ABSTRA CT

Explicit form and robustness of martingale representations

Jean Jacod

, SylvieMeleard y

and Philip Protter z

ABSTRACT

Stochastic integral representation of martingales has been undergoing a renaissance due to questions motivated by Stochastic Finance theory. In the Brownian case one usually has formulas (of diering degrees of exactness) for the predictable integrands. We extend some of these to Markov cases where one does not necessarily have stochastic integral representation of all martingales. Moreover we study various convergence questions that arise naturally from (for example) approximations of "price processes" via Euler schemes for solutions of stochastic dierential equations. We obtain general results of the following type: let

U

,

U

n _{be random variables with decompositions:}

U

=

+Z 1 0

s

dX

s+

N

1

U

n₌

_n+Z 1 0

ns

dX

ns+

N

n 1

where

X

,

N

,

X

n_,

N

n _{are martingales. If}

X

n !

X

and

U

n !

U

, when and how does

n!

?

1 Introduction

1)

Consider a sequence

X

n _{of square-integrable martingales, which converge to another}

square-integrable martingale

X

: this convergence may hold in a strong sense (as in

IL

2)

and all the

X

n_{'s and}

X

_{are on the same probability space, or it may hold in the weak}

sense (convergence in law) and each

X

n _{is dened on its own probability space. Let also}

be a bounded continuous functional (say, on the Skorokhod space of all right continuous functions with left limits), and set

U

n_{= (}

X

n_{) and}

U

_{= (}

X

_{), so that}

U

n _{converges to}

U

. Suppose in addition that we have the martingale representation property for each

X

n

and for

X

, so we can write

U

n _and

U

_{as stochastic integrals as follows:}

U

n ₌

_n+Z 1 0

ns

dX

ns

U

=

+Z 1 0

s

dX

s

(1.1)

Laboratoire de Probabilites, Universite Paris 6, 4 place Jussieu, 75252 Paris, France (CNRS UMR

7599)

yUniversite Paris 10, MODAL'X, JE 421, 200 av. de la Republique, 92000 Nanterre, France

zMathematics and Statistics Departments, Purdue University, W.Lafayette, IN 47907-1395, USA.

where

nand

are random variables measurable w.r.t. the relevant initial

-elds, and

n

and

are predictable processes. Then an important theoretical problem is to nd whether the sequence

n _{converges in law, for a suitable topology, to}

_.

This problem has also much practical relevance. For example in nancial mathematics, suppose that

X

models the price of a stock, and

U

is a claim based upon this stock, and for simplicity the riskless bond has constant price 1. If the model is complete and with no arbitrage opportunity, then

X

is a martingale under the unique risk neutral equivalent measure, the price of the claim is the expectation

E

(

U

) =

under this measure, and we have the martingale representation property w.r.t.

X

: then the process

in (1.1) is the so-called hedging strategy. Now, for computational purposes we might want to take a discrete time approximation for

X

: e.g. a binomial approximation

X

n _{which thus}

converges in law to

X

, or an Euler approximation

X

n _{which thus converges strongly to}

X

when this process is the solution of a stochastic dierential equation. If one also has the martingale representation property for the discrete time models (as is the case for the binomial approximation), then it is important to know whether the \approximate" hedging strategies

n _{do converge in some sense to}

_{. Such questions have been touched}

upon in 6] for example.

The above brief description immediately gives rise to two kinds of problems. The rst one comes from the fact that the martingale representation property quite often does not hold: it holds under reasonably general conditions when the basic martingale

X

is continuous, but it is usually lost as soon as

X

as jumps, and in particular in the discrete time setting (except for the binomial model).

The second problem is to nd an adequate topology for which the

n_{'s might converge.}

This is not obvious, because these processes have a priori no regularity in time (they are predictable, but otherwise neither right continuous nor left continuous in general).

