A lemma of variational distance between maximal ...

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A lemma of variational distance between maximal functions with application to the skorokhod problem in a nonnegative orthant with statedependent reflection directions Malkhaz Shashiashvili To cite this article: Malkhaz Shashiashvili (1994) A lemma of variational distance between maximal functions with application to the skorokhod problem in a nonnegative orthant with state-dependent reflection directions, Stochastics and Stochastic Reports, 48:3-4, 161-194, DOI: 10.1080/17442509408833905 To link to this article: http://dx.doi.org/10.1080/17442509408833905

Published online: 04 Apr 1994.

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Sfochastics and Sfochastics Reports, Vol. 48, pp 161-194 Reprints available directly from the publisher Photocopying permitted by license only

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A LEMMA OF VARIATIONAL DISTANCE BETWEEN MAXIMAL FUNCTIONS WITH APPLICATION TO THE SKOROKHOD PROBLEM IN A NONNEGATIVE ORTHANT WITH STATE-DEPENDENT REFLECTION DIRECTIONS MALKHAZ SHASHIASHVILI Laboratory of Stochastic Analysis and Statistical Decisions, Tbilisi State University, University Str. 2, 380043, Tbilisi, Georgia (Received 30 September 1992)

We consider the Harrison-Reiman problem of constructing the reflected Brownian motion on an ndimensional nonnegative orthant in the case of state-dependent reflection directions. The corresponding modified Skorokhod problem is posed for n-dimensional right continuous functions with left-hand limits on [0, co) and the sufficient condition, that requires a restriction on the directions of reflection is found, which together with the local Lipschitz condition on reflection directions (for each (n - 1)-dimensionalhyperplanes that form the boundary of the orthant) gives the existence and the uniqueness of the corresponding solution. The main tool in proving the uniqueness result turns out to be a certain lemma of variational distance between the maximal functions. Thanks to uniqueness the continuity of the corresponding solution mapping (in the Skorokhod topology) is established. Then by simply substituting an n-dimensional Brownian motion in place of arbitrary continuous function in the Skorokhod problem we obtain the reflected Brownian motion on an orthant with state-dependent reflection directions.

KEY WORDS: Skorokhod problem, maximal functions, variational distance, solution mapping, reflected Brownian motion.

1. INTRODUCTION We consider the Skorokhod problem in the n-dimensional nonnegative orthant in this paper. This problem has been studied earlier by J. M. Harrison and M. I. Reiman [I] in the case of constant reflection directions for each (n - 1) dimensional hyperplane, that form the boundary of the orthant. Though Tanaka [2] solved the Skorokhod problem for general convex domains in the case of normal direction of reflection, the case of oblique reflection investigated in [I] turns out to have special features, namely the possibility of either non existence of the solution, or the existence of more then one solution. In the case ofnormal reflection the solution to the Skorokhod problem always exists and is unique (for the case of continuous functions). To illustrate this let us consider the Skorokhod problem in the nonnegative quadrant with constant reflection directions. Let

M. SHASHIASHVILI

162

be the nonnegative quadrant and the vectors

be the reflection directions (respectively) on the boundary lines X, = 0,X2 = 0. For arbitrary continuous function X, = (X: ,Xf), t 2 0 with X; 2 0, X i 2 0 it is required to find the pair of continuous functions z,=(z:,z:),

Y,=(y:,Y:)

t 2 0,

which jointly satisfy the following conditions:

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1) Zf>O,t>O,i= 1,2,

2) Z:=X:+Y:+q,Yf

t20,

z;=x;+ y;+qly: 3) Y: , Yf are nondecreasing functions, such that

YA=O,Yi=O

and

1,~,~,d~f=O,t20,

i = l,2,

that is, Yf increases only at those times t where Z: = 0. If such a pair exists, then it is called the solution of the Skorokhod problem in the nonnegative quadrant for the given function X, = (X: ,X:), t 2 0 (and with the given reflection directions). a) Suppose first that q, < O,q, c Olq, I.Iq,l> 1. Let us take an arbitrary continuous function X, = (X:, X f ) with XA 2 0, X i 2 0 such that at some t > 0 X: < 0, X; < 0. If there exists the above mentioned pair Z , , Y,, then (at the same t > 0) we must have

but from the definition of Y:, Y: we have Y: 2 0, Yf 2 0, and hence Y: > 0, Yf > 0. Now multiplying the previous two inequalities we obtain Y: Yf > (- 9,) (- 4,) Y: Y: ,i.e.,

which is a contradiction. Thus in the case (a) there doesn't exist a solution of the Skorokhod problem. b) Suppose now

VARIATIONAL DISTANCE AND THE SKOROKHOD PROBLEM

Let us take the function X, = (X:, X:) with

x:

= - qt.

