.Voniineor Analws. Tkheory Methods & Apphcnnom. Vol. 9. No. 1. pp. 39-51. 1985.

Printed in Great Bnrain.

0362-546x185 53 W - .W Q 1985 Pergamon Prrss Lrd

A SYSTEM OF PARABOLIC VARIATIONAL INEQUALITIES ASSOCIATED WITH A STOCHASTIC SWITCHING GAME

NAOKI YAMADA

Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe, 647 Japan

(Receiued 1 November 1983; received for publication 15 February 1984)

Ke.v words and phrases: Variational inequalities, stochastic game.

1. INTRODUCTIOS

IN THIS paper we are concerned with a system of parabolic variational inequalities each of which is subjected to two obstacles, That is, we shall prove the existence and uniqueness of a solution to the following system:

~~+~(~,t)-~k~~(~,r)IuP*‘(x,t) -K a.e.inQ,

-!J-g+ AP(t =f” if uP+~-~<~P<uP~‘+K,

-$+AP(r)uPsfP if up=&‘-l-K,

(1.1) if UP = UP - 1 - k

LlPI~ = 0, uP(x, T) = Lqx) x E R.

forp = 1,. .,m, whereu”” = ul.

Here Q = $2 x (0, T) is a bounded cylindrical domain in 22 xR:, TX is the lateral side of Q,

N

AP(Ou = - & a$(~, 0 $fCP(X,l)U.

I

p=l.. . ., m, are second order elliptic operators, fJ’, LP are given functions, and k, K are positive constants.

Motivated by Evans and Friedman [3] and Lions [5] who treated the Bellman equation by an analytical method, we already studied an analogous system of elliptic variational inequalities in [7]. Evans and Lenhart [4] and Lions [6] have treated the parabolic Bellman equation by an analytical method similar to those in [3] and [5].

Our method for treating the system (1.1) is also quite similar to the previous one [7]. After stating assumptions and formulating bilinear and strong problems in Section 2, we consider, in Section 3, an approximate system for the problem and solve it. In Section 4 we establish some apriori estimates for approximate solutions. Section j is devoted to proving the existence of solutions for the bilinear problem by passing to the limit in approximate systems. In Section 6 we shall prove that the solution obtained in the preceding section has indeed W’.‘-regularity

39

10 N. YAMADA

in the space variables. This result shows not only the solvability of the strong problem, but also the applicability of the Ito formula in the next section.

In Section 7 we introduce a stochastic switching game and represent the solution component as a value of a saddle point of this game. By means of this representation we have the uniqueness of solutions. Our main result is stated in theorem 7.2.

2. ASSUMPTIONS AND FORMULATION OF THE PROBLElM

Let Q be a bounded domain in R” with smooth boundary. Let T > 0 and write Q = R X (0, T), C = JR x (0, T). The space W’*‘.r(Q), 1 5 r Z =, is the totality of functions such that

We denote by C’.‘(Q) the space of continuously differentiable functions in Q up to order 2 with respect to x and up to order 1 with respect to t. Other spaces appearing in this paper are the usual function spaces. For U, u E L’(R), (u, u) denotes the inner product.

Letm>lbeafixedinteger.Forp=l,..., m, we consider second order elliptic operators

.v

AP(r)u = - ;,Fl a$(.~, t) & + j bB(x, r) $ + cP(x, t)u ” I 1

(2.1)

and the corresponding bilinear forms

where

We impose the following assumptions:

(A. 1) There exists a positive number (Y such that

for allp = 1,. . ., m, (x, t) E Q and EE RN. (A.2) tj~E C’.‘(Q) and

for some constant M > 0, where tj~ = a$, b+‘, cp, fp, p = 1, . . . , m, i, j, r = 1, . . . . N (fp are given functions of (x, t) E Q). Moreover, we assume cp(x, r) ZZ 0.

Consider two positive constants k, K and functions rip, p = 1, . . ., m, Of x E G.

A system of parabolic variational inequalities associated with a stochastic switching game 41

(A.3) tip E C;(a) for p = 1, . , m and there exists some constant k! > 0 satisfying

for all p = 1, . . . , m and r, s = 1, . . N. Moreover, we assume that

S-‘(x) - k S tip(x) 5 tip+‘(x) + K, x EQ,p = 1,. .,m,

where ti” -’ = ti’. Finally, we impose the following condition:

(A.4) k m-q

K+-- 4 forall 4= 1,. . .,m - 1.