2)

To begin with, let us consider the rst problem described above. Let

X

be a locally square-integrable martingale on a ltered space (

F

(Ft)t

0

P

) having F =

W

tFt, and

U

be a square-integrable random variable. Using the theory of \stable" subspaces gener-ated by a martingale (see Dellacherie and Meyer 5], or 15] or 9] for this fact, as well as for all results on martingales and stochastic integrals), we have the decomposition

U

=

+Z 1 0

s

dX

s+

N

1

(1.2)

where

=

E

(

U

jF

0) and

N

is a square-integrable martingale (i.e. a martingale such that

suptj

N

tj is square-integrable), orthogonal to

X

and

is a predictable process, and this

decomposition is unique up to null sets: it comes in fact from the (unique) decomposition of the square-integrable martingale

M

t=

E

(

U

jFt) as a stochastic integral w.r.t.

X

, plus

an orthogonal term. Recall also that two locally square-integrable martingales

M

and

N

are orthogonal if their product

MN

is a local martingale, and this is denoted by

M

?

N

.

Observe that

and

N

are dened uniquely up to a

P

-null set, while

is dened uniquely up to a null set w.r.t. the following measure

on

IR

+. Here,

h

XX

i denotes the \angle" (or predictable) bracket. We will denote

the process

by

(

XU

), which is square-integrable w.r.t.

Q

X.

Section 2 of this paper is devoted to nding an \explicit" expression for the process

above: rst in the discrete time setting, where it is very simple next in some Markovian situations, when

U

has the form

U

=

f

(

Y

T) for a xed time

T

and an underlying Markov

process

Y

and

X

is a locally square-integrable martingale on this Markov process. We thus extend the well known Clark-Haussmann formula, usually given for Brownian motion, in two directions: the Brownian motion is replaced by a rather general Markov process, and we do not assume the martingale representation property. But of course we are limited to variables

U

of the form

U

=

f

(

Y

T) or more generally of the form

U

=

f

(

Y

T1

:::Y

Tk) for

xed times

T

1

< ::: < T

k.

Let us come back to the nancial interpretation of (1.2): if the martingale representa-tion property w.r.t.

X

does not hold, the variable

N

1in (1.2) is in general not equal to 0.

We are in the incomplete model case, and the process

is shown to be a risk minimizing strategy for hedging the claim

U

: see Follmer and Sondermann 7].

3)

Let us now turn to convergence results. To get an idea of what to expect as far as convergence results are concerned, here is a trivial special case: we have a sequence

U

n

of random variables tending to a limit

U

in

IL

2(

P

), and a xed locally square-integrable

martingale

X

. Writing

M

n_,

n_,

n _and

N

n _{for the terms associated with}

U

n _and

X

in (1.2), the three variables

n ;

, R

1 0 (

ns

;

s)

dX

s and

N

n 1

;

N

1 are orthogonal in

IL

2(

P

) and add up to

U

n

;

U

, so they all go to 0 in

IL

2(

P

). Since the expected value of

(R 1

0

s

dX

s) 2

is

Q

X(

2), we deduce in particular that

U

n !IL 2

(P)

U

)

(

XU

n) !IL 2

(Q X

)

(

XU

)

:

(1.4)

This leads us to consider rst the case where all locally square-integrable martingales

X

n _and

X

_{are dened on the same space (}

F

(Ft)t

0

P

) with F =

W

tFt. The simplest

result one can state in this direction is as follows:

Theorem A

Assume that

X

n _and

X

_{are locally square-integrable martingales on a}

ltered space, such that h

X

n;

XX

n;

X

it ! 0 in probability for all

t

2

IR

+, and that

U

n _{converges to}

U

_in

IL

2(

P

). Then

n converges to

in

Q

X-measure.

We also give a series of other results, which are more dicult to state, and which mainly concern discrete time approximations of a given martingale

X

, of various kinds: stepwise approximations, or Euler schemes when

X

is the solution of a stochastic dier-ential equation. All these results are proved in Section 3.

4)

Section 4 is devoted to weak convergence results. First, we take advantage of the explicit results of Section 2 in the Markov case to show that if

X

n _{is the solution of the}

equation

dX

_nt =

g

n(

X

nt;)

dZ

nt and

X

is the solution of a similar equation with

g

and

Z

,

where

Z

n _and

Z

_{are Levy processes, and if}

g

_n !

g

and

Z

n converges in law to

Z

, then

topology, when

U

n₌

f

₍

X

_{nT) and}

U

₌

f

₍

X

_{T) and}

f

_{is a dierentiable function (typically}

Z

is a Brownian motion, but the

Z

n_{'s are not, so we have the martingale representation}

property w.r.t.

