IS

_{,IQ}

### UNIVERSITAT DE

### BARCELONA

AN EXTENSION OF

### ITÓ’S

FORMULA FORANTICIPATING PROCESSES

by

Elisa Alós and David Nualart

AMS

_{Subject Classification:}

_{60H05, 60H07}

### An extensión of Itó’s

### formula for

_{anticipating}

### processes

Elisa Alós and David Nualart 1
Facultat de _{Matemátiques}

Universitat de Barcelona
Gran Via 585
08007 _{Barcelona,}

_{Spain}

Abstract

In thispaper weintroduceaclassofsquareintegrableprocesses,

de-noted_{by}_{LF,}_{defined}_{in}_{the canonical probability}_{space}_{of}_{the}_{Brownian}

motion, which contains both the adapted proceses and the processes in the Sobolev space L2,2. The processes in the class LF verify that

for _{any} _{time} _{t,} _{they} _{are} _{twice} _{weakly differentiable} _{in} _{the}_{sense} _{of the}
stochastic calculus of variations in _{points} _{(r,}_{s)} _{such} _{that} _{r} _{V} _{s} _{> t.}
On the other _{hand,} processes belonging to the class LF are

Skoro-hod integrable, and the indefiniteSkorohod integral verifies properties

similar tothose of the Itó_{integral. In} _{particular}we prove a

change-of-variable formula that extends the classical Itó formula. Those results
are _{generalization of similar} _{properties} _{proved in} [7] _{for} _{processes} _{in}

L2'2.

^upported by the DGICYT grant number PB93-0052

1

> to

O

1 Introduction

A stochastic _{integral for} _{processes} _{which} _{are} _{not} _{necessarily adapted} _{to} _{the}
Brownianmotionwasintroduced_{by Skorohod}_{in} _{[10]. The Skorohod}_{integral}

turns out to be a_{generalization of the classical Itó integral. In} _{[3] Gaveau}

and Trauber _{proved} _{that the Skorohod integral coincides with the adjoint}
of the derivative_{operator}_{on} _{the Wiener} _{space.} _{Starting} _{from this fact,} _{one}

can use the _{techniques of the stochastic calculus} _{of variations, introduced}

by Malliavinin [5], inorder to study the Skorohodintegral. More precisely,

the Sobolev _{space}

_{L1,2}

_{is}

_{included}

_{into}

_{the}

_{domain of the Skorohod integral,}and in this

_{space}

_{the}

_{Skorohod integral verifies}

_{some}

_{of}

_{the usual}

_{properties}of the classical Itó

_{integral like the quadratic}

_{variation property}

_{and the}local

_{property.}

_{Results of this}

_{type}

_{were}

_{proved by Nualart and Pardoux}

in _{[7], where} _{a} _{change-of-variable} _{formula} _{was} _{established for the Skorohod}

integral ofprocesses in the space

### L2,2

(twice weakly differentiable). So, weknow that a stochastic calculus can be _{developed for} _{processes u} _{in} _{the}

Brownian filtration which _{belong} _{to} _{one} _{of the} _{following} _{classes:}

(i) The class ofadapted processes such that

### /J

### v%dt

< oc a.s.(ii) The class ofprocesses that are locally in the Sobolev space

### L2,2.

The _{purpose} _{of this} _{paper} _{is} _{to} _{introduce} _{a} _{class of} _{processes} _{included}

in the domain of the Skorohod _{integral,} _{that contain both the} space of

square integrable adaptedproceses

### L2(fl,

T,P), and the Sobolev space### L2,2.

This class will be denoted_{by}

_{LF.}

_{A}

_{process}

_{u}

_{=}

_{{ut,t}

_{€}

_{[0,T]}}

_{in}

_{LF}

_{is}

requiredto havesquareintegrable derivatives Dsut and

### D%sut

intheregions {s >### t}

and### {sVr

>### i},

respectively. We### will

show that the### L2-norm

ofthe Skorohod _{integral of} a process u in the space

### LF

is dominated by the L2-norm of the process uand the derivatives Dsut and### D^sut

in the regions {s >### t}

and### {s

Vr >### ¿}.

Using### this fact,

we### will

### establish the local

propertyand the_{quadratic variation} _{property} _{for the} _{Skorohod integral of}a process

u in LF Afterwards we discuss the existence of a continuous versión for

the indefinite Skorohod _{integral of} _{a} _{process} _{in}

_{LF}

_{using}

_{the techniques}introduced

_{recently}

_{by Hu and Nualart}

_{in}

_{[4],}

_{and}

_{we}

_{establish}

_{a}

_{}change-of-variable formula for theSkorohod

_{integral of}processes in

### LF

verifying some additional_{properties.}

_{These}

_{results}

_{generalize similar}

_{properties}

_{proved}

_{in}

[7] for processes in the spaces

### L1,2

or### L2,2.

The paper is organized as follows. In Section 2 we present some pre¬ liminares on the stochastic calculus of variations on the canonical Wiener

space, we introduce the class

### i/

of stochastic processes, and we show themain _{properties} _{of} _{the Skorohod integral of}_{processes} _{in} _{this class. Section}
3 contains the _{change-of-variable formula for the Skorohod integral, which}
extends both the cassical Itó formula and the Itó formula for the Skorohod

integral. Finally in Section 4 we introduce a class of processes containing

the _{space}

_{L2([0,}

_{T\}

_{x}

_{Q) for which the forward integral (see}

_{[9])}

_{exists}

_{and}

generalizes the Itó integral.

2 A class of Skorohod

_{integrable}

_{processes}

Let Q =

### C([0, T])

be the_{space}of

_{continuous}functions from [0, T]

_{into}

_{1}

equipped with the uniform topology, JF the Borel cr-field on ÍI and let P

be the Wiener measure on _{(fb}_{F).} _{The canonical} _{process} _{W} _{—} _{{Wt,t} _{e}

[0,Tj} defined by Wt{u>) = u(t) is a standard Brownian motion. Let

### F?

=### 0’{W.s)O<s<í}

andsetTt =### J\í,

where### Ai

isthe class of P-negligeablesets. Let H be the Hilbert _{space}

_{L2([0,}

_{Tj).}

_{For}

_{any}

_{h}

_{€}

_{H}

_{we}

_{denote by}

W{h) the Wiener integral

W{h) =

### i*

h{t)dWt.Jo

Let S be the set ofsmooth and _{cylindrical} _{random variables of the} _{form:}

F = f(W(hl)t...,W(hn)), (2.1)

where n > _{I,} _{/} _{€}

_{Cg°(Mn) (f and all}

_{its}

_{derivatives}

_{are}

_{bounded), and}

hi,...,hn e H. Given a random variable F of the form (2.1), we define its

derivative as the stochastic _{process} {DtF,t _{€}

### [0,TJ}

_{given}by

D*F = * e [o,t].

j=zl u }

In this way the derivative DF is an element of

### L2([0,T]

x### Q)

=### L2(Q;

### H).

More_{generally,} _{we}_{can}_{define}_{the}

_{iterated derivative}

_{operator}

_{on}

_{a}

_{cylindrical}

random variable _{by}

_{setting}

### DI

### tnF

—### Dti

• • •### DtnF.