For q E Hi(n), we define a set X6( ~JJ) in H& S2) by

X(y) = {u E H&R); y(x) - k 5 u(x) 5 I/J(X) + K, a.e. inn}

and use the notations

v+(x) = max{y(x), 0}, v-(x) = max{-q(x), 0).

Using these notations, we may state two formulations of the problem.

Bilinear problem. Find

cPEL2(0, T;H$Q)),~EL2(0, T;L2(S2)),p =l,. . .,m,

satisfying

- T(t), u - UP(L)) + aP(t; UP(f), u - UP(f)) 2 VP(t), u -UP(f))

for a.a. t E (0, T) and all u E X(G-l(t)).

uP(x, 7) = tiP(x),x E 5;2,

forp = 1,. . . , m, where ~l~-‘i = ui.

Strong problem. Find up E W2~‘~r(Q), 1 5 r < =, p = 1, . . . , m, satisfying (1.1).

(2.3)

(2.4)

(2.5)

Remark. In the following the letter iM denotes various constants which depend only on the constants in the assumptions. We choose co > 0 so large that bilinear forms

aP(t; u, u) f c&, u), p =l,. . .,m,t E(0, T)

become coercive and fix it.

3. APPROXIlMATE SYSTEMS

In this section we construct penalty equations which approximate (2.5) and prove the solvability of these equations.

Throughout this section we always assume (A.l)-(A.3).

We choose the penalty function p E Cz(X) such that P’(t) 2 0, P”(t) E 0, ,6(t) = 0 if t 5 0 and - 1 5 ,&(t) - @L(r) S 0, where ,&(r) = ,!z?(r/c), E > 0.

For each E> 0, we consider the following approximate system:

u{ E W’,‘.‘(Q), 1 5 r < z.

- -$ + Ap(t)~$ + PE(u{ - UC -’ - K) - PE(n$-’ - k - n:) = fP a.e. in Q. (3.1)

$1~ = 0, u$(x, 7) = S(x) if x E Q,

forp = 1,. . .,m, whereu:-’ = 11:.

To prove the existence of solutions of (3.1), we use the method of successive approximation. In the following of this section we omit the subscript E for simplicity.

First, we define 1(6(x, r) = S(x) for all p = 1, . . . , m. f E (0, 7). For n 2 1 and p = 1, . . , m, we define uf: inductively by the solution of

- f$ + AP(f) + ,6( uP,-uP~I~-K)-P(uP,I~-~--_UP,)=~~ a.e.inQ, (3.2)

z&ix = 0, uP,(x, T) = U”(x) if x E n, forp = 1,. . .,m, whereu;:ii =uA_i.

It is well known that there exists a solution L(, E W”.l,r(Q), 1 I r < TJ (for example, we refer to Bensoussan [2, theorem IV.4.1, p. 1571).

LEMMA 3.1. Let us, p = 1, . . . , m, be a solution of (3.2). We have

I/ ~%.‘(QJ 5 $a& {ecOTii fp%.“~Q~/~O~ eco’l/ ~pIb~d~ (3.3)

Proof. Let y be the right-hand side of (3.3). If we define Lz{(x, t) = ecO’@ (x, 0, then, by (3.2), the tif: satisfy equations:

- eco’p( e -w,{ I ; - k - e-‘@$‘) = &@fP,

ti(lx = 0, fi{(x, T) = ecoTtiP(x) if x E R,

forp = 1,. . .,m, wheretiT.?l’= ti!,_i.

(3.4)

First we show that ti$(x, t) S y by induction on n. Clearly, the case n = 0 is valid. We assume r.i{_iS y for p = 1, . . ., m. Multiply both sides of (3.4) by (ti$ - y)’ and integrate over

51 x (t, I”). Using the coercivity of the form ap(s; u, u) + c&, u) and the induction assumption.

we get

(P(e-cPti$Ii - k - e-c@Cpn), (tit - y)+) = 0,

111 (G(t) - r) + /I&_) 5 0 a.a. t E (0, T).

A system of parabolic variational inequalities associated with a stochastic switching game 13

Hence we have uf: 5e-“” y 5 y. In the same way we can prove the - y ~5 US: part. n

LEMMVIA 3.2. For each E> 0, there exists a solution UC of (3.1) belonging to C’-“.‘-‘““(@. where O< 6< 1.