X

, but not w.r.t.

X

n_{). We also give a discrete time version of this result.}

Finally, we give an analogous convergence result when

U

n _{= (}

X

n_{) and}

U

_{= (}

X

₎

for a continuous bounded function on the Skorokhod space, when

X

is the solution of an equation as above with

Z

a Brownian motion, and the

X

n_{'s are discrete time solutions}

of dierence equations converging to

X

. As an example, particularly relevant in nancial applications, let us mention the case where

X

ni+1 =

X

ni+

g

(

X

ni)

Y

ni+1

where for each

n

the (

Y

ni)i1 are i.i.d. bounded variables, centered with variance 1

=n

.

Then the processes

X

_nt=

X

n

nt]converge in law to the solution of

dX

t=

g

(

X

t)

dW

t, where

W

is a Brownian motion, as soon as

g

is Lipschitz. In this situation, with

U

n _{= (}

X

n₎

and

U

= (

X

) with as above, the processes

n _{(naturally dened as some sort of}

interpolations of the discrete time processes

(

X

n

U

n_{)) do converge in a suitable sense to}

, in law.

2 Explicit representations of the integrand

In this section our aim is to give an \explicit" form for the integrand

(

XU

) in essentially two specic cases: one is the discrete-time case, with an extension to the discretization of a continuous-time process the other is a Markov situation. It seems hopeless to obtain such an explicit form in general, but other cases are found in the literature, essentially on the Wiener space and using Malliavin calculus: see e.g. the book of Nualart 14] and the references therein.

Before starting we wish to make precise the various notions of (locally) square-integra-ble martingales used in this paper, since they play a crucial role. As said in the introduc-tion, a process

X

given on a stochastic basis, either with discrete or with continuous time, is called a square-integrable martingale if it is a martingale and if the supremum of

X

over all time is square-integrable: then the limit

X

1 exists and is a square-integrable variable.

X

is called a locally square-integrable martingale if there is a sequence

R

n of stopping

times increasing to +1, such that the process

X

stopped at any

R

nis a square-integrable

martingale. In between, we say that

X

is a martingale square-integrable on compacts if the process

X

stopped at any nite deterministic time is a square-integrable martingale: for example the Wiener process is a martingale square-integrable on compacts in this sense.

2.1 The discrete-time case

In this subsection, time is discrete: we have the basis (

F

(Fi)i

2IN

P

) with F =

W Fi

and with a given locally square-integrable martingale

X

. We also have a square-integrable variable

U

. For any process

Y

we write !

Y

i =

Y

i;

Y

i

;1. In this discrete-time case, (1.2)

becomes

U

=

+ 1 X i=1

where the series converges in

IL

2 and

i is Fi

;1-measurable and

N

is a square-integrable

martingale orthogonal to

X

. Here the orthogonality of

X

and

N

amounts to say that

E

(!

X

i!

N

ijFi

;1) = 0

8

i

1

:

(2.2)

The above conditional expectation is to be understood in the generalized sense, since the variable !

X

i!

N

i might be not integrable: it is however integrable on each Fi

;1

-measurable set f

R

n

i

g (where

R

n is as above), while nf

R

n

i

g = . The same

comment applies below.

Proposition 2.1

Assume that

X

is a locally square-integrable martingale, and let

M

i =

E

(

U

jFi). Then a version of

=

(

XU

) is given by

i =

E

(!

X

i

U

jFi ;1)

E

((!

X

i)2 jFi

;1) =

E

(!

X

i!

M

ijFi ;1)

E

((!

X

i)2 jFi

;1)

:

(2.3)

Proof

. By denition of

M

and by the property

E

(!

X

ijFi

;1) = 0 (where again the

conditional expectation is in the generalized sense), the last equality in (2.3) is obvious. Dene

by (2.3). The measurability condition is obviously met. Set

!

