View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Stochastic Processes and their Applications North-Holland 18 (1984) 67 67-79 A MULTIPARAMETER STOCHASTIC INTEGRAL AND FORWARD EQUATIONS zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED E.M. CABAr;JA zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Department of M athematics, Unicersidad Simrin Bolivar, Apartado Postal No. 80654. Caracas, Venezuela Received 4 March 1983 Revised 27 October 1983 Integrals of the form 5 f(s) d Y (I, wl xl) are defined for nonanticipating processes f with respect to the composition of regular functions p and a Wiener process with parameter in (R’ )‘I. Using such integrals, a forward diferential equation (23) is established for the density CJ a Wiener process killed when it reaches one or two constant barriers. stochastic integrals * multiparameter Wiener process * diffusion equariony zyxwvutsrqponmlkjihgfedcbaZY 1. Introduction It has been shown in [t] that the probability density p(x, y, z) at the point z, of a two-parameter Wiener process w(x, y) (x, y E Iw+) killed when it reaches 3 constant barrier u (>O) (this statement will be made precise in paragraph 4) satisfies the forward inequality (I) which reduces to zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA D*tD*pa 0 with t = xy and D” = d/dt- i(d”/dr*), because p(x, y, z) only depends on x. y through t!le product t = xy. The proof of (1) given in [l] relies upon a di-ect computation of expectations of ‘1 sum zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA of double increments of an arbitrary function of X, y and ~v(x, y). We wish 1.0 present here a simpler proof based on the ur.e of a particular type of stochasfic integral developed in Section 3. This gives d probabilistic interpretation to the nonnegative left hand side of (l), related to the expectation of one of such integrals ‘see (22) and (25)). The present proof generalizes a procedure that establishes the well-known forward equation for one-dimensional-parameter Wiener process, and provides a forward equation (23) for any value of the dimension of the parameter space d. 03t)4-4149/Y4/$3.00 @ 1984, Elsevier Science Publishers B.V. (North-Holland) E. M. Cuba& / Multiparameter zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB forward equations 68 When d = 1, the well known It6 Formula for stochastic differentation provides a simple way of defining the integral of a nonanticipating process f with respect to the composition of a sufficient1.y regular function !P( f, Z) (t E R+, z E R) and a standard Wiener process w: I (;Rr) d’J+-, W(T)), by means of the rule dY(. a?P W(T)) = $2, ,a2Y ( 7, w dT r+))+zaz’ +$T.W(T)) dw(7) (2) which leads to I ,: f(r) d ‘U 7, W(T)) = w(r)) d7 (3) with lf E I:, f’( 7) dr < COand, for simplicity, the derivatives of (I, are bounded, then the stochastic integral I,!,f( ~)(;tV/ar)( T, W(T)) dw( 7) has zero expectation and the formula E f(r) d’P(TT w( 7)) = E ‘f(7)DP(7, I0 w(T)) dr (5) holds. This formula may be used as a tool to derive the forward diffusion equation for the transition density of the Wiener process with barriers. We shall notice here for further reference that (3) and the inequality (A + B)’ s LA’+ 2 R’ also imply Instead of using the It& Formula, an alternative jfc 7)dw7,W(T)), is to extend the rule procedure to define the integral zyxwvutsrqpon EM. Cabaria / Multiparameterforward equations 69 valid for the indicator function lta,bj of any interval [a, b), by using additivity and linearity. When the dimension of the parameter space is d> 1, the extension of the It6 Formula becomes complicated (see [3] for instance). Nevertheless, the analogue of (5) which only involves the expectations but does not require an ex.plicit description of the integrals (as the one given by Wong and Zakai in [3] by means of different kinds of multivariate stochastic integrals) can still be obtained by the alternative procedure. We describe it in the following and establish the forward equation for the transition density of a d-parameter Wiener process, killed by one or two barriers. 2. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Multi-dimensional-parameter Wiener process and multiple increments The aim of this section is to introduce some notation, and to establish (9) and (12) in preparation for development of the stochastic integral in Section 3. The partial orderings < and Q are introduced on !