Proof. The assertion follows easily from

fp - ,d,(up, - Z.&I; - K) + &(u;I i - k - up,) E L ‘(Q)

by lemma 3.1. n

1. A PRIORI ESTIMATES

In this section we shall derive some a priori estimates. which are independent of E. on the solution u{, p = 1, . . ., m, of (3.1). Let the assumptions (A.l)-(A.3) be always satisfied. Write a/ax, = ai and use the summation convention for simplicity.

LEMMA 4.1. We have

Proof. This is nothing but lemma 3.1. n

LEMMA 4.2. (1) We have

I grad @Y, 4

(2) We have

/zM foryE&2,rE(O,T) and p =l,..., m.

,T) S-M for xER and p=l,..., m.

Proof. (1) Since aS2 is assumed to be smooth, the exterior sphere property holds, i.e. there exists a positive number p such that for each (y, s) E aS2 x (0, 7’) we can find ,’ E R6?zB satisfying

{zEIWN;/2-3idP}n~={y}.

Let p > 0 be a number to be determined later and consider

+, r) = e-rPz - er(x-112+if-si’).

Since a simple calculation shows that we can take p > 0 so large that the inequality

(-a, + Ap(t)) w(x, t) 2 A

holds for some A > 0 and for all p = 1, . . . , m, (x, t) E 0, we have

(-a,+Ap(t))(Mw)-AP(t)UfP on (i,p=l,...,m,

MwZO on X U(SJ x {T}), MNY, s) = 0,

for some constant M > 0.

(4.1)

Next, we show

i~~(~,t)-_~(~)/~~f~(x,r) for (x,r)EQ,p=l,.... m. (3.2)

In fact, to prove the part @(x, t) z -Mw(x, r) T ii”(x), suppose that the minimum of u$(x, r) + Mw(x, t) - a(x) with respect to p and (x, r) is attained at po. 1 Z:po Z m, (x0, to) E Q. If (xa, r~) E Z U (Q x IT}), then the assertion is obvious by (4.1) and the terminal and boundary conditions.

Consider the case (x0, to) E Q U (G x (0)) and suppose the contrary. Then, from the maximum principle, we get

0 2 ( -8, + AP”(ro)) ( up + Mw - P)(X~, fg). (4.3)

On the other hand, the right side of (4.3) is strictly positive by the definition of PO, the compatibility condition on tip and (4.1). This is a contradiction.

Repeating the same argument on u$(x, f) - rip(x) - Mw(x, f), we have (4.2). To conclude (l), it is sufficient to note that Igradu{(y , s) / = / (d/&z)u{(y, s) / where !I is the

outer normal vector to aR at y E S2. (2) For ,u > 0, consider the functions

9(x, t) = rip(x) T CL(T - r) and _wP(x, f) = P(x) - ,u( T - r).

Ifwechoose,u>Osolargethat/fP-AP(t)liP1<,~for(x,t)EQandp=l,. .,m,thenwe have, by the comparison theorem, K/(X, t) 5 uP(x, t) 5 G(x, t). Hence we get

which concludes (2). n

LEMMA 4.3. We have

Proof. Without loss of generality we may assume that the up belong to C3.‘(Q). Let ,u > 0 be a constant to be determined later and consider

(4.4)

Suppose that z, P attains its maximum with respect to p and (x, t) at p 0. 1 Z;O 5 m, (x0, to) E Q. If (~0, to) E C or to = T, then the assertion is already obtained in lemma 4.2. Consider the case (~0, to) E Q. In the remainder of the proof, we omit the subscripts ~0, E and write u = r.@+l, t = to.

Using the differentiated relations of both sides of (3.4) with respect to t and xi, we have

( - a, + A(r))2 5 - UZ - 2CXe’“(didfU)(did,t4) - 2ae”(diaju)(did,U)

+ 2e’~(a,Uij)(aiaju)(a,u) -2@(8tbi) (d,U) (a&)

- te”(d,c)(a,u)u + 2ePt(a,f)(a,u)

A system of parabolic variational inequalities associated with a stochastic switching game 45

+ 2eYa~U,j)(aia;K)(aSK) - 2eP'(aib,)(a,u)(a,u)

- 2ew(djc)(d:u)u + 2eYaJ)(d:K)

- 2ew(p'(K - u - K) f p’(u - k - K)){(d$ - d&>(~&>

+ (d;U - d;U)(d;K)}.