N

i =

E

(

U

jFi);

E

(

U

jFi ;1)

;

i!

X

i = !

M

i;

i!

X

i

and

N

i = P

ij=1!

N

j. Then

N

is a square-integrable martingale with (2.2). That (2.1)

holds is then obvious. 2

2.2 Discretization in time

Here we have a basis (

F

(Ft)t

0

P

) such that F =

W

tFt. We consider a

square-integrable martingale

X

. We also consider a locally nite subdivision

of

IR

+, consisting

of an increasing sequence

= (

T

i:

i

2

IN

) of stopping times such that

T

0 = 0

T

i

<

1 )

T

i

< T

i

+1

lim

i

T

i =1 a.s. (2.4)

The discretized process is then

X

i =

X

Ti

i

2

IN

(2.5)

which makes sense even on the set f

T

i = 1g. Then the sequence (

X

i)

i2IN is a

square-integrable martingale w.r.t. the discrete-time ltration (FT i)i

2IN. If

U

2

IL

2, we then

have the two decompositions (1.2) and (2.2), namely

U

=

8 < :

+R 1

0

s

dX

s+

N

1

+P 1

i=1

i!

X

i+

N

1

(2.6)

where

N

is a square-integrable martingale w.r.t. (Ft)t

0 orthogonal to

X

, and

N

is a

square-integrable martingale w.r.t. (FT i)i

2IN orthogonal to

X

, and

=

E

(

U

jF

it is natural to call the discretized version of the integrand

the following continuous-time process:

0

t =

i if

T

i;1

< t

T

i

i

1

:

(2.7)

In a sense, this process

0 naturally occurs if we replace

X

by the discretized version along

the subdivision

.

Our aim here is to compute

0 in terms of

. This is simple, after recalling that the

process

is square-integrable w.r.t. the nite measure

Q

X dened by (1.3), and after

introducing the

-eld P

0 on ~ =

IR

+ which is generated by the sets

D

(

T

i

T

i +1],

where

i

2

IN

and

D

2FT i:

Proposition 2.2

Assume that

X

is a square-integrable martingale. With the above no-tation we have

0 =

Q

X(

jP

0) (the conditional expectation of

w.r.t. P

0 for the nite

measure

Q

X).

Proof

. Set

A

= h

XX

i and

B

t = Rt

0

s

dA

s. If

M

t =

E

(

U

jFt), then (1.2) yields

M

t =

+Rt

0

s

dX

s+

N

t, hence

h

XM

i=

B

+h

XN

i=

B

because

X

and

N

are orthogonal. So

an application of Proposition 2.1 yields the following explicit form for

:

i =

E

(

B

Ti ;

B

T

i;1 jFT

i;1)

E

(

A

Ti ;

A

T

i;1 jFT

i;1)

:

(2.8)

For

D

2FT

i;1 we have

Q

X(1D(T i;1Ti

]

) =

E

(1D(

B

T i

;

B

T

i;1)) =

E

(1D

E

(

B

Ti ;

B

T

i;1 jFT

i;1))

:

By (2.8) and (2.7) this is equal to

E

(1D

i(

A

Ti ;

A

T

i;1)) =

Q

X(1D (T

i;1T i]

0)

:

Since

0 is obviously P

0-measurable, this implies the result. 2

Remark 2.3

Exactly the same result (with the same proof) holds if we assume that

X

is a locally square-integrable martingale, such that each stopped process (

X

Ti

t =

X

Ti ^t)t0

is a square-integrable martingale. 2

2.3 A Clark-Haussmann formula for Markov processes

In this subsection we give an alternative form of the Clark-Haussmann formula giving the integrand

(

XU

): see Nualart 14] for a general form of this formula.