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It is easy to verify that the conditions of the Skorokhod problem are satisfied by the following two pairs of functions

Thus to obtain the existence of the solution (or existence and uniqueness) it is necessary to impose some restrictions on the reflection directions. It is well-known (and will be clear from the subsequent presentation), that the existence and the uniqueness will hold under the following condition

which is equivalent to the assumption that there exist positive numbers a,, a, and a with 0 < a < 1, such that a, Iq, I < aa,, a,lq,( < aa,. Now let us formulate the Skorokhod problem in a nonnegative orthant for continuous functions. Consider n-dimensional nonnegative orthant R: = {XI,...,X,,):Xi>O, i = 1,..., n). The faces of the orthant will be denoted by Fiand the regular part of the face Fi-by Fi:

F i= {(X,,. . ., X,):Xj 2 0, j = 1,. . . ,n.Xi = 0), i = 1,. . ., n

It is assumed that on the faces Fi there are given continuous functions qij(x), i , j = 1,. . .,n, j # i. These functions in the sequel will be extended to the whole orthant and we shall do this in an obvious way putting

The Skorokhod problem in an n-dimensional nonnegative orthant R; with the reflection directions on the regular part Fiof the Faces Fi at the point x€Figiven by the vector (qil(x),. . ., 1, .. .,qin(x)) (the 1 being in the ith position) and for continuous function X = X,, t 2 OEC(R+ ,R") is posed in the following manner: we seek a pair of n-dimensional continuous functions Z = Z,, Y= Y,, t 2 0, which jointly satisfy the following conditions:

M. SHASHIASHVILI

1 64

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3) every component Y f of the function Y, = ( Y: ,.. . , Yy) is nondecreasing with Yb = 0 and

that is Yf increases only at those times t where Zf = 0. To get the results on the existence (or existence and uniqueness) of the solution of the Skorokhod problem we follow the same path as in Tanaka [2], that is we generalize this problem for the case of functions X = X,, t 2 0 belonging to the space D(R+,R,), the space of n-dimensional right continuous functions with left-hand limits on time interval (0, a), and consider the modified Skorokhod problem: for the function X = X,, t 2 0, belonging to D(R+,Rn) we seek a pair of functions Z = Z,, t 2 0, Y= Y,, t 2 0, belonging also to the space D(R+,Rn),which jointly satisfy the following conditions:

3) every component function Yi, of Y, is nondecreasing with Yb = 0 and satisfies

In this setting the word modified is related to the terms qki(Z,-)instead of natural reason, why we consider this modified problem instead of natural one lies in the fact that the uniqueness of the solution (Z,, Y,) is lost even for smooth qij(x) t 2 0 being step-functions. Indeed suppose, that for some i there exist and with X = two points Z, Z E F ~such , that the reflection rays (with the origin at the points Z, 2) (qil(Z),. . . , 1,. . .,qin)Z))(qi1(2),. . . , 1,. . ., gin@))if extended in the opposite direction (outer direction) have an intersection at some point X$R:, that is - qki(Zs).The

3,

or equivalently

-

- -

Z j - XI = /?qij(Z), where qii(X)= 1,

B

(note that if Z i = Zi = 0, then p = = - Xi). Now for arbitrary X,ER;, T > 0, take the following step-function

VARIATIONAL DISTANCE A N D THE SKOROKHOD PROBLEM

Then it is easy to see that two distinct solutions of the Skorokhod problem are X,,

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"={z,

OdtT

Y,=(Y: ,..., Y:) Yj= 0, j i,

with

+

Therefore to get the uniqueness results for all X = X,ED(R+,Rn) some unnatural conditions should be imposed in that natural problem. Here it is worth noting that for general convex domains G in Rn and for any X = X,ED(R + ,Rn)as shown by Tanaka [2] for the Skorokhod problem with normal reflection the uniqueness of the solution Z = Z,, t 3 0 always holds. In this case, the equation