The iterated derivative operator Dn is aclosable unbounded operator from

S with _{respect to} _{the} _{norm} _{defined by}

II F

_{Ii;,2=|l}

### F

### ||2t,(n)

+### ]T

### ||

### D>F

### ll!1(|0iT|,xn)

./=L

For _{any} _{Borel subset A of} _{[0,}

_{Tj}

we

### will

### denote by Ta the cr-field

gen-erated _{by the random variables}

_{{/0}

_{1}

_{s(s)dWs,B}

_{e}

_{B({0,T}),B}

_{C}

_{.}

_{i}. The}

following result is proved in [7, Lemma

### 2.4]:

Proposition 2.1 Let A be a Borel subset o/[0,T] and _{consider} a random

variable Fe ®1,2 which isTA-measurable. Then DtF =0 almost everywhere in Ac x fh

For _{any}_{smooth random variable F} _{e} _{S and for}_{any} _{he} _{H} _{we can}_{define}

DhF = (DF,

h)H-Wehave that, for all h e H, D/> isaclosed unbounded _{operator} _{from}

### L2(Q)

into

_{L2(Cl),}

_{and}

_{we}

_{will denote}

### D¿’2

_{the closure of S by}

_{the}

_{norm}

II F

_{\\l2,h=\\}

### F

### ||22(n)

+### ||

### DhF

### \\lHU)

.When h = 1a, with A e _{5([0, T|)} we will simply write Da and

### ¡B>^’2.

### If

F e### B>¿2,

_{the derivative}

_{process}

_{{DtF,t}

_{e}

_{A}}

_{is}

_{well defined}

_{as an}

_{element}

of

_{L2(A}

_{x}

_{Ü).}

We denote _{by 6 the adjoint of the derivative}_{operator}_{D that}_{is}_{also called}
the Skorohod _{integral with} _{respect} _{to} _{the Brownian} _{motion} _{{Wt}. That} _{is,}
the domain of 6 _{(denoted by}_{Dom¿)} _{is} _{the} _{set}_{of}_{elements}_{u} _{e}

_{L2([0,}

_{T]}

_{x}

_{Q)}such that there exists aconstant c verifying

E

### f

_{DtFutdt}

Jo <c|| F

for all F e S. If u e _{Dom<5,} _{6(u)} _{is} _{the element} _{in}

### L2(Q)

_{defined by}

_{the}

duality relationship

E{6(u)F) = E

### F

DtFutdt, FeS.Jo

We will make use of the following notation:

### Jq

### utd,Wt

_{=}

### 6(u).

The Skorohod _{integral}_{is}anextensiónof the Itó integral inthesensethat
the set

_{L2([0,}

_{T\}

_{x}

_{fi) of}

_{square}

_{integrable and adapted}

_{processes}

_{is}

_{included}

intoDomó and theoperatorÓ restricted to

### L2([0,

T\ xQ) coincides with theItó stochastic _{integral} _{(see} _{[7]).}

We will make use of the _{following lemma that} _{can} _{be proved easily by}

duality.

Lemma 2.2 Consider a_{process} _{u} _{G}

### L2([0,

_{T]}

_{x}

_{ü)}

_{such that there}

_{exists}

_{a}sequence {un,n > 1} C Domó satisfying:

(i) un —> u as n tends to infinity, in

### L2([0,

### T]

x Q).(ii) There exists a randorn variable A G

### L2(Q)

such that for all F g S,### E\6(un)F\

converges to E[AF}.Then we have that u _{belongs} _{to} _{the domain of 6,} _{and} _{A} _{=} _{6(u).}

The next result is _{proved} in ([7j).

Lemma 2.3 Let h G

### L2([0,T]),

_{and}

_{F}

_{G}

### D¿’2.

_{Then the}

_{process}

_{{Fh(t),t}

_{€}

[0,T]} belongs to Domó and

S(hF) = F6(h) - DhF.

Let Ln’2 =

### L2([0, T¡;E)n'2)

equipped with the_{norm}

n

I! v

### lln,2=ll

v_{llL2([o,r]xn)}

_{+}

### II

### Hl2([o,rp+1xO)

• j=iWe recall that L1,2 is included in the domain of ó, and for a process u in
L1,2 we can _{compute} _{the} _{variance} _{of}_{the Skorohod integral of}_{tí as} _{follows:}

### E(6(u)2)

= E### í

### u2dt

+ E### í

### f

### DsUtDtusdsdt.

### (2.2)

Jo Jo Jo

We will make use of the following notation

### AÍ

=### {(s,í)

€### [0,T]2

: s >### t},

### A^

=### {(r,

s,t) €### (0,

### T]3

: rVs >### t}.

Let _{St be the} setofprocesses ofthe form ut =

### Fjhj(t),

where F¿ €Sand _{hj} _{G} _{H. We}

_{will}

_{denote by}

_{L1,2’^}

_{the}

_{closure of St by the}

_{norm:}

I! u lli’2/= E

### f

### uídt

+### E

### [

### (DgUtfdsdt,

### (2.3)

and the _{space}

_{LF}

_{will be defined}

_{as}

_{the}

_{closure}

_{of St}

_{by}

_{the}

_{norm:}

II u IIf=II u II12/

### +E

### [

### (DrDsUt)2drdsdt.

### (2.4)

That _{is,}

_{L1,2’^}

_{is}the class of stochastic processes {vt,t €

### [0,

Tj} thatare differentiable with _{respect to} _{the Wiener} _{process} _{(in} _{the} _{sense} _{of the}

stochastic calculus of _{variations)} _{in} _{the future. For} _{a} _{process u} _{in}

_{L1,2^}

we can define the _{square} _{integrable kernel} _{{Dsut,s} _{>} _{t} which belongs}

to

_{L2(Aj}

_{x}

_{fí). Similarly,}

_{LF}

_{is}

_{the class of stochastic}

_{processes}

_{{ut,t}

_{e}

[0,

### T]}

such that for each time t, the random variable ut is twice weakly differentiable with respect to the Wiener process in the two-dimensionalfuture _{{(r, s),r}_{V} _{s} _{>}

_{t}.}

_{We}

_{also observe that}

_{i/”}

_{coincides with the class}of processes u €

### L1,2’^

such that_{{Z?sU(l[0,s|(t),}

_{t}

_{€}

_{[0,T]}}

_{belongs}

_{to}

_{L1,2}

as
a _{process} _{with valúes} _{in} _{the Hilbert} _{space}

### L2([0,

_{Tj).}

Remark: Notice that if u €

### L1,2^,

then_{f^Usás}

€ ### for

any 0 < a <b <T.