Estimating each term by using Schwartz’s inequality and the maximality of z{O(xa, lo), we have, for some constants Mi, lMz,

( - a, + A(t))z 5 (- ,u + M1)z + M~ew. (4.5)

On the other hand, since z$” attains its maximum at (.~a, to), we have, by the maximum principle, that the left side of (4.5) is nonnegative. Hence if we choose the constant ,K such that p > Mi, then we have the boundedness oft{. n

5. PASSAGE TO THE LIMIT

In this section we shall prove the existence of solutions of the bilinear problem which belong to W’.‘.-(Q).

THEOREM 5.1. Suppose (A.l)-(A.3). There exists a solution up, p = 1,. . . , m, of the bilinear problem each of which belongs to W’~‘~~(Q).

Proof. If we set GP(.x, r) = eco’uJ’(x, t), then the problem is equivalent to

2’ E W1.‘.-(Q),

( - Z(t), b - iCP(t)) + aqt; tip(t), i, - i(“(t))

+ (c$dP(t), U - P(r)> Z (ecdfP(t), i, - tip(t))

for a.a. t E (0, T) and all d such that eeCNd E 7~(e-C@tiP-1),

fip(p(x, t) = 0 if (x, t) E Z,

fip(x, T) = eCOrfi+) if x E S2,

for p = 1,. . . , m, where time1 = ic’.

(5.1)

Since the estimates given in the previous section are valid also for ti$! =ecd& we can find subsequences (denoted again by E) of Q and lip EW’.‘.~(Q), p = 1,. . , , m, such that for everyp = 1,. . ., m, ti$ converges to tip in the weak-star topology of W’.‘.=(Q). Moreover, by virtue of Sobolev imbedding theorem, we see that Lip EC(Q) and ti!j converges to rip in the strong topology of L”(Q).

Let t E (0, T) be such that (a/&)a E t”(S2) exists. Let u E X(U$*’ (~9) and fix it. By lemma 5.1 in [7] there exists u~E?X(K$+~) such that uE converges to u in the strong topology of H’(S2) as E+ 0. We put 6 = eco’v and d, = eQ’u,.

We shall now show that the 2, p = 1,. . . , m, satisfy (5.1). Since KS satisfies (3.1), ii{

36 N. YAMADA

satisfies

- 2 + AP(t)iC$ + c&I + e’@&(u$ + uj -’ - K) - ecti@Ju$ -’ - k - UC) = ecDIfP.

Multiply this by g(t) - CE and integrate over Q. Using the boundedness of u, in H&R) and a priori estimates for ti$ and noting the convergence of rt{ and u$-‘, vve obtain up([) E X( rP+‘(t)) for everyp = 1,. . . , m.

Next, consider the form

X!(f) = &(t; @(r) - LjE, @z(t) - 6;) + c&$(t) - fiE. G{(f) - a,)

+ eco’(P,(uC(t) - u{“(f) - K) - p& - @-l(t) - K), C{(f) - 6,)

- ecO’(P,(u{-i(t) - k - c&Z(t)) - &(K{-'(I) - k - uE). U:(f) - U,).

Noting that X$‘(r) 2 0, substituting (5.1) and passing to the limit as E--+ 0, we obtain

t

anp -- , i, - tip at >

+ aP(r; 0, I, - UP) + co@, V - tip) 2 eco’(fP, b - fip) (5.2)

for a.a. f E(0, T), 0 E X(U~-~(~)). Since u EX(&“i(t)) and BE (0, 1) imply 60 + (1 - 8)cP E YC(P’l), we may choose

6% + (1 - B)cP as u in (5.2). Dividing both sides of this inequality by 8 and taking the limit as 8+ 0, we obtain the inequality in (5.1).

It is clear that the 11P satisfy the other conditions in (5.1). n

6. W’,‘-REGULARITY WITH RESPECT TO SPACE VARIABLES

In this section we prove the W’,‘-regularity of the solution which was constructed in the previous section. By means of this regularity, we see that the solution given in theorem 5.1 is indeed a strong solution.

THEOREM 6.1. Assume (A.l)-(A.4). The solution ~8, p = 1,. . . , m, of (2.5) given in theorem 5.1 belongs to W2.‘(Q) for a.a. t E (0, T) and r, 1 $ r < 2.