(where X$G and n(Z) is a unit inward normal at the boundary point Z 4 G ) has a unique solution-the nearest point to X of the boundary. The intuitive reasoning behind the modified problem consists in the following: when the continuous function X = X,E C(R + ,Rn) is approximated uniformly (on bounded intervals) by step-functions Xn = X:, then the jumps AX: become small as n + co,this causes closeness of qki(Z:-) to qki(Z:), thus the modified problem approximates the true problem of reflection. In applications, such as networks of queues (M. I. Reiman [3]) it appears that the components Yf, i = 1,. . . , n, are continuous (even absolutely continuous) by the very structure of the problems and then the integrals

coincide. So nothing is lost in considering the modified Skorokhod problem and there is one essential gain-the uniqueness of the solution (in assumption of local Lipschitz condition on functions qij(X)),which will give to us 1) the continuity in the Skorokhod topology of D(R +, Rn)of the mapping X +(Z, Y), which enables one to prove weak convergence results in applications (M. I. Reiman [3]), 2) the possibility to prove the existence of homogeneous (in time) Markov process, which behaves like a n-dimensional Brownian motion on the interior of the orthant and reflects from the Faces F , of the orthant in the directions determined by the vectors (at the point XEF",)(qil(X), . . . , 1,. . . , qin(X)).

It should be noted here, that the uniqueness of the solution of Skorokhod problem with oblique reflection for general continuous X = X,EC(R+,Rn) has been an open question in P. L. Lions, A. S. Sznitman [4]. Now it is solved for the case of nonnegative orthant by virtue of a Lemma about variational distance between maximal functions, which will be presented in Section 2.

166

M. SHASHIASHVILI

The precise formulation and proofs of main results will be given in Section 3, where our principal assumption is the following: there exist positive constants a,, a,, .. .,a,, a with 0 < a < 1 such that for any XER'!+we have

Finally in Section 4 we will construct reflected Brownian motion on an orthant by simply substituting an n-dimensional Brownian motion in place of arbitrary continuous X = X,, t > 0 in the Skorokhod problem (1).

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2. LEMMA O F VARIATIONAL DISTANCE BETWEEN MAXIMAL FUNCTIONS As mentioned above the main technical tool in proving the results concerning uniqueness of the solution of the Skorokhod problem is the variational distance lemma proved for the case of continuous functions in the author's work-M. A. Shashiashvili [ 5 ] . In the present paper we shall need to use this lemma for the case of functions X = X,€D(R+,R)-the space of the right continuous functions with left-hand limits on the time interval [0, co). Therefore we have to prove it for this case and in turn it will be necessary to recall the one-dimensional Skorokhod problem for functions X = X,cD(R+,R) (For completeness we include the corresponding proofs). Consider the function X = Xi,t > 0, belonging to the space D(R+,R) with the initial condition X o 2 0. By the solution of the one-dimensional (in the domain (0, + co))Skorokhod problem there is understood the pair of functions (Z,, Y,), t 2 0-belonging to D(R+,R), which jointly satisfy the following conditions:

2, = X ,

+ Y,,

t 20,

(4)

3) Y, is a nondecreasing function with Yo = 0, such that

Let us show at first the uniqueness of the solution and then prove the existence. Suppose (Zf,Yf),i = 1,2 are two solutions. We have

VARIATIONAL DISTANCE AND THE SKOROKHOD PROBLEM

Now taking into account the third requirement of the problem

we get and thus

Consider now the pair of functions (Z,, Y,), t 2 0, given in an explicit form

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Y,

= sup max (0, - X,),

Z, = X, + Y,.

sQt

It is obvious, that Yo = 0, Y, is nondecreasing right continuous function, where Z, 2 0. It remains to prove (and this is an essential step) that

For this purpose at first notice that

Y, - = sup max (0, - X,), hence S 0, AX, = 0)

We have

y s > -xs, where the function X, is continuous at a point s, therefore E > 0 can found such that Y, >

sup

(- Xu).

s-eQuQs+c

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But from the very definition of the function Y, we have

Y, -,,sup max (0,- Xu) s-e 0, AX, = 0)' As regards the second set (s: 2,> 0, AX, Z 0), it is clear that this set is countable, i.e.equals (s,: Z,, > 0, AX," # 0). But we already know that Zan> 0 +AYSn = 0, and hence

finally we have

S

dY, = 0

(s: z, > 0)

VARIATIONAL DISTANCE AND THE SKOROKHOD PROBLEM

169

Thus the one-dimensional Skorokhod problem h as a unique solution for any X = X,ED(R+ ,R) with X, 2 0 and is given in an explicit form