The setting is as follows: we have a quasi-left continuous

IR

d_{-valued strong Markov}

process

Y

on (

F

(Ft)

P

x), where

P

x is the probability measure under which

Y

0 =

x

a.s., and we assume also that

Y

is a semimartingale under each

P

x. Let

be the jump

measure of

Y

, and (

BC

) its characteristics: we refer for this to 9], and also to 3] for the following structural results, showing in particular that (

BC

) do not depend on the starting point: there exist a continuous increasing additive functional

A

, a Borel

transition measure

F

from

IR

d _{into itself integrating}

z

j

z

j

in

y

, with all partial derivatives being continuous.

Next, our basic locally square-integrable martingale

X

is of the form

X

t =

X

0+

where

Y

c _{denotes the continuous martingale part of}

Y

_{, and \}T" denotes the transpose,

and

= (

i₎ Observe that under (2.11),

X

is well dened and is a locally square-integrable martingale under each

P

x and

P

, with angle bracket h

XX

it= We use here the traditional convention 0

0 = 0, since when

a

s= 0 the numerator in the

right side of (2.12) is also 0. Observe that the process

does not depend on the measure

m

in

P

m:=

P

=R

By hypothesis,

g

is once dierentiable in

t

and twice dierentiable in

y

with continuous partial derivatives. By It^o's formula,

+Z t

The rst three integrals above are predictable process of nite variation. The last integral may be rewritten as the sum of the stochastic integral w.r.t. the measure martingale

;

,

plus the integral w.r.t.

, which again is a predictable process of nite variation. Since

M

is a martingale, the sum of all predictable processes of nite variation must equal 0, and after a simple transformation we get

M

t =

+

In view of (2.11), this becomes

h

NX

it =

of

is thus given by (2.12), and we have proved the claim. 2

The class DT of functions for which (2.12) holds is rather restrictive. It might be of

interest to enlarge this class. To this eect, for each

x

2

IR

d we introduce the setD 0 T of all

functions

f

for which there is a sequence

f

n2DT (called an \approximating sequence")

The measure

Q

X associated by (1.3) with

X

(and relative to

P

) is here

Q

X(

d!dt

) =

P

(

d!

)

dA

t(

!

)

a

t(

!

).

Finally, letD 00

T be the subset of all

f

2D 0

T such that

y

P

t

f

(

y

) is dierentiable for

0

< t

T

, and for which there is an approximating sequence

f

n in DT such that for all

t

2(0

T

] and

y

2

IR

d we have

P

t

f

n(

y

) !

P

t

f

(

y

)

@

@y

i

P

t

f

n(

y

) !

@

@y

i

P

t

f

(

y

)

(2.13)

and that for

Q

X-almost all (

!t

) with

t

T

we have Z

F

(

Y

t;(

!

)

dz

)

(

!tz

)(

P

T;t

f

n(

Y

t;(

!

) +

z

) ;

P

T

;t

f

n(

Y

t;(

!

))) !

Z

F

(

Y

t;(

!

)

dz

)

(

!tz

)(

P

T;t

f

(

Y

t;(

!

) +

z

) ;

P

T

;t

f

(

Y

t;(

!

)))

:

(2.14)

Observe that (2.13) implies (2.14) as soon as

f

, the

f

n's and the_@y@i

P

t

f

n's are uniformly

bounded for each

t

, by virtue of (2.11) and of the fact that R

F

(

ydz

)(j

z

j 2

^1)

<

1.

Corollary 2.5

a) If

f

2 D 0

T with the approximating sequence

f

n, then a version of the

process

(

Xf

(

Y

T)) is the limit of

(

Xf

n(

Y

T)) in

IL

2(

Q

X).

b) If further

f

2D 00

T, then a version of the process

(

Xf

(

Y

T)) is given by (2.12) for

s

T

, and by 0 for

s > T

.

Proof

. The claim a) readily follows from (1.4). Assume now that

f

2D 00

T, with the

ap-proximating sequence

f

n. Then if

0is given by (2.12), on the one hand

(

Xf

n(

Y

T))s(

!

)!

0

s(

!