@ (or (@)d) by (x1,. . . , xd) < W)(Yl, - - . 7 yd) iff Xi-=I(s)Yj for j = 1,2, . . . , d, and notation such as (a, b]={x~R~: u<xs 6) will be used for generalized intervals. Given any function g : D(c W”) + R and any interval ments of g on the interval are defined and denoted by (i=i,z,...,d, (x - 6, x], the partial incre- X=(Xl,...,Xd), 8=(61,...,&)j, and the multiple increment by provided all points where g has to be evaluated belong to D. A process w : (lR’)d X f1+ IR on a probability space (0, & P) is a Wiener process, when the properties (i) {U(.x,,Iw:04 xs y} is a Gaussian family, (ii) JXI,,,,~W = 0, E0i.x ,~~wU(,~,,~~w = A((.& y]n (x’, y’l), (iii) w(x) =0 when []yz,‘xj=O, are satisfied, A being the Lebesgue measure on R”. (When (i) to (iii) hold fcx a different measure A, w is said to be a A-Wiener process.) Given a C”: function ~:(R+)“xR4Fq(x,z)=(x,,. . . ,X&++P(X,2)), let us consider the composition To w(x) = F( x, w( x j) as a function on (Rs )” to the space of random variables. In order to simplify the notation, all increments below will be computed on intervals contained in zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJI [0,1Id.Moreover, it will be assumed for E. M. Cabaiia / Multiparameter forward equations 70 simplicity that all derivatives of W are bounded. This avoids technical complications and implies no restriction in the case that P has compact support, as in Section 4. Since th:: process wi defined by 4(y).= W(X,, . . . , -+I, y, xj+l, * . * 7 4 is a (one-dimensional-parameter) &zjx,r-Wiener process, that is, @‘,= MYI’, is a standard Wiener process, and ,the partial increment l~x_-b,xl!P’o w (HMj XII)-“? is the increment of ‘Y(x,, . . . , +_I, y, xi.+1,zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK . . . , &, w’,(y )) on xj - 8j < y s xj, w e may write I:x_-bi,xl Wow in the form j;~_, d@x(y, G’,(y)), where l/2 ~~(Y,z)=y(x,,...,xj-,,Y,Xi+,,...,xd) ( > fl Xh 2. zyxwvutsrqponmlkjihgfedcbaZ h#j As a consequence, (5) becomes (7) with and The subindex (x - 6, x] will be omitted below, to abbreviate notation. Let us apply now (7) to Z”V instead of 9, with k # j. Since Lj depends on x (more precisely, on the coordinates of x other than the j-th one., hence in particular. on xk ). (7) turns into i EPt”‘b w= E I I”LjP’ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA W ivherc it will be noticed that I’. operates after Lj in the right-hand member. Using (7) again. E. M . Cabah 1 M ultiparameter forward equations 71 By using the same substitution that was used to obtain (7) from (5), the foliowing bound for the second order moment of I’Pa w can be obtained from (6): because x is assumed to be in [0, lld and hence n,,, j xh d 1. Again, with k f j and Z”(Po w) instead of !Po w in (ll), we get zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK E(lilk’h wJ2a2 and since then zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA E(ljZk!Po W)‘a 2’E [jl” ((Lk+;)(L,+;)Pow)’ and the generalization E(Oww)2sE with 5 (M!Pow)* (12) (13) follows at once. 3. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Stochastic integral with respect to *o w. L.et us consider a given family (&X),rt(R+jl’of a-fields satkfying (u,,) !!or any x (1(R+)d, ~4~c Sp, (a,) x s y implies ,d, c ,tiY (_u,y f (lR+)”1, (x,,...&)h(v (&)=([W+)“, (k) ‘%,?.=~.~n~~ ,,..., y,,)=(x,f\y,, .... xd * yd)) for each XE (R’)d, {w(y): ye x} are &,-measurable and {Cl,c,,~,~w: a K x}, are independent of d.X. An example of such a family is obtained by choosing tiI as the a-field generated by {w(y): y c x}, but the a-fields in the family could be richer than those particular ones. (a-,) E. M . Caban’a / M ultiparameter forward equations 72 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA A zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA prCJCeSS f : (R’)’ X a--, W is said to be nonanticipating, when f(x) is tiX-measurable for each X. Given a C“ function P: (R’)d X IR + R and a nonanticipating process zyxwvutsrqponmlkjihgfed f such that E J f’(5) [O.l 1” dT--‘, we shall proceed to define the integral ,I ~;I(Od’J’(& w(O)= ‘f d*ow I0 forOSxS(l,l,.. .,l). This restriction of the domain of the integral to [0, lid is arbitrary and unimportant. The construction in any other bounded interval is the same. As a first step, we define the integrals of indicator functions of intervals in the obvious way, that is, if A = (a, h] is an interval and Ii, Cl ;I denote [nrilO.rl, •Ar,[o,x~, respectively, then After this, the definition is extended to sirnpk nonanticipating processes as follows. For each n = 1, 2,. . . , we consider the partition of (R+)” into cubes 9, = { ( h - S,,, h]: 2”h E (E+)“}, a,, = (2_.