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Before formulating the variational distance lemma in its generality, let us consider it for the case of sequences. Suppose there are given two sequences of real numbers X i , X,Z, k = 0,1,2,.. . . Now define the following maximal sequences

Consider the increments

Then we assert that the following variational inequality is true (1 < n < N )

Indeed, for two arbitrary real numbers a and b the following equality holds

From the definition of the sequences Y,', YZ we have

Y,'

= max (Y,'-, ,X,'),

hence

AY,' = max(O,X,'- Y,'-,),

AY,Z = max (0, X i - Y,ZY,'

-

X,' = max (0, Y,'-, - X,');

Y; - X i

= max (0, Y,2-

- X:);

M. SHASHIASHVILI

170

If now we introduce the notation

then it is obvious that

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Thus the above-mentioned equality admits the following (impressionable) form

From this equality denoting 2, = (Y,' - x,')- (Y; - x;)

and using the obvious inequality

Summing these relations, we come to the desired variational inequality (of course in fact these two inequalities are equivalent). Consider now two functions X : , X: t 2 0 belonging to the space D(R + ,R). Let us choose two arbitrary numbers Y;, Y; with Y; 3 x;, Y; 2 x; and define the following maximal functions

(Note that for s < t, Y:

= sup max (Y,' ,Xi), s s u s t

LEMMA(ofmaximalfunctions)

Y,2 = sup max (Y:, Xi)). s$u$t

Suppose that on some time interval (s, t ] the difference

VARIATIONAL DISTANCE AND THE SKOROKHOD PROBLEM

Xu= X,' - X i has a bounded variation. Then the following inequality holds

Proof First note that if the function Y,ED(R+,R) is of locally bounded variation, then there exists a Borel measurable function U, with two values 1, such that

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+

Indeed, write the decomposition of Y, as the difference of two nondecreasing (right continuous) functions

Consider the function

Ft = Y;

+ Y?.

Obviously the measures dY:, dY,2 are absolutely continuous with respect to the measure d Y,.Therefore by the Radon-Nikodym theorem there exist nonnegative Borel measurable functions I;, I:, such that

From these relations we have

hence

where

and it is clear that Us takes on only two values

+ 1.

M. SHASHIASHVILI

172

Taking into account the established relation in our case for the difference Y, = Y,? - Yf we have

where Vuis a Bore1 measurable function with two values f 1. Consider the one-dimensional Skorokhod problem (in the domain (0, a))for the functions Yh - X,?, Y i - X:, t 2 0, and denote the corresponding pairs of solutions by

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we have

z,~=Y~-x:+?:,

where

?,2=supmax(O,-Y;+Xf), sst

and also

Thus we get (note the obvious equalities Y,? = Yh + P:, Y: = Y i + p:)

z:=y:-X2 where

S:

I(Z: z 0)d Y: = 0,

I

I(Z: > 0)d Y:

= 0.

Introduce now the notation

then it is clear that the following implications hold: Zt > O=Z,? > 0,Z, < 0+Z,2 > 0. Therefore taking into account the last relations we get

These relations give the possibility to establish the following key representation for the variation of the difference of maximal functions Y:, Y,2

VARIATIONAL DISTANCE AND THE SKOROKHOD PROBLEM

where

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Indeed, we have

but the last expression exactly coincides with the following one

Therefore the representation (13) is valid. From the assumption made in Lemma that the difference X: - X: is of bounded variation on time interval (s, t ] we conclude, that the same property holds for the function Z , , hence

Now let us bound the expression

For this purpose we need a formula of differentiation of the functionf ( Z , ) , where f ( x ) is a convex function (P. A Meyer [ 6 ] )

In this formula f ' ( x ) denotes the left derivative of the function f (x).

M. SHASHIASHVILI

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Rewrite this formula in the form, convenient to us

Let us take, respectively

f (x) = x'

= max (0, x)

f (x) = x-

= max (0, - x),

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then as (x+)' = I(,,,),

(x-)' = - I

(x - Xi, yi 2 0, thus y i 2 rnax (0,- 2'). Suppose in this relation that strict inequality holds: y ' > rnax (0,- f i ); that is yi > 0, yi> - Zi, then we should have y i > 0 have yi > 0, zi > 0, which contradicts the conditions zi.yi = 0. Hence we get yi = rnax (0,- f ').Suppose now (z,y) is a solution to the problem (20).We have y i = rnax (0,- Xi),zi = X i + y ' = rnax (0,f i ) , therefore zi 2 0, yi 2 0, zi.yi = 0; thus the pair (z,y) is a solution to (19). We have established the equivalence of two problems (19)and (20)and what remains to prove is the fact that (20) has a unique solution. For this purpose consider the space R: with the metric