) for

Q

X-almost all (

!s

), and on the other hand (1.4) holds: hence

0 =

(

Xf

(

Y

T))

Q

X-a.s., and we have b). 2

2.4 A particular case

Theorem 2.4 and its corollary are not quite satisfactory, because they give

(

XU

) for a variable

U

of the form

U

=

f

(

Y

T), while one would like to have it for

U

=

f

(

X

T). It

becomes more satisfactory when

X

itself is Markov. We give in some detail a simple case of this situation, namely when

X

is the solution of the equation

dX

=

g

(

X

;)

dZ

, where

Z

is a 1-dimensional Levy process and

g

a smooth enough coecient.

Since we wish

X

to be a locally square-integrable martingale, it is natural to assume rst that the Levy process

Z

, which is dened on some space (

F

(Ft)t

0

P

), is a

locally square-integrable martingale itself. This in fact implies that it is then a martingale square-integrable on compacts, and its characteristic function has the form

E

e

iuZt

= exp

t

;

cu

2

2 +

Z

F

(

dx

)(

e

iux;1;

iux

) !

(2.15)

where

c

0 and the Levy measure

F

integrates

x

2. We set

~

c

=

c

+Z

soh

ZZ

it= ~

ct

, and of course we assume that ~

c >

0 (otherwise

Z

= 0 and what follows is

empty). Next we have a continuously dierentiable function

g

with bounded derivative

g

0,

and for any

x

we consider the solution

X

x_{of the following stochastic dierential equation:}

X

xt =

x

+Z t 0

g

(

X

xs;)

dZ

s (2.17)

A classical argument (see (5.2) in the Appendix) yields that

X

x _{is a square-integrable}

martingale over each nite interval 0

T

]. Further, the solution of the following linear equation

X

0xt = 1 + Z t

0

g

0(

X

xs ;)

X

0xs

;

dZ

s

(2.18)

is also a square-integrable martingale over each nite interval 0

T

]. So for each measurable function

f

with at most linear growth we can set

P

t

f

(

x

) =

E

(

f

(

X

xt))

Q

t

f

(

x

) =

E

(

f

(

X

xt)

X

0xt

)

:

(2.19)

Observe that (

P

t) is the semi-group of

X

x, which is a Markov process. We then have:

Theorem 2.6

Assume that

g

has a continuous and bounded derivative.

a) For any

T

2

IR

+ and any di erentiable function

f

with bounded derivative

f

0 the

variable

f

(

X

_xT) is square-integrable, and a version of

(

X

x

f

₍

X

_{xT)) is given by}

s(

X

x

f

(

X

xT)) =

(

sX

xs;)1

0T](

s

)

(2.20)

where

(

sy

) =

Q

T;s

f

0(

y

) + 1~

c

Z

F

(

dz

)

z

2 Z

1 0

(

Q

T;s

f

0(

y

+

g

(

y

)

zu

) ;

Q

T

;s

f

0(

y

))

du:

(2.21)

b) The same holds when

f

is the di erence of two convex functions, with a right derivative

f

0

r bounded and

f

0 above replaced by

f

0

r, provided we have

P

t(

y:

) has no atom for all

t

2(0

T

],

y

2

IR

.

In the last claim one can of course replace the right derivative

f

0

rby the left derivative

f

0

l. The last condition is obviously satised when

P

t(

x:

) has a density: this is the case

when

c >

0 as soon as

g

does not vanish (or, does not vanish in the set in which the process

X

x _{takes its values). When}

c

_{= 0, one can nd conditions implying the existence}

of a density in e.g. 2].

Proof

. 1) Our assumptions always imply that

f

and

g

have at most linear growth. Then the property

f

(

X

_xT)2

IL

2(

P

) follows from (5.2) in the Appendix.

