for the processes ξ and η we have such property of their occupation times: We denote function ( ) From [21] it follows the existence of unique strong solution of Equation ( 9).
Because processes ( ) t η and ( ) x t are different only at point 0 and both processes ( ) t η and ( ) x t spend zero time at point 0 so we have equality For the functional we may use ( [22], theorem 13, p. 149).From it and formulae ( 8)- (10) we get the statement of Theorem 5. □ Remark 3. It follows from Theorem 1 that for Wiener process ( ) We can compare this result with analogous one for skew Brownian motion.It follows from Theorem 5 that and for Proof.We firstly consider the case 0 x > .By the multiplication rule of probability, definition of T M and property (3): Now consider the first multiplicand in the right hand side of equality (13). From The last equality follows from properties of skew Brownian motion ( [22], p. 169).So from (13) it follows the first equality in (11).
Further we use the formula for transition probability density function of skew Brownian motion (22, theorem 6, p. 168).Then for The result for { } can be proved analogously.
Corollary.Now we compare results of Lemma with known results about distributions of maximum and minimum of Wiener process ([2], theorems 3.15, 3.17) and conclude that distribution of maximum (minimum) of skew Brownian motion not depend from skewness coefficient β and coincide with distri- bution of maximum (respectively minimum) of Wiener process.
Proof of Theorem 6.By the Markov property of process ξ , Chapman-Kolmogorov equation, properties (3), ( 4) and results of the Lemma we have: ∫ Journal of Applied Mathematics and Physics So we get (5).□ Proof of Theorem 7.

{ }
Analogously to proof of the Lemma for processes ( ) ( ) ( ) So statement of Theorem 7 can be proved in the same way as analogously proof of such result for Wiener process (for example see [24], theorem 5.26).□ Proof of Theorem 8.By the Markov property of process ξ , Chapman-Kolmogorov equation, properties (3), ( 4) and results of the Lemma we have:

∫
Later one received other similar results where appear arc-sine function.They were called the arc-sine law, particularly result of the theorem 1 was named "the first arc-sine law".One consequence of the work[1] is the following result:

0
,T and for θ -the first instant when for functional of maximum (and minimum) each of them coincide by Corollary with distribution of maximum (and minimum) of Wiener process ( ) w t with ( ) Applied Mathematics and Physics