Lemma 2.4 The _{space}

_{L2([0,T]}

_{x}

_{Q)}

_{is}

_{contained}

_{in}

_{LF.}

_{Furthermore, for}all u €

### L2((0,Tj

_{x}

_{íí)}

_{we}

_{have DsUt}

_{=}

_{0}

_{for almost all}

_{s}

_{> t,}

_{and, henee,}

II u IIf=II u

### lli2([o,rixH)

_{•}

_{(2-5)}

Proof: We will denote by Sj- the class of elementary processes ofthe form

N

^ =

### EFAb,b+1l(í)i

### (2-6)

3=0

where0 = ío _{<} íi _{<} _{■}_{• •} _{<} í/v+i _{=} T and, for allj _{=}0,...,N, Fj is_{a}smooth

and

_{^-measurable random}

_{variable. The}

_{set}

_{Sf}

_{is}

_{dense}

_{in}

_{L2([0,}

_{Tj xíl).}

On the other hand, we have Sf _{c}

### L1,2’^

### and for

any v### of the form (2.6)

we _{have,} _{using} _{Proposition 2.1,} _{Dsvt} _{=} _{0,} _{for}_{any s} _{>} _{t.} _{This allows} _{us} _{to}

complete the proof. QED

The next two _{propositions} are extensions of known resuls for the space

ID1’2 _{(see} _{[6, Proposition} _{1.2.2} _{and Proposition}

_{1.3.7)).}

Proposition 2.5 Let tp : Km —> IR be a continuouslp differenciable function
with bounded _{partial} _{derivatives.} _{Suppose that} u =

### (u1,...,um)

is an m-dimensional random_{process}

_{whose}

_{components}

_{belong}

_{to}

_{the}space

### L1,2’^.

Then_{u)}

_{€}

_{¡L1’2’^,}

_{and}

Ds(ip{ut)) =

for all (s,t) 6

### Af.

Proposition 2.6 Letu €

### L1,2’^

and A £ E, such that_{ut{uj)}= 0 a.e. on the

product space [0, T) x A. Then Dsut = 0 for almost all (s,t,u>) _{in}

### A^

_{x}

_{A.}

The _{following} _{result} _{provides} _{an} _{isometry} _{property} _{for the Skorohod}

integral ofaprocess u in the space

### Ll,2,/

satisfying an additional condition.Lemma 2.7 Consider a _{process u} _{in}

### ¡L1,2^.

_{Suppose that for almost all}

9 £ _{[0,}

_{T\,}

_{£)¿ii¿l[O}

_{0j}

_{belongs}

_{to}

_{the domain of 6 and,}

_{moreover,}

E _{DgUsdWs}

2

d6 < oo.

Then

_{wl[o,t|}

_{belongs}

_{to}

_{the domain of 6 and}

E u2ds +2E _{DgusdWs)d9.}

(2.7)

(2.8) Proof: Suppose first that u has a ñnite Wiener chaos expansión. In this

case we can write:

E

_{u2sds}

_{+}

_{E}

### [

### f

_{DsUgDgUsd9ds}Jo Jo rt rO

### u2sds

+ 2E### /

DsugDgusd9ds Jo Jo u2ds+2E### J

### Ug

### DgUsdWs^j

d9.Now, let us denote by un the sumof the n first terms in the Wiener chaos

expansión of u. It holds that un converges to u in the norm || • ||as n

tends to _{infinity. For} _{each} _{n we} _{have}

E

2

= E 2ds _{+}2E

Deunsd\Va)dO. (2.9)

It suffices to show that the _{right-hand side of (2.9)} converges to the

right-hand side of _{(2.8). This} _{convergence} _{is} _{obvious for the first} _{term.} _{The}
convergence of the second summand follows from condition (2.7). QED

Remark 1: In the statementofLemma 2.7 the _{assumptions} areequivalent

to _{saying} _{that} _{u} _{£}

### L1,2’^

_{is}

_{such that}

_{(s),}

_{s}

_{£}

_{[0, T]}}

_{belongs}

_{to}

Remark 2: Lemma 2.7_{generalizes the}_{isometry properties}_{of the Skorohod}

integral for processes in the spaces

### L2([0,

T\ x fi) and### L1’2.

The _{following} _{estímate} _{of} _{the ZT-norm of the Skorohod integral} _{is} _{a}

consequence of Corollary 2.2 of [4].

Lemma 2.8 Let p € (2,4), a — Consider a process u in n

La([0, T] x fl). Suppose also that, for each interval [r,6\ C [0,T\,

### Dsul^j

belongs to the domain of 5, and, moreover,

E _{DeusdWs}

2

dO < oo. _{(2.10)}

Then

_{¿(ul[r í¡)}

_{belongs}

_{to}

_{LP for}

_{any}

_{interval}

_{[r, í]}

_{C}

_{[0,T]}

_{and}

_{we}

_{have:}

### EI

### i'

usdWs### P

< Cp{t —### r)?~l{E

### f

### \us\ads

### +

### E

### f

### \

### f

### DeusdWs)\2de},

\Jr Jr Jr Jr

(2.11)

where _{Cv} _{is} _{a} _{constant} _{depending only} _{on p} _{and T.}

Proof: We deduce from Lemma 2.7 applied to

_{ul¡r}

T] ### that ul¡r>fj belongs

to the domain of 6 for any interval

### [r,t\

C### [0,T].

Now using Corollary 2.2of _{[4]} we deduce that (2.11) is true in the set Vt ofprocesses u of the form:

N

”í =

### E01|.,..j«

### |(0.

### (212)

j=0

where 0 = to _{<} • • • < í/v+i = T and for all j — 0,...,N, Fj are smooth

randomvariables of the form_{(2.1),} _{/ being}apolynomialfunction. We know

that _{Vt is}_{dense}in _{LQ([0,}T] xQ) . So, wecanget a sequence {un,n > 1} of

processes in Vt so that un -+ u in

### LQ([0,T]

x íl). Moreover, ifwe consider the Ornstein-Uhlenbeck_{semigroup}

_{{Tt,t}

_{>}

_{0},}

_{we}

_{know}

_{that, for all}

_{t,}

_{Ttu}

is also an element ofVt, and _{we can} easily _{prove} that, for all [?\

### t\

_{C}

### [0,

T\_{:}

limlim

### EÍ

_{|Tiu"}

_{-}

_{iq¡|ads}

» _{fc} Jo
*
lim lim E
n _{k}

### J*

### |

### j\De(Tiuns)

### -

### Dgus)dWs\2de

0, 0,which allows us to _{complete} _{the proof.} _{QED}

The next _{proposition} _{provides estimates} _{for the}

### L2

_{and LP}norms of the

Proposition 2.9

### l/

C Dom¿> and we have that, for all u in### LF,

### E\6(u)\2

< 2 ¡| u### \\2F

. (2.13)Consider _{p} _{€} _{(2,4)} _{and} _{a} = If furthermore, u belongs to the space