By virtue of Theorem 2.4 applied to

Y

=

X

x_{, it suces to prove that}

f

2 DT and

that (2.12) reduces to (2.21) in our situation. The rst property is proved in Lemma 5.1 of the Appendix. So it remains to identify (2.12) with (2.21). With

Y

=

X

we have

b

= 0,

c

(

y

) =

cg

(

y

)2,

A

t=

t

and

F

(

y:

) is the image of the measure

F

under the map

z

7!

g

(

y

)

z

,

while in (2.10) we must take

s= 1 and

(

sz

) =

z

. So if

a

(

y

) =

cg

(

y

)2+ R

F

(

dz

)

z

2

g

(

y

)2,

(2.11) becomes

a

s =

a

(

X

s;), while

r

P

t

f

=

Q

t

f

0 by (5.3). Then (2.12) yields that we

have (2.20) with

0 instead of

, given by

0

(

sy

) =

g

(

y

)2

cQ

T;s

f

0(

y

) + R

F

(

dz

)

zg

(

y

)(

P

T;s

f

(

y

+

g

(

y

)

z

) ;

P

T

;s

f

(

y

))

g

(

y

)2(

c

+ R

F

(

dz

)

z

2)

if

g

(

y

) 6= 0, and

0(

sy

) = 0 if

g

(

y

) = 0. By Taylor's formula and again the property r

P

t

f

=

Q

t

f

0 and (2.16) we note that

0(

sy

) equals

(

sy

) as given by (2.21) when

g

(

y

) 6= 0. Finally the process

(

X

x

f

(

X

xT)) is unique up to a

Q

X

x-null set, and the set f(

!t

) :

g

(

X

xt

;(

!

)) = 0

g is

Q

X

x-negligible: hence (2.20) gives a version of

(

X

x

f

(

X

xT)).

3) Second, we prove the result when

f

and

g

are once continuously dierentiable with bounded derivatives

f

0 and

g

0, and further

f

is bounded, and when

Z

is a locally

square-integrable martingale.

First we replace

Z

by a sequence of Levy processes

Z

n _{obtained by truncating the}

jumps of

Z

of size bigger than

n

. More precisely, we may write

Z

t =

Z

ct+ Z t

0 Z

IR

z

(

;

)(

dsdz

)

where

Z

c _{is the continuous martingale part of}

Z

_{(of the form}

cW

_{, where}

W

_{is a Wiener}

process),

is the jump measure of

Z

and

(

dsdz

) =

ds

F

(

dz

). Then we set

Z

nt =

Z

ct+Z t 0

Z

IR

z

1fjzj ng(

;

)(

dsdz

)

:

We have h

ZZ

it = ~

ct

where ~

c

=

c

+ R

F

(

dz

)

z

2, and

h

Z

n

Z

nit = ~

c

n

t

where ~

c

n =

c

+ R

F

(

dz

)

z

21

fjzj ng, and obviously

h

Z

;

Z

n

Z

;

Z

nit ! 0

:

(2.22)

Next, let

be an innitely dierentiable function with compact support and inte-gral equal to 1. Then we replace

g

by

g

n(

x

) = R

n

(

ny

)

g

(

x

;

y

)

dy

and

f

by

f

n(

x

) = R

n

(

ny

)

f

(

x

;

y

)

dy

. Thus

g

nand

f

nare innitely dierentiable with bounded derivatives

of all orders, and there exists

K

such that for all

n

:

j

g

n(0)j

K

j

g

0

n(

x

)j

K

j

f

n(

x

)j

K

j

f

0

n(

x

)j

K

(2.23)

and moreover

g

n !

g g

0 n !

g

0

f

n !

f f

0 n !

f

0 locally uniformly

:

(2.24)

Now, denote by

X

nx _and

X

0nx the solutions of (2.17) and (2.18), with

Z

and

g

f

nsatisfy the conditions of Step 2,

n=

(

X

nx

f

n(

X

_Tnx)) is given by (2.20), with

ngiven

By virtue of (2.22) and (2.24), stability results for stochastic dierential equations (see 10]) imply that (

X

nyn

X

0ny

n) converges locally uniformly (in time) in probability

to (

X

y

X

0y) for any sequence

y

in the next section (not based upon the present theorem, of course), namely Theorem 3.3: this theorem asserts that

n _{converges to}

₍

X

x

f

₍

X

_{xT)) in}

Q

_{X-measure. Then, since}

ns=

n(

sX

snx;)10T)(

s

) and since

X

nx_{converges locally uniformly in time, in probability,}

to

X

x_{, we deduce (2.20) from (2.26).}

4) For (a) it remains to consider the case where

f

has a continuous bounded derivative but is not bounded itself. We can nd a sequence (

f

n) of bounded continuously

dier-entiable functions such that j

f

0

n(

x

)j

K

for some constant

K

and

f

n(

x

) =

f

(

x

) for all j

x

j

n

and j

f

nj j

f

j. Then

n =

(

X

x

f

n(

X

xT)) is given by (2.20) with

n given by

(2.21), where

f

is substituted with

f

n. That

f

n(

X

xT) !