Then consider the mapping Y Yof this space into itself defined by the formula ( Y Y ) i = m a x ( O , - - x i - ~ q k i Y k )i,= l , ..., n, k f i

(here notice that the point x = (x', . . . ,xn)is fixed). Let us show that this mapping is a contraction. Take Y, "YE R; ; we have

Y Y)' = rnax (0,- xi - C qkiY k ) , (Y t)i = rnax (0,- xi k#i

1 qkiFk), k+i

and hence

Now after multiplying this inequality by ai and then taking the sum, we get (here and

VARIATIONAL DISTANCE AND THE SKOROKHOD PROBLEM

afterwards we use the obvious equality

C:=,Ck,ibik= C;= C i Zbik) k

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that is d ( Y Y , Y k ) < ad(Y, y). Thus a unique solution of problem (20) (and therefore of problem (19))does exist. Consider the step-function x = x , e D ( R + ,Rn) with x,ER; ,i.e. there exist times 0 = to < t l < ,. . ., < t , < ,. . . , with t , + oo,such that

If we can construct the solution of the modified Skorokhod problem at times t,, then as the function x, is constant over the time intervals [t,, t,+ ,), obviously we can define ( Z , , Y, = Gm? Yt,) if

Put ( Z , , Y o )= ( X , , 0) and suppose the solution ( Z , , Y, ) is constructed at times to,,. . . ,t,- ,. To extend the solution to time t , means to satisfy the following conditions Z i m > O , i = 1,..., n , A Y f m > O , i = l ,..., n.

where obviously

The latter conditions can be rewritten in the following form

But this is precisely the problem (19),which as we already have established has a unique solution. Therefore the solution of the above-mentioned problem (2) exists for the case of step-functions.

M. SHASHIASHVILI

178

THEOREM 2 Suppose the condition ( 3 ) holds. For any solution (Z,, Yt) of the modified Skorokhod problem the following bounds hold for (s < t )

1 a i ( y j - ~ f 0 and X = (X,)(,,,, define

then by the definition of the Skorokhod problem (1) and the uniqueness theorem 3 we easily obtain

thus Z T + ,= z: = Z(X*),, where the process X* =(X:)(,,,,

is defined by

Now as Z, is F,-measurable and as the process (X,+, - X,), t 2 0 is independent of the a-algebra F , (having the probability distribution also independent of parameter T ) we assert that Z = (Z,),,,,, is indeed a Markov process with stationary transition probabilities. Remark By similar methods as above it is possible to construct a unique pair of continuous processes (Z, Y) = (Z,, Y,)(,,,, satisfying the system (34) for the case of state-dependent drift b(Z) = (bl(Z), . . ., bn(Z)),which is supposed to be locally Lipschitz and satisfying the linear growth condition. The resulting process will remain Markov with stationary transition probabilities. References [I] J. M. Harrison and M. I. Reiman, Reflected Brownian motion on an orthant, Annals of Probability, Vol. 9, No. 2 (1981), pp. 302-308. [2] H. Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J., 9 (1979), pp. 163-177.

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[3] M. I. Reiman, Open queueing networks in heavy traffic, Mathematics of operations research, Vol. 9, NO. 3 (1984), pp. 441-458. [4] P.L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Communic on Pure and Applied Math., Vol. XXXVII, No. 4 (1984), pp. 51 1-537. [S] M. A. Shashiashvili, On the variation of the difference of singular components in the Skorokhod problem and on stochastic differential Systems in a half-space, Stochastics., Vol. 24 (1988), pp. 151-169. [6] P. A. Meyer, Un cours sur les inttgrales stochastiques, Seminaire de Probabilities X,-Berlin etc. Springer-Verlag, Lect. Notes Math., Vol. 511 (1976) [7] T. Lindvall, Weak convergence of probability measures and random functions in the function space D[O, co)J. Appl Probab., Vol. 10 (1973), pp. 109-121. [8] S. Anulova and R. Liptser, Diffusion approximations for process with normal reflection, Theory of Probability and its Applications (in Russian), Vol. 35, No. 3 (1990), pp. 417-430.