La([0, T] x Q) we have that, for all [r,t\ C [0,T],

_{£(til¡r>t])}

### is in

### LP and

### E\6(ul[rA)\p

### <

### Cp(t

-### r)?-1

### ^

### £

### E\us\ads

### (2.14)

p0 p0 p0 \

+

### EJ \Dgus\2ds

### +

### EJ

### J \DaDgus\2dsda\,

where _{Cv is the} _{constant} _{appearing in}

_{(2.11).}

Proof: Because u €

### LF

we have that### {Aj«<í1(o,0)(s),

s €### [0,T]} belongs

to L1,2 C Domó for each 6 €_{[0,T],}

_{and}

_{furthermore}

_{we}

_{have}

E

### [T

_{|}

_{DeusdWs\2d9}

_{<E}

### T F(Dgus)2dsd9

_{(2.15)}

Jo Jo Jo Jo

pT p0 p0

+E

_{/}

_{/ /}

_{(DaDgus)2dadsdO.}

_{(2.16)}

Jo Jo Jo

rT r0

Then, applying Lemma 2.7 we obtain that u is Skorohod integrable and

### E\6(u)\2

= E### F

### u2ds

+ 2E### [T

### ue(

### f9

### DeusdWs)dd.

Jo Jo Jo

Using now the Cauchy-Schwartz inequality we have

### E\8(u)\2<eJ^

### u2ds

### +

### 2y

### u2$d9\JE^ |jí

### DeusdWs\2d9,

and _{using} _{the fact that, for} _{all} _{a,}_{b} _{€} _{IR,} _{2ab} _{<}

_{a2}

_{+}

_{b2}

_{it}

_{follows that}

### E\8{u)\2

<2E### fT

### u2ds

+E### F

### \

### [°

### DeusdWs\2dO.

Jo Jo Jo

Now _{applying} _{(2.15)} _{we} _{obtain}

### £|ó(u)|2

### <2EÍ

### u2ds

+### E¡ í

### (DgUsfdsdO

Jo Jo Jo

pT p0 p0

+ E

### / / (DaDgUs)2dodsdQ

Jo Jo Jowhichproves (2.13). Similarly, using Lemma 2.8 we can prove (2.14). QED

Note that _{uelf implies}

_{ul[r,í|}

_{6}

_{LF}

_{for}

_{any}

_{interval}

_{[r, í]}

_{C}

_{[0,}

_{T], and,}

by Proposition 2.9 we have that

_{ul|rt]}

6 ### Domó.

We have the _{following} _{commutative} _{relationship between the derivative}
and the Skorohod _{integral for} _{processes} _{in}

_{LF}

_{(see}

_{[6]}

_{,}

_{property}

_{(3)}

_{of the}Skorohod

_{integral).}

Lemma 2.10 _{Suppose that} _{u} _{=}

_{{ut,t}

_{€}

_{[0,T]} belongs}

_{to}

_{LF.}

_{Then the}

process

_{{ó(ul¡o,í]),í}

_{€}

_{[0,T]}}

_{belongs}

_{to}

### L1,2’^

_{and, for all (s,t)}

_{€}

### AF

_{we}have

### Ds(ó(ií1[o,£|))

=### ¿(DíUljo.íj).

### (2.17)

The next result_{provides sufficient conditions for the continuity of the}

process

### ó(iil[o,£j)

=### J

### usdWs

where u € LF.Theorem 2.11 Let u — {ut,t e

### [0,T]}

be a process in### LF

such that forsome _{j3} _{>} _{2,} _{E}

### /0r

### \usfds

_{<}

_{oo}

.

### Then the

process### Xt

:=### /0(

### «sdiys has

a continuous versión. More_{precisely,}

_{for}any0 < 7 < , there is a random

variable _{C7 such that}

\Xt - X,\ <

### C7|i

- s|T (2.18)A/3 'T' 2p

Proof: Letp := We have E

### f0

### |us|4-pds

< oo,### and applying

Proposi-tion 2.9 we obtain

E\Xt - X,\p <

### Cp(t

-### s)*~l{j‘

### E\ue\0d6

+

### i*

### E\Drue\2drdQ

+### í

### f f

### E\DaDrue\2d<Tdrd9}.

Js Je Js Je Je

Therefore, there exists a nonnegative function A : [0, T] —> 1+ such that

### Jq Ardr

< oo andFor _{any} _{2} _{<} _{a} _{<} _{applying} _{Fubini’s theorem,} _{we}_{obtain}

Henee the random variable

is finite almost _{surely. By} _{the Garsia-Rodemish-Rumsey lemma} _{(see [2]),}

we have that for _{any 7 :=}

### -—-4^'—,

_{there}

_{is}

a

### random

constant### C7

### almost

surely finite such that

\Xt — Xs| < C7|í— s|7.

Noting that 2 < a < is equivalent to 0 < 7 < we prove the

theorem. _{QED}

We have the _{following} _{local} _{property} _{for the} _{operator} _{6:}

Proposition 2.12 We consider u a process in and A € F so that

Ut{u>) = 0, a.e. on the product space

### [0,T\

x A. Then 6(u) = 0 a.e. on A.Proof: Consider the sequence ofprocesses defined by

2m —1

2m rTj2~m

UT —

### ~7r(

### /

### 'W«d5)l(rj2-m,r(j+i)2-mi(í)-jrí

### 1 Jt(í-i)2-m

Itis _{easy}_{to}_{show that for all} _{m} , u —► umis anlinear boundedoperatorfrom

¡Lf into Lf whose normis bounded _{by}

_{^}

_{and}

_{that limm_oo}

_{\\um}

_{—}

_{u}

_{||f=}

_{0.}

Using now Proposition 2.9 we have that S(um — u) tends to zero in

### L2(íí)

as m tends to 00. _{But,} _{on} _{the} _{other hand, using}_{Lemma} _{2.3} _{we} _{have}

rTj2~m

### I

### Usds)(WT(j

+l)2-m -### WTj2-m)

ro-i)2-™and _{by the} _{local}_{property} _{of the} _{operator} _{D} _{in} _{the}_{space}

_{L1,2’^}

_{(Proposition}

2.6) we have that this expression is zero on the set

### {J^

_{u^ds}

= ### 0},

whichcompletes the proof.