f

(

X

xT) in

IL

2(

P

) and that (2.25)

holds with

Q

tinstead of

Q

nt are obvious by the previous estimates on

f

nand

f

0

nand (5.2)

of the Appendix. It follows as above that

n !

pointwise, where

is given by (2.21).

Then by (1.4) we have (2.20).

5) It remains to prove (b). We set

f

n(

x

) = R

n

(

ny

)

f

(

x

;

y

)

dy

as in Step 3. Then

f

n !

f

locally uniformly, while

f

0 n !

f

0

r everywhere except on an at most countable set

D

, and we still have j

f

0

as in Step 3, we have

f

n(

X

yt)!

f

(

X

yt) and

f

proof follows as in Step 4. 2

Remark 2.7

When

c

= 0 and

F

is a nite measure, the process

Z

is a compensated compound Poisson process, and the situation is much simpler. One can show with the same methods used in Steps 3 or 4 above that the result hold without any dierentiability condition. We need

f

to be continuous, and both

f

and

g

with linear growth, and (2.21) takes the following simple form (with 0

0 = 0):

Of course in this simple situation we could also write an \elementary" proof which looks like the proof of Proposition 2.1. This is no surprise, since the compound Poisson case has a \discrete" structure.

Remark 2.8

We have considered above the \homogeneous" situation, where the coe-cient

g

does not depend on time. Similar formulas would obviously hold when the coe-cient depends on time, and also when the process

Z

is a non-homogeneous process with independent increments.

Remark 2.9

Let

f

be a function on

IR

k _{(say, with linear growth) and 0}

< T

1

< ::: <

By iteration of the previous result we then get that

s = Xk

Of course we need some smoothness conditions of

f

to do that: that

f

is continuously dierentiable with all partial derivatives bounded is enough, in which case we need to reproduce the proof of the previous theorem.

We do not have an explicit form for

(

XU

) when

U

is a function of the whole path of

X

x _{over 0}

T

_{]. But the variables of the form above are dense into the set of}

to the Clark-Haussmann formula in the Wiener case, see e.g. Nualart 14]: in this case one has an \explicit" form for

(

X

x

U

_{) for variables}

U

_{that are smooth in the Malliavin}

sense (and thus include the variables

f

(

X

_xT1

:::

) for smooth

f

's). But this approach is

limited to the Wiener space, and the explicit form involves the not so explicit Malliavin derivatives and predictable projections of such derivatives.

3 Strong convergence results

3.1 Discretization of a process

Here we consider the situation of Section 2.2, and we look at what happens when we have a sequence of subdivisions whose meshes go to 0. More precisely, we have a square-integrable martingale

X

on a basis (

F

(Ft)t

0

P

) with F =

W

tFt, and for each

n

a subdivision

n= (

T

(

ni

) :

i

2

IN

) satisfying (2.4). The sequence (

n) satises

sup_i

1

(

T

(

ni

)^

t

;

T

(

ni

;1)^

t

) ! 0 a.s. as

n

!1

:

(3.1)

Then set

X

ni=

X

T(ni): for each

n

the sequence (

X

ni)i

2IN is a square-integrable martingale

w.r.t. (FT (ni))i

2IN. Finally, let

U

2

IL

2(

P

) be xed. We have the rst decomposition

(2.6), and the second one for each

n

with the process

n_{, and we associate with}

n _and

_n

the process

0n by (2.7). Then we have:

Theorem 3.1

Under (3.1), and if

X

is a square-integrable martingale and

U

2

IL

2(

P

),

the functions

0n tend to

in

IL

2(

Q

X).

Proof

. For each

n

we endow the space ~ with the

-eld