“, 2 -)I, . . . , :I-“), WV and define / to be a sir;@ pruccss, when there exits n such that a representation f = 2: ‘“*fA 1,.\ ,I. .n,, (15) holds. with suitab!e random variables {f/\:AzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH E S,,}. When f is given by (15), we define and it is clear that different representations of the s;lme f lead to the same result in (16,). As a second step, we shall state a theorem which summarizes some properties of the integral, and prove it first for simple J zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH Theorem 1. 71x3 integral of a ncvlanticipating pm-em f such that 73 E. M . Caban’a / M ultiparameter fo:- ward equations (iv) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA there exist constants C*, & such that, for every f, E ([“r d~-&G[~fl~2, / IJO E(fW2~ &llf 112. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Remark. The condition !Pf C” is unnecessarily strong, but adequate for our purposes. It could be replaced by a weaker one, namely, that the derivatives of !P appearing above exist and are bounded, with no change in the proof. Proof of Theorem 1 for simple fi The well-known a.s. continuity of w readily implies the a-s. continuity of {i f d!P 0 w as a function of x. The equality (ii) follows by computing expectations in ( 16) by means of E(f,O:;~ow)=E(f,E(OXAly”w/~~)) with ti~to,hl= s&, and using property (a3) of d. and E(Cl”,?Pow/.dJ=E XLP~w/&, (I A (9) to derive . zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON > This same property implies that if An(O,x]=kl, ITIfLW0 w/ dX) = 0, so that y 2 x implies the martingale N(YW d then equality E(l%fdVow- =h) which proves (iii). In order to obtain the first estimate in (iv), let us write E(s,:f d@w)’ s 2E(f(x))‘+2E (S,:fL’row)2 and notice that since the derivatives of P are assumed to be bounded and hence E( 5,: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA f L!Pow)‘SconstantX jIf[l’, it only remains to establish the second estimate. To do that, let us write z f /\( Cl” (.,.,. A‘I’ ~~-[/~L~cJq E(f(x))‘=E I,+$, = J,~~~~w)(~w- zyxwvutsrqponmlkjihgfedcbaZYXWVU and compute first the expectation of terms with A = (a’, a”] # f3 = (h’, b”]. In this case, either a’K b’ or h’ < a’; if b’ $4’ then the expectation can be computed as Il/ow- J;~Ylow)+ow- J;Lqow,.$,,)) E.M . Caban’a / M ultiparameferforward equations 74 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA where iju is the a-field generated by (A$: h’ K y}, and the conditional expectation vanishes because of (a,). The same happens when a’ K b’. The sum reduces then to the terms with B = A, and the use of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF (a,), (12), and the inequality Var(X/ &,,) d E(X21 ~4~) applied to X = Cl 2 !Po w, lead to fkEc(u~~ow)2/~~)~~nCyl SEATfi t =E fX f. n ” x(A4!P~w)2 J /I zyxwvutsrqponmlkjihgfedcbaZYXWVUT ~ f’(iMI~ w~ “~ constantxII/I~ ’ J zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ends the proof of the theorem. Our third and final step is to define the integral for general nonanticipating f with finite f.’ norm (17). Such a process can be approximated in the norm as close as required by simple nonanticipating ones, so that, given an arbitrary sequence (a,) of positive numbers, there exists a sequence f,, of simple nonanticipating processes such that whic h Definition. The sequence f,, of simple rrorzanticipatingprocesses is said to converge rapidly lo f (11f I(( CC), when Ren;drk. The previous observation shows that there exist sequences of simple nonanticipating processes that converge rapidly to a given nonanticipating process f of bounded norm, by taking C arf,” < r in ( 18). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON T he ore m 2 . If f,, is a sequence of simple nonatlticipating processes c’ot~ccrging rapidly m the notlutttic.i~atinl:process f, 11 f jjK w. then the sequerzce t 20) s E [Cl.11” with probability otle wherr ti tetids to iti,fitiity. De finit ion. 7 Y lciritcgrul I,‘,f d P 0 w is the limit of (2 0 ) w 1 ie ntl goes to ittfitlity, whets the cr.wiitnptim~0.f Theorem 2 hold. E. M . Cabaria / M ultiparameter forward equations 75 (If f!,, f:’ converge rapidly to f, then f:: f:‘, f:, fi’, . . . also converges rapidly to the same limit, hence the integral is uniquely defined). zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP Proof of Theorem 2. We shall show that the series with general term ji (f,, f,,_,) d??o w converges absolutely, and that the convergence is uniform in x. Let us consider first the series 1 ji (f” -f,_l)Llyo w. Since where Kg, is a bound of If.?Po WI,the series in the left-hand member of (21) converges when 181I[O,,,dIf,, -fl does, and by Schwarz’s Inequality, this one converges when C,, l,o,,,ff ( fn -f)” < CO, that is, almost surely (because 1 /If,, -fII”* < 00 implies c zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA II~~-fII” =E~.5(f,,-f)‘ <~). According to this, to show that I:__, Id (fn -fn-_,) d!Po w converges uniformly, it is enough to show that the series P ‘(J&,)dPow’ (frl-fn_,) LPow = f &xl-a,..,) Il=l > * LT, c (5 0 I0 converges uniformly in x, where f, is the martingale associated to f,, as f is associated to f in Theorem 1 (iii). The Cairoli inquality [2]: AP{suplX( 3 A}< zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA A,, sup E[IX/(log+lXI)“--‘I+ & holds for martingales X wth continuous paths, where the sup is over the parameter set [Cl, 11” of X, log’ x = (log x) v 0, and zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG Ad, ,Bd are constants. Hence, if u is any positive number, and Cc1= max,,. , t --“? log’ z)~‘--‘,we have *p(sup/XI at} s A,, sup E((u-~“IXI”“)C~, + B,, d A,,C<i~-“‘2(~~pEIX(2)3hzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF B,,. This implies that, given any /3 > 0, P{suplXI 3 p} c p f K&‘(sup ElXIZ)3’J, with KJ = A,,C,,V’B,,. In particular, with f,, --x1..-, in the place of X, and B~~,,~~~~fi~-f~l~+~If,~-~-f~l’~‘~’~l~fi,-fl/’~”+llf,~-~-fil’~~. we have E.M. Caban’a / Multiparameterforward equations 76 (use Theorem I (iv)). Using now 11fn - fn-l It*&, we get 4supIJ,,-f,_,l~p”}~(1+23’4Kd~~J)p,l, and the a.s. uniform convergence of C Ifn(x) -fn_,(x)I follows from (19) and the Bore1 Cantelli Lemma. This ends the proof of Theorem 2. zyxwvutsrqponmlkjihgfedcbaZYXWVUTS Sketch of the Proof of Theorem 1, for general 6 The uniform convergence of the approximating integrals of simple processes I,”fn dtY0 w to 5,”f dW0 w, and the continuity of the integral for simple f,,, imply (i). From Fatou’s Lemma, we have which proves the first inequality in (iv). The other one is similar. Therefore we may write The proof of (iii) follows in the same way. and this implies (ii). 4. Forward equation for Wiener process Given a d-parameter Wiener process w, let us introduce the related processes of maxima and minima and consider the probability distribution of w killed when it reaches the level u < 0 or b > 0 \vhose density is denoted P(X, zt by =+x. z z). The constants a or h can possibly be chosen of one-sided barrier t. oblems. To each C ’ functiorl equal to --cc” or +IX: to cover the case u/ : (Et+)” x R + R with compact support contained in (0, 1)” X ;1~3sociate the expectation of the integral over [O, 11” with respect to d’I’-’ ~t’(.r 1 of the indicator function f of the random set 111. bi. ivc E. M . Caban’a / M ultiparameter forward equations thus obtaining a linear mapping (l....,l) h:‘P- +E fdW w. I 0zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA By Theorem 1 (ii) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA and the boundedness of the derivatives of P, h is a continuous linear functional of !P, and using the notations of distribution theory, we may write (l,....l) E I fdPow= h(x, z)W (x, lr e 0 z) dx dz. (22) (O.l)dx(a.h) Let us finall;? observe that Theorem h(x,y)!P(x.z)dxdz=E 1 (ii) and equation (22) lead to the identity j+fL’P= Kl,l)“x(a,h) j- j- p(x,z)L!P(x,z)dxdz (C,ljdx(n.b) valid for arbitrary ?P,so that, integrating by parts in the right-hand ter-p, considering p as a distribution, the forward equation L”p= h (23) follows, with This is all WCcan say in general, but the cases d d 2 still deserve some comments. For d = 1 and d = 2, h reduces respectively to zero and to a nonnegative measure. in fact, when d = 1, I,‘f d’P~w=[~jr*’ d?Pow= ‘P(T/\ 1, W(TA I))--‘P(O,O), where T = inf{x: w(z) = Q or b}. The assumption that q has a compact support in (0, 1) X (a, b) implies zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA P(0, 0)zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA = W( 1, w( 1)) = 0, and, since w( T) = a or h, also implies V( T, w ( T)) = 0. This proves h = 0 for d = 1, As for the case d = 2, let us call zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC f,, the simple process defined by (15) with ([x’, f[x’..r,‘, = 1 (M (x’).