In particular, lip, p = 1, . . . , m, is a solution of the strong problem.

Proof. Let 1 < r < = and to be such that (a/&)uP(ro) E LX(Q), p = 1. . . . . m. exist. First we note that for any x0 E R there exists p such that

up- ‘(x0, to) - k < uP(xo, ro) < up -‘(x0, to) + K. (6.1)

Indeed, if not so, we have for all p = 1,. . . , m,

uqxo, to) = llp-~(xo, to) _L Kp,

where KP = K or -k. Summing up these equations from 1 to m with respect to p, we get Z:,“=l~P =O. But this contradicts the assumption (A.4). and thus we have (6.1).

Changing the number p if necessary and considering the continuity of the solution, we may assume that

r?(x, to) - k < u’(x, to) < u’(x, to) + K (6.2)

A system of parabolic variational inequalities associated with a stochastic switching game 17

in some neighborhood Ga ={x E Q; /x - x0/ < 6). Since u1 does not touch the obstacle U? - k or u2 + X, we have ui(ra) E W’,‘(Gs).

Next. we show ~~(t,j) E W’, r(Gb). By the uniform convergence of u{ to up, we may suppose

uf(x,t)-kkuu:(x,t)<rt’Xx,t)+K in Gbx(rI,t~

for sufficiently small E and some interval (ti, tr) 3 fo. For such u{, we have

- % + A’(t)ud = f’ in Ga x (ti> rz).

Hence A’(f)uf. and 11; are bounded in L’(G6) and W’.‘(G,), respectively, for a.a. f. Put &(fo) = uf(ro) -k, cP,(fo) = ua(ro) + K and note that Am”&, Am(fg)QE(h) are also

bounded in L’(Ga). It is sufficient to show that uy(to) are bounded in W&J(Ga). Choose [E C=(Q) such that supp CC Gg, 510, and fix it. Multiplying both sides of (3.1)

for p = m by c’p:- ‘(~4” -DE) and integrating over S2, we have

- I o$; (x, to)Ux)P:-‘@Xx, co) - @4x, to)) dx

+ I

@‘VO)U:(X, to) Y(x)/%-‘(4x, lo) - 04x7 lo)) dx R

Note that the fourth term on the left-hand side is equal to zero. Writing the second term on the left as

am(to; u’: - @E, ~‘K’(u~ - @A) + I

(A”(~o)%)U%-$4’ - ac) dx, R

we can estimate each term as follows:

am(fo; IJJ, ~‘P:-‘(~JJ)) = r ,/ fi:-‘r-‘a~(ai~>(~,,‘> dx n

+ R (P:-2P:c){(r - l)a~(ai~)(aj~) + &“(a,r&& + c”‘$} dx I

- R P:-?‘(P;v - PJ(@“(Q4 + c”~) dx I

2 - M R c”-‘Igrad [Ifl:-‘dx - r~ oP:(v)f’dx - M I I

(6.3)

for any r] > 0. where y(ro) = $‘(t,) -aE(rO),

i R (A”‘(ro>@,),“‘/3:-‘(u’: - OJ d.r 2 - rj n r;/3:(u’: - QE) dx - .Ll j ;‘IAm(ro)OE r d.u.

i R

Noting that (d/df)(u?(ta)) EL=(Q) in the first term on the left-hand side of (6.3) and combining above estimates we get

J ;r/3:(u:(ro) - Q’,(Q) dx 5 M. R

By a similar argument we get

These show the boundedness of uT(to) in &,(Ga). Passing to the limit as E+ 0 and repeating this argument inductively with respect to p, we can show that for any x0 E Q, there exists 6 > 0 such that

KP(t”) E W,?d,l(Gn)

forallp=l,..., m,andr, l<r<x.

Considering at each point x0 E R and noting the boundedness of Q, vve conclude that

rlP(t,j) E W;:(Q) for all p = 1, , m, and r, 1 < r < x. (6.4)

On the other hand, since UP does not touch obstacles at dQ x (0, T). UP must have W’,‘- regularity in some neighborhood of it. From this and (6.4) we obtain that UP E W2J(Q). n

7. STOCHASTIC REPRESENTATION AND UNIQUENESS OF THE SOLLTION

In this section we shall introduce a stochastic switching game and represent the solution component Kp of our problem as the value of this game. As a result of such representation, we shall obtain the uniqueness of solutions of the strong problem.