QED

Ifwe have a subset L C

### L2([0,T)

_{x}

_{fí)}

_{we can}

_{localize}

_{it}

_{as}

_{follows.}

_{We}

will denote _{by Lioc the} _{set} _{of} _{random} _{processes u} _{such that there} _{exists} a
sequence {(Qn,un),n > 1} C Tx L with the following properties:

(i) t D, a.s.

(ii) u = un, a.e. on

### [0,T]

x Qn.We then say that {(Dn, un)} localizesu in L. Ifu €

### L^,

by Proposition2.12 we can define without _{ambiguity} _{S(u) by} _{setting}

$(u)|nB=5(ti")|nB,

for each n > 1, where

### {(Qn,un)}

is a localizing sequence for u in### LF.

Thefollowing result says that the operator Ó on Lis an extensión of the Itó integral.

Lemma 2.13 Letu be an adapted_{process} verifying

### /0T

### u2sds

_{<}

_{oo a.s.}

_{Then}

u _{belongs}

_{to}

### L^.

_{and}

_{6(u) coincides}

_{with the Itó integral.}

Proof: For any integer k > 1 consider an infinitely differentiable function
: 1 —> IR such that <Pfc(x) _{=} _{1} if |a:| _{<} k, and <Pk{x) _{=} 0 if |a?| _{>} k _{+} _{1.}

Define

«t =

### (jf

### ulds)

,

and

Dfe =

### u*ds

< kThen we have _{íí* T} _{a.s., u} _{=}

### uk

_{on}

_{[0,T]}

_{x}

_{and}

### uk

_{G}

### L2([0,

_{T\}

_{x}

_{D)}

because uk is _{adapted and}

### j

### (uk)2dt

### =

### J

### ufol

### {^J ulds'j

### dt

### <

### k

### +

### 1.

QED

The next result will show theexistence ofa nonzero_{quadratic} _{variation}

(2.20)

Theorem 2.14 _{Suppose that}u is aprocess of the space LThen

in _{probability,} _{as} _{¡7r|} _{—♦} _{0,} _{where} _{ir runs over} _{all finite} _{partitions} _{{0} _{=} _{¿o} _{<}

ti <■■■< tn = T} of [0, T}. Moreover, the convergence is in

### Ll(U)

ifubelongs to

### LF.

Proof: We will describe the details of the proof only for the case u G

### LF.

The_{general}

_{case}

_{would be deduced by}

_{an easy}

_{argument}

_{of localization. For}

any process u in

### LF

and for any partition 7r = (0 =to < h < •• • < tn = T}we define

^

v*(u) =

### i£(ft'+1usdWs)2.

i=o Jti

Suppose that u and v are two processes in

### LF.

Then we have£(1^» - V"»|) <

### (j2(Jtt,+

### \us-vs)dWs

x

### ^E

### (/

### +

### (u*

### +

### vs)dWs

< 4 _{||} _{u-v} _{||f||} _{u}_{+}_{v} _{||f} _{•} _{(2.21)}

It follows from this estímate that it suffices toshow the result for a class of processes u that is dense in

### LF.

Sowe can assume thatm—1

Ut =

### E

+

J=0

where _{Fj}_{is} _{a}_{smooth random variable for each}_{j,} _{and}_{0} _{=} _{so} _{<} _{• ■ •} _{<} _{sm} _{=} _{T.}
We can assumethat the _{partition}_{7r}_{contains} _{the}_{points} _{{so,..., sm}.} _{In this}
case we have
V*(u) =
m~l
/ _{fti+1}

### E

### E

_{(Fjmu+Ú-Wiu))-}

_{DsFjds}

j=0 _{{t■.sj<ti<sj+1}}

### K

### Jti

m—1 =### £

### E

### d(M/(‘.+>)

-j=° -2(W(íi+i) - W{U))### ft'+1

### DsFjds

### +

### Qtl+1

### DsFjds')

With the _{properties} _{of the quadratic variation of the Brownian} _{motion,} _{this}
converges in

### Ll(ü)

to m—1_{.j'}

### 2

### Fj(si+1

-### Sj)

=### /

### Ulds>

j=0 Jo as ¡ 7r| tends_{to}

_{zero.}QED

As a _{consequence} of this result, if _{u} _{G}

### L^.

_{is}

_{a process}

_{such that the}

Skorohod_{integral} _{usdWs has}acontinuous versiónwith bounded variation
paths, the u = 0.

3 Itó formula for the Skorohod

_{integral}

Our purpose in this section is to prove a versión of the change-of-variables

formula for the indefinite Skorohod _{integral. For} _{a process} _{X} _{€}

_{L1,2’^}

we
will denote D~X the element of

_{Ll(\Q,T]}

_{x}

_{fi) defined}

_{by}

lim

### í

_{sup}

_{E\DsXt}

_{-}

_{(£>~X)s|ds}

_{=}

_{0,}

_{(3.22)}

”

(s-±)V0<t<s

i

2 /

provided that this limit exists. We will denote by _{L1_}

_{the class of}

processes
in L1’2’-^ such that the limit _{(3.22} _{)} _{exists.} _{It is} _{easy} _{to}_{show} _{that} _{a} _{process}
of the form

Xt — Xo +

### /

usdW3 +### /

vsds,Jo Jo

where _{Xo} €

### JD>¡^,

_{u}

_{G}

### L^.,

_{and}

_{v}

_{6}

_{belongs}

_{to}

_{the}

_{class}

### Lj’2’^

_{and}

(D X)t - DtA"0 +

### í

Dtvsds +### /

DtusdWs.Jo Jo

We will also denote _{by}

_{Lf}

_{the}

_{space}

_{of}

_{processes u}

_{€}

_{LF}

_{such that}

||

### Jo

### u2ds||oo

< oo.Theorem 3.1 Considera_{process} _{of the form Xt} _{=} _{Xo}_{+}

_{Jq}

_{usdWs}

_{+}

_{vsds,}

| n n \ 0 f

where X0 € «e

_{(LO,*:,}

_{and}

_{v}

_{e}

_{Lfa'.}

_{We}

_{will}

_{also}

_{assume}

_{that}

_{the}

E be a twice _{continuously} _{differentiable function. Then} we have

F(Xt) = F(XQ) +

### f

### F'{Xs)dXs

_{+}

### ±

### /*

### F"(Xs)u2ds

Jo 2 Jo

+

### f

_{F"(XS)(D~X)susds.}

_{(3.23)}

Jo

Proof: Suppose that

### (fí’1,1,

A"”).### (Qn,2,un)

and### (Qn’3,vn)

are localizing se-quences for Xo, u, and v, respectively. For each positive integer k let ipkbe a smooth function such that 0 < _{ipk < L} _{ipk(x)} = 0 if |x| > fc + 1, and
i>k{x) = 1 if |a;| < k. Define

### Vt’k

= v?ipk{### í

### \v"\2ds).

Jo

Set

_{Xfn,fc}

_{=}

_{X¿1}

_{+}

_{Jq u”dWs}

_{+}

_{Jq}

_{Vg'kds,}

_{and consider the family of}

_{sets}

Gn,k = Qn,1 nnn,2R 3 R_{{} _{gup} _{^} _{<} _{fc}} _{n}_{{}

### fT

### \v”\2ds

_{<}

_{k}.}

te[o,i]

### Jo

Define also Fk = F'ipk■ Then it suffices to show the result for the _{processes}

X£, un, and

### vn'k,

### and

for### the function

### Fk.

### In this

way we can assume that Xo G E>1,2, u G### I/, ¡q

### u2ds

_{<}

_{M,}

_{for}

_{some}

_{constant}

_{M}

_{>}

_{0,}v G

### L1’2’^,

### ¡q

### |i>.¡|2ds

<### k, and that the functions F,

### F'

### and

### F"

are### bounded.