~~h.,?l(x’)- - .cr~ and V,, the random set where x”) E % ,>, 1, hence V,, 1 V for each H, and n,, V,, is the closure of V. For each x E V (or V,) the interval [0, x] lies in V (or V,, j, anu consequently the intersection C (or C,,) of the border aV (or dV,) with (0, !)’ is an arc of curve (xl(t), x2(t)) (respectively (x\“‘(t), x$“‘(t))), parametrized by t. Without loss of generality, we may take xl and xi’*)to be nondecreasing and x2 and x 2” to be nonincreasing. Since V,, is a union of squares, C,, has alternatively horizontal and vertical sides (see Figure 1); the sets of vertices where a horizontal side ends will be called Vz, and the vertices corresponding to the transition from fn = 78 E. M. Cabaria / Multiparameter forward equations vertical to horizontal sides, constitute (I.11 J dY0 w= f,; 0 It Vi. It is easy to zyxwvutsrqponmlkjihgfedcbaZYXWVUT verifythat cv:, w& ww- cv,. w, w(x)). (24) xi The curve C may also have horizontal sides followed by vertical ones, intersecting in points that constitute the random set that we shall call zyxwvutsrqponmlkjihgfedcbaZYXWVU V+. (See Figure 2). When .Kis the starting point of an horizontal side, the ending point of a vertical one, or a point where both xl and x2 increase, then we must have w(x) = a or w(x) = b. For each w, the continuity of w implies that the level sets {w(x) = a} and {w(x) = b} are surr’ounded by neighbourhoods in which !P(x, w(x)) vanishes. Therefore, when M goes to infinity the points in Vi will be eventually absorbed by those neighbourhoods, so that !i_z 1 *y(x, w(x)) = 0. *1 \,,” In the same way, each point in V’ will be approached by one in Vz, and the remaining points in Vi will be absorbed by the same neighbourhoods, leading to 2: Wx, w(x))= lim !I + x t ( ,’ ;, C Y(x, W(X)). .xt v‘ From (24) and the definition of the integral, we obtain the following representation ll.ll I 2: f dYow= r, II W(x. w(x)), \” which gives another way of writing h(x, r)Y(x, h: z) dx dz = E C P(x, w(x)). Y6\‘+ (25) :tt.1 t‘ *lcr.h! 1 x2 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Fig , I. +: po ints in V,:; -: po ints in V,;. EM . Cahan’a / M ultiparameterforward equations 79 Fig. 2. +: points in V’. shows in particular that h is nonnegative, and this implies that it is the formal density of a positive measure. When d = 1, (23) reduces to the very well-known fact that zyxwvutsrqponmlkjihgfedcbaZYXWVUTS p satisfies the heat equation, and when d = 2, we obtain the inequality I.*p a 0. This last inequality gives equation (l), which agrees with the result in Theorem 2 (ii) of [l], when the fact that p depends on (x,, x2) only through the product xl x,? is taken into account. When d 3 3, it is no longer true that the integral I:I”“‘*”f d!Po w can be written as a sum of values of ?P(x, w(x)) all with the same sign, extended to a suitable set of values of x. The following trivial example, though unlikely, illustrates the sort of difficulties that can occur. Let d = 3, 0 < (x,, x2, x3) -=c( y,, y,, y3) < ( 1, 1, f 1, ..nd suppose that the only roots of w(x)=a or w(x)= b in zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP [0, zyxwvutsrqponmlkjihgfedcbaZYXW 13” are (x,, y,, y,), (y,, x7, YJ and (yr. ~5x4. Then Equation (25) (1.1.1) J0 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA f d!Pow= ~‘OM’(y,,X~,X~)+ly~~~(x,.yz,x~)+~~w(~,,-~7,yj) - !ifoWl, y2, yJ. The appearance of contributions with different signs persists in more realistic situations and constitutes E difficulty in dealing with the case ti 3 3. zyxwvutsrqponmlkjihgfedcbaZ References EM. Cabafia and M. Wschehor, The two-parameter Brownian bridge: Kolmogorov inequalities and upper and lower bounds for the distribution uf the maximum, Ann. Probab. IO (1982) %0--3()?. ‘;d;c.~ multiples et scs applications, S6minaire dc R. Cairoli, Une inegalit6 pour martingales i I, probabilitCs IV. Universitd de Strasbnurg (Springer. Berlin, 137iij pi;” !--27. E. Wang and M. Zakai. Martingales and stochastic integrals for processes with a multidlmensi~m:~l parameter. Z. Wahrsch. Yerw. Geb. 29 (1974) 1CY4-12L.