Let (S?, 5, P) be a probability space and let w(f) be an N-dimensional Brownian motion. We denote by S, the u-field a(w(s), 0 5 s 5 t) in % which is generated by {w(s); 0 % s 5 r}.

Let aP =[o$(x, t)] be a nonnegative matrix satisfying up = (1/2)aP(oP)* for each p = 1 m, where (of’)’ is the transposed matrix of up and ap =[a$(~, t)] is the coefficient mat;;; of the principal part of AP(t). Let bP =(bf(x, t), . . . , &(x, t)) be an N-vector of coefficients of the first order terms of Ap(f).

We may assume that aP and bP are extended to the whole W” x [0, T) preserving Lipschitz continuity.

Consider a system of stochastic differential equations

dEP(t) = - bP(tP(t), t) dr + op(~P(t), t) dw(t), p = 1,. . . , m. (7.1)

In the following, we only construct a stochastic switching game associated with the solution component ui for the sake of simplicity.

Choose (x, t) EW” x (0, r) and fix it. Let q= (r]i. ~2, . . .) be a sequence of 3,stopping

.J, system of parabolic variational inequalities associated with a stochastic switching game 49

times satisfying

and for a.s. o E 6. there exists n( IX) such that q,,(,,)( w) 2 T. (7.2)

By making use of solutions of (7. l), we define a continuous process Z(S) = &. r,l(s) as follows: foranyintegerI2Oandp=l,....m,

&.r.l(s) = Zp(s) with. ~P(q~m_p_l) = fp-‘(q~,,,-p-l) if ~l~,+~_~ 5 s 5 v/~,_~. (7.3)

where we put go = em 5 ,rlo=r. We use the notation

f(;L%X.I(4) =P(z%)) and c( E,,.,. @>) = cp( Ep(s)) (7.4)

when &.,Y.I(s) =Ep(s). Let 7 be the exit time of th_e process &.,.( from the domain Q. N_ote thal there exists almost surely the number q(T), 1 S q(T) S m, such that $,,..,.,(T) = p(T).

For two sequences 0 = (O,,), t = (G) of stopping times satisfying (7.2). we consider the following cost functional:

J.i(r; 0, r) = E, [lrA’exp (- [‘c(E(r),r) dr)f(E(s),s) ds

+ @“(f(T)) exp (- j’c(E(r), r) dr)X(rZ T)]. I

where ‘I = (q,J is defined by 7, = 8, A rn, E(s) = 5,rl..~,,(s) and x(A) is the defining function ot a set A.

We are interested in the value of the stochastic switching game:

sup inf Ji(r; 8, r) or inf sup J,t(t; 8, r) e T T e

where 8, t range over the set of all sequences of stopping times satisfying (7.2). Let up E W’.‘,r(Q) n C(Q), p = 1,. . . , m, be a solution of the strong problem. Let, for each p = 1, . . . , m,

9(s) = {x E !5; uqx, s) = up+ ‘(x, s) - k}, (7.6) P(s) = {x E n; UP(X, S) = r&+(X, S) + K}

and define sequences 6 = (g,,) and 5 = (a) of stoppi .:g times associated with 9(s) and ?(p(s). p=l,..., m, respectively, as follows:

6,m+p = inf{s 2% BI,~,- 1; Ep(s) E Sp(s), fP(81,LP_ t) = ~P-‘(&m+p_ r)}. (7.7)

$m+p = inf{s Z Z,m_p_l; tp(.s) E Tp(s), Ep(&_p_l) = ~P-‘(Zlm+p-l)}. (7.8)

50 N. YAMADA

The next theorem asserts that any solution component u1 E W’.‘.‘(Q) of the strong problem can be represented as the value of our stochastic switching game and that the sequences 8, t of stopping times constructed above are a saddle point of this game.

THEOREM 7.1. Assume (A. l)-(A.4). For any solution up E W’.‘.‘(Q), p = 1, . . . , m, of (2.6), we have

Ui(X, t) = Jj(r; e, ?)

J,i(r; 8, f) 5 U’(X, r) 5 J.J(t; e, r)

for all sequences 8, r of stopping times satisfying (7.2).