### Moreover,

we can assume that the _{process} Xt has _{a} _{continuous versión.}

Set t" = 0 < i < 2n. Applying Taylor development _{up} _{to} the second

order we obtain

F(Xt) = F(Xo) +

### £ F'(X(t?))(X(i?+1)

_{-}

### X(%))

i=0

+

### 2¿'

### ií"(Xi)(X(í?+1)

_{-}

### XK))\

_{(3.24)}

i=0 ¿
where_{X¿} _{denotes} arandom intermedíate pointbetween X(t") _{and}

_{X(t"+1).}

Now the_{proof}_{will} _{be decomposed} _{in} _{several} _{steps.}

Step 1. Let us show that

### 2¿'

### F"(Xt)(X(tr+1)

-### X(í?))2

-### f

### F"{Xs)u2ds,

### (3.25)

in

_{L^Q),}

_{as n}

_{tends}

_{to}

_{infinity.}

The increment

_{(A"(í”+1)}

_{—}

_{X(tf))2}

_{can}

_{be}

_{decomposed}

_{into}

Using theboundednessof

### F"(Xt),

theproperty### /0'

### |us|2ds

< k, and Theorem2.14 we can show that the contribution of the last two terms to the limit

(3.25) is zero. Therefore, it suffices to show that

2n—1

### £

UsdWt_{)2}

_{F"{Xs)u]ds,}

_{(3.26)}

in

_{Ll(fl),}

_{as n}

_{tends}

_{to}

_{infinity.}

_{Suppose that}

_{n}

_{>}

_{m,}

_{and}

_{for}

_{any}

_{i}

_{=}

1,...,2n let us denote _{by}

### í-m)

_{the}point of the mthpartition which is closer

to t” from the left. Then we have

### E

### Fn(Xi)(

### /t,+l

### usdWs)2

-### f

### F"(Xs)u2ds

~n Jtn Jo E < E ¿=o 2"-l

### E

### [F"m

~### f"(x

### (t.

•ai ¿=o 2m—1 h+i### MWS)2

+F_{E}

_{E}

j=o
( ### [

### ,+1

### usdWs)2

—### [

### ,+l

### u2ds

Jt? Jt? !"‘-l m_{(}

### ^

### F"(X(í™))

### /

### ,+1

### u2ds

-### /

### F"(Xs)u2sds

### j^o

### JtT

### Jo

+E bi +b% + 63.The term _{63} can be bounded _{by}

E

### (

sup

### |F"(XS)

-### F"(Xr)|

### f

### u2ds)

,\|s-r|<t2-m JO )

which convergestozeroasm tendstoinfinity by thecontinuityof theprocess

which tends to zero as m tends to _{infinity uniformly} _{in} _{n.} _{Indeed, if} _{we}
write
Am = sup

### \F"(XS)

-### F"(Xr)\,

\s-r\<t2~m and 2n —1_{tn}

### r

### U‘dW»)2'

t=0 Jtithen, for any K > 0 we can write

E{AmBn) < KE(Am)

_{F2\\F"\\0OE(Bnl{Bn>K}),}

and this _{implies that}

lim lim _{sup} _{E(AmBn)} _{=} _{0.}

A'__{oom-oon>m}

Finally, the term 62 converges to zero as n tends to infinity, for any fixed m,

due to Theorem 2.14.

Step 2. Clearly the term

vsds)

converges a.s. and in

### Ll(ü)

to F'{Xs)vsds, as n tends to infinity. Step 3. From Lemma 2.4 and Proposition 2.6 wededuceftn ftn

### F'(X(0)

### ,+1usdWs=

### f

_{1+1}

_{F'(X(t?))usdWs}

Jv> Jt?
4- ### [ti+l

_{Ds[F'(X(t^))]usds}

### Jt:

=### fti+l

### F'(X(t?))usdWs

### Jt?

### +F"{X(t?))

### /t,+1

DsX(t?)usds.### Jt?

Notice that ft1} ftn ftn rt1}### /

### *+1

### DsX{t™)usds

=### /

### (DsXq

+### /

### *

DsurdWr +### /

### ’

Davrdr)u$ds.Jt? Jt? Jo Jo

We will show that 1 _{F'^-X^í”))}

_{ír+1}

_{DsX(tf)usds}

_{converges}

_{in}

_{L1}

_{to}

### fo

### F"(XS)(D~X)ausds

as n### tends

to### infinity.

In### fact,

we have### £

### F"(X(i?))

### I

### DsX(t?)usds

-### /

### F"(XS){D~X)susds

¿=o + i=0 '"■i Consequently, we obtain / 22z} r* E### £

### F"(X(í?))

### [i+1[DsX(t?)-(D-X)s}usds

### *to

•'í?### fi+l{F"{X(t."))

-### F"(XS)\(D~

### X)susc

### £

### F"{X{t?))

### [

### i+1

### DsXWusds

-### f

### F"(XS)(D~X)susds

### í=o

### ^

<

### v^?||F,/||oo

### (f^1

### /t,+1

### |DaX(C)

-### (F>-X)s|2ds)2

l »=o ■'‘r J

+F

### (

_{sup}

_{|F"(Xa)}

_{-}

_{F"(Xr)|}

### C\(D-X)aus\ds)

_{=}

_{di}

_{+}

_{d2.}

\|s-r|<2-" ^0 /

The term d2 converges to zero as n tends to infinity, because

E

### í

_{\(D~X)sus\ds}

_{<}

_{oo.}

Jo

For the term _{di the} _{convergence} _{to}_{zero} _{follows from the} _{estimates}
2n —1
F V'