(7.9)

(7.10)

Proof. Let (x, t) E Q. First we note that fi = (&) and ?= (-tJ defined in (7.7) and (7.8), respectively, satisfy the condition (7.2). Indeed, if 4, < Tfor all n for some event w E h, then 4, must converge to some 6, 5 T. By the continuity of the path j-(3) and 1~~. there exist the limits E(&) = lim,, -E( &) E 0 and uP(_t(&), 8,) = lim,, &(c( &), 0,) for each p = 1 , . . . > m. Since up(;‘(&), 6,) =up”(~(&), &) - k for n = lm + p, we get

r/(&k), 8,) = ~P+i(@z), 6,) -k

forallp=l,..., m. but this is impossible. In the following we assume C+‘(X) = c (constant) for simplicity. Write the cost functional as a form

+ K e-C(“~m+p-i)x{qj,,,tp A TA 7 = tlm +p} (7.11)

- ke-c(~h*p-‘)x{q-~m+p A TA T = &,,+,}]

+ E,[dn({(T)) e-C(‘-r)x{TZ T}].

We shall show (7.9). Let 4 = dr\ ?. Since &) E 9(s) u fJ’(s) whenever @,,Tp_, <s < QI,,,+~, we have

- $ (cp(s), s) + A~(s)u”($~(s), s) = fP(;P(x), s) (7.12)

in this interval. Applying Ito’s formula to (7.11) we get

Jj(t; 4, 9 = &[ut(&fi,,/\ TA T), &A TAT)]

+ ,iOpil .C[{- u”(~~(%,+~A TA T), lj/m+p” TAT)

+ uP+‘(~P+‘(?~~~_~ A TA T), Ij,,,,+p A TA q

+ ~xhn+p A TAT= $,+,} - k,y{ijln+pA TAT= 8,m+p}} e-C(ih-pATA’-‘)]

+ E,[icq(n(ij(T)) e-c(r-‘)x{T2 T}] (7.13)

A system of parabolic variational inequalities associated with a stochastic switching game 51

where rjO = t. To conclude (7.9), it is sufficient to show that each

u(l,p; 6, f) = -u”($%m& TA T), Ij,m+pA TA 7)

+ u~+~(~~+'(~~~+~~TTA~),~~,~,_~ATAT)

+ KXvllm+p ATAT= %n,,}- ik~{fj~~_~A TAT= &m+p} (7.14)

is equal to zero. This is shown by considering each of the possible cases of ?j determined by ?jrm_p A T A\T= t&, t,,,,ip, T and 7:. For example, consider the case fi,,,, +,, /r~ T A 7 = Zimrp < 61~+~. Since

5P(Z/m-p) = s’P*‘(Er,,,+p) E T ( -p L-p)r

we see

- ~~(~~(~,,,z+~~ TA T), 4~~~ TA n + ~P-'(5Fp~~(4,~i~/\TA'~),4~,i,r\T/\i)+ K=O

and

x{fpm-$ TAT= &,,,,,} = 0.

Thus we obtain U(l, p; 0, 2) = 0. The considerations of other cases are omitted. Since it is quite similar to show (7.10), the proof is also omitted. n

Since this representation implies uniqueness of solutions, we may state our main result in the following theorem.

THEOREM 7.2. Assume (A.l)-(A.4). There exists a unique solution of the strong problem each of which belongs to

w?&(Q) ” W’.‘.“(Q)

foranyr; lIr<x.

REFERENCES 1. BENSOUSSAN A. & LIO?;S J. L., Applications des InCquations Variationnelles en ContrBle Stochastique, Dunod,

Paris (1978). 2. BEMOUSSAN A., Stochastic Control by Functional Analysis Method, North-Holland, Amsterdam (1981). 3. EVANS L. C. & FRIEDMAN A., Optimal stochastic switching and the Dirichlet problem for the Bellman equation,

Trans. Am. math. Sot. 253, 365-389 (1979). 4. EVANS L. C. & LENHART S., The parabolic Bellman equation, Nonlinear Analysis 5, 765-773 (1981). 5. Lross P. L., RCsolution analytique des problemes de Bellman-Dirichlet, Acta Math. 146, 151-166 (1981). 6. LIONS P. L., Le probleme de Cauchy pour les Cquations de Hamilton-Jacobi-Bellman, Ann. Fat. Sci. Toulouse

3, 59-68 (1981). 7. YAMADA N., A system of elliptic variational inequalities associated with a stochastic switching game. Hiroshima

Math. J. 13, 109-132 (1983).