### ÍÍ+1

_{/}

_{DaUrdWr}

_{-}

_{i}

_{DsUrdWr}i=0

### Jt?

### J°

### J°

2"-1_{I}/•*

### I2

=### eJ2

_{\}

_{DsUrdWr\}

### ds

t=o •'*" r*" I 1 rS 2"~} ft?+1 r4 fs### /

### /

### |F>sur|2drds

FE ¡### / /

### |-De.Dst¿r|2d0drds,

«/t? •/(?_{-/t;*}

_{•/«?}

_{■A'»}¿=o 's

and a similar estímate for
2n—1 _{-en} _{I}
r^ixi r*’i
• X *
J'* I /*{. /*£
£ V

### /

### *+1

### /

_{Dsvrdr}

_{—}

### /

_{Dsvrdr}

### £o

K0### •/o

ds.Step 4■ Finally, we will show that for all t e (0, T], the process

$$ — F1

### {Xs)us1\q¿\(s)

belongs to the domain of 6 and that

t rt

### F'(Xs)usdWs

= F(Xt) - F{Xo) -### /

### F'{Xs)vsds

### (3.28)

oTo _{get} _{this} _{result} we will apply Lemma 2.2 to the sequence ofprocesses

2" —1

i=0

We have that $n converges in

### L2([0,T]

x fí) to $ as w tends to infinity.From _{Step} _{1,} _{Step 2} _{and} _{Step 3,} _{we} _{have that}

converges in

### L^Q)

to a random variable A equals to the right-hand sideof _{Equation} _{(3.28).} _{Then in} _{order} _{to} _{complete the proof} _{it} _{suffices} _{to} _{show}
that A _{belongs} _{to}

_{L2(Q).}

_{This}

_{follows from the fact that}

_{u}

_{€}

_{Lf,}

_{Xo}

_{€}

_{ID)1,2}

and v € L1,2’^. _{QED}

Remarks:

1.- The _{assumptions} _{of Theorem} _{3.1} _{can} _{be slightly modified. For} _{instance,}

in order to insure that the process uadWs has continuous paths we can

assume that u e

### (Lf

_{Pi}

### L,P([0,T]

_{x}

_{ü))ioc}

_{for}

_{some}

_{0}

_{>}

_{2,}

_{due}

_{to}

_{Theorem}

2.11. On the other hand, notice that the fact that

_{jjf}

_{u2sds}

_{is}

_{bounded}

_{is}

only used to insure that the right-hand side of Equation (3.29) is square

integrable. Then, instead of assuming that the process u is locally in the

space

### Lf

we can assume that theprocesses u, v and the initial condition Xoverify locally the followingproperties: (i) u €

### LF

and_{£(/0r}

_{u2ds)2}

< oo.
(iii) v €

### Ll’2J

and_{£(/{s>t}}

_{IDsvt\2dsdt)2}

_{<}

_{oo.}

### 2.-

If_{Xo}

_{is}

_{a}

_{constant,}

_{and}Ut and vt are adapted processes such that

### ¡q

### \ut\2dt

< oo,### Jq

### |vt\2dt

< oo a.s.,### then these

processes satisfy theas-sumptions of the theorem because u €

### (TL[)ioc

_{by Lemma 2.13,}v e (thispropertycan beproved byalocalization procedure similartothat used

in the _{proof}_{of Lemma 2.13), and the} _{process} _{Xt} _{=} _{Xo}_{+} _{usdWs}_{+}

_{/0‘}

_{vsds}

has eontinuous _{paths}

_{because}

_{it}

_{is}

_{a}

_{continuous}

_{semimartingale. }

Further-more, in this case (D~X)S vanishes, andweobtainthe classical Itó formula.

## 3.-

Instead of_{working with}

_{processes v}

_{€}

_{we}

_{can}

_{rewrite}

_{the}

_{proof}of Theorem 3.1

_{assumig}

_{that}

_{v}

_{is}

_{a process}

_{verifying}

_{locally}

_{the}

_{following}

properties:

0)

### Jo

<k-(ii) The process V =

### {/q

### vsds,t

€### [0,T]}

### belongs

to### the

spaceIn this case the _{proof} _{is} _{the} _{same,} _{changing the} _{term}

### f¿¡

_{Dtvsds by}

_{the}

term _{(D~V)t■ Notice that if} _{we assume} _{this hypothesis for} _{v,} _{taking into}
account Remark 2, the change-of-variable formula for the Skorohod integral

established in Theorem 3.1 is a _{generalization of the classical Itó’s formula,}
in the sense that v is an _{adapted} _{process} _{satisfying} \vt\dt _{<} _{oo} _{a.s.}

4 Forward

_{integral}

The forward _{integral introduced} _{in}

_{[1]}

_{is}

_{an}

_{extensión}

_{of the}

_{Itó integral}

_{to}

nonadapted processes which differs from the Skorohod integral. In

### [9]

this typeofintegral is defined using the followingapproximation approach. Letu = {ut,_{t} _{€}

### [0,T)}

be_{a process}in

### L2([0,T]

_{x}ü). Then for

_{every m}

_{>}

_{1}

_{we}can introduce the term

2m fT

Fm(u) = —

### yo

### Ut(W{t+T2-”>)AT

### -

### Wt)dt

Definition 4.1 We_{say}_{that}_{a process u} _{—}

_{{ut,t}

_{€}

_{[0,T]}}

_{in}

_{L2([0,T]}

_{xíí)}

_{is}

forward integrable if the sequence Fm(u) convergesinprobabilityasm —> oo,

and in this case the limit will be denoted _{by}

### fj

_{utd~Wt.}

The next result _{gives} _{us} _{the relationship} _{between the forward} _{integral}
and the Skorohod _{integral:}

(4.29)

Theorem 4.2 Let u e

### (LF_

_{)ioc.}

_{Then}

_{u}

_{is}

_{forward}

_{integrable and}

Proof: By a localization argument we can assume that u e

### Lf_.

### Thanks

to Lemma 2.3 we have _{that, for} _{every m} _{>} _{1,}

'(t+T2-m )AX

Drutdr)dt.

Now the _{proof}_{will be}_{descomposed} _{into} _{two steps.}

Step 1. Let us show that

(4.30)

in

_{L2(f2),}

_{as m}

_{tends}

_{to}

_{oo.}

_{Using the Fubini theorem for the}

_{Skorohod}

integral (see

### [6],

Exercice3.2.8) we have thatwhere

It is easytoshow that for allm, u —► Um is anlinearbounded operatorfrom

LF into LF whosenormis bounded_{by}

_{^}

_{and that limm_oo}

_{II Um}

_{—u}

_{||p=}

_{0.}

Using now Proposition 2.10 we have that 6(Um) tends to S(u) in

### L2(ü)

asm tends to oo.

Step 2. Let us prove that

(4.31)

tends tozero as m tends to oo.

We have that

(t+T2~m)AT

2"l rT r{t+T2-m)/\T rT

### +E\jr

### J

### (y

### (D~u)rdr)dt-

### JQ

### (D~u)rdr\

<

### í

_{sup}

_{E\Drur}

_{—}

_{(D~u)r\dr}

Jo _{(r—T’2_m)V0<t<r}

/*T rr rT

+E|—

### /

### /

### {D~u)rdtdr

—### /

### (£>-u)rdr|.

1 _{Jo} _{J}_{(r—T2~}_{m)VO} _{J}_{O}

Now,thankstothedefinition of the class

_{Lf_}

### and the dominated

convergencetheorem we have that these two terms tend to zero as m tends to oo, and

now the _{proof} _{is} _{complete.}

QED
Remark: The forward _{integral is} an extensión of the Itó integral. In
fact,noticethat

### L„([0, T]

xü) C### Lfl

### and that for

a processu €### £„([(), T) xü)

the forward _{integral coincides with the Itó stochastic integral.}

References

[1] M. A. Berger and V. J. Mizel: An extensión ofthe stochastic integral.

Ann. Probab. 10 _{(1982) 435-450.}

[2] A. Garsia, E.Rodemich, and H. Rumsey: A real variable lemma and the
continuity of paths of some Gaussian processes. Indiana Univ. Math.
Journal20 _{(1970/71)} _{565-578.}

[3] B. Gaveau and P. Trauber: L’intégrale stochastique comme opérateur
de_{divergence dans l’espace fonctionnel. J. Functional} _{Anal.} _{46} _{(1982)}
230-238.

### [4]

Y. Hu and D. Nualart: Continuity of some anticipating integral pro¬cesses. _{Preprint.}

[5] P. Malliavin.: Stochastic calculus of variations and hypoelliptic

oper-ators. In: Proc. Inter. _{Symp.} on Stock. Diff. Equations, Kyoto 1976,
Wiley 1978, 195-263.

### [6]

D. Nualart: The Malliavin Calculus and Related Topics, Springer, 1995.### [7]

D. Nualart and E. Pardoux: Stochastic calculus with anticipating inte-grands. Probab. Theory Reí. Fields 78 (1988) 535-581.### [8]

D. Oeone and E. Pardoux: A generalized Itó-Ventzell formula. Appli¬cation to a class of_{anticipating} stochastic _{differential} _{equations.} _{Ann.}

Inst. Henri Poincaré 25 _{(1989) 39-71.}

[9] F. Russo and P. Vallois: Forward, backward and symmetric stochastic integration. Probab. Theory Reí. Fields 97 (1993) 403-421.

### [10]A.

V. Skorohod: On a generalization of a stochastic integral. Theory Probab._{Appl. 20}

_{(1975)}

_{219-233.}

Relació deis últims

_{Preprints}

_{publicáis:}

— 197 A Gentzen

system equivalent to the BCK-logic. Romá J. Adilion and Ventura Verdú.

Mathematics_{Subject} _{Classification:} _{03B20. January 1996.}

— 198 On Gentzen

systems assoaated with the finite linear MV-algebras.

### Ángel

J. Gil, AntoniTorrens Torrell and Ventura Verdú. Mathematics _{Subject} _{Classification: 03B22,} _{03F03,}
03G20, 03G99. February 1996.

— 199

Exponentially small splitting of separatrices under fast quasiperiodic forcing. Amadeu

Delshams, Vassili Gelfreich,

### Ángel

Jorba and Tere M. Seara. AMS classification schemenumbers: 34C37, 58F27, 58F36, 11J25. February 1996.

— 200 Existence and

regularity ofthe densityfor Solutions to stochastic differential equations with

boundary conditions. Arturo Kohatsu-Higa and Marta Sanz-Solé. AMS 1990 SCI: 60H07, 60H10, 60H99. March 1996.

— 201 A

forcing construction of thin-tall Boolean algebras. Juan Carlos Martínez. 1991 Mathe¬

matics _{Subject Classification:} _{03E35, 06E99, 54612.} _{March 1996.}

—

202 _{Weighted} _{contínuous metric} _{scaling.} _{C. M. Cuadras and} _{J. Fortiana. AMS Subject Classi¬}

fication: _{62H25, 62G99. Apiil 1996.}

— 203 Homoclinic orbits _{m} the complex

domain. V.F. Lazutkin and C. Simó. AMS Subject Clas¬

sification: 58F05. May 1996.

— 204

Quasivarieties generated by simple MV-algebras. Joan Gispert and Antoni Torrens. Mathe¬

matics _{Subject Classification:} _{03B50, 03G99, 06F99, 08C15. May 1996.}

—

205 _{Regularity ofthe lawfor}aclassofanticipatingstochastic differentialequations. Caries Rovira

and Marta Sanz-Solé. AMS _{Subject} _{Classification: 60H07, 60H10. May 1996.}

—

206 _{Effective} _{computations} _{in}_{Hamiltonian dynamics. Caries} _{Simó. AMS Subject Classification:}

58F05, 70F07. May 1996.

—

207 Small_{perturbations} _{in} _{a} _{hyperbolic stochastic partial differential} _{equation.} _{David }
Márquez-Carreras and Marta Sanz-Solé. AMS _{Subject Classification: 60H15, 60H07. May 1996.}

— 208

Transformed empincal processes and modified Kolmogorov-Smimov tests for multivariate

distributions. A. Cabaña and E. M. Cabaña. AMS _{Subject Classification:} _{Primary: 62G30,}

62G20, 62G10. Secondary: 60G15. June 1996.

—

209 _{Anticipating stochastic Volterra} _{equations.} _{Elisa Alós and} _{David Nualart.} _{AMS Subject}

Classification: _{60H07, 60H20. June 1996.}

— 210 On the

relationship between a-connections and the asymptotic properties of predictive dis¬

tributions. J.M. Corcuera and F. Giummolé. AMS 1980 _{Subjects} _{Classifications: 62F10,}

62B10, 62A99. July 1996.

—

211 Global _{efficiency. J.M. Corcuera and} _{J.M. Oller.} _{AMS} _{1980} _{Subjects Classifications: 62F10,}

62B10, 62A99. July 1996. "

—

212 Intrinsic _{analysis of} _{the} _{statistical} _{estimation.} _{J.M. Oller and J.M. Corcuera. AMS 1980}

Subjects Classifications: 62F10, 62B10, 62A99. July 1996.

—

213 A characterization _{of}monotone and regular divergences. J.M. Corcuera and F. Giummolé.

AMS 1980 _{Subjects} _{Classifications: 62F10, 62B10,} _{62A99. July} _{1996.}

—

214 On the _{depth of the fiber}_{cone} _{of}_{filtrations. Teresa Cortadellas and Santiago Zarzuela.}

Subject Classification: Primary: 13A30. Secondary: 13C14, 13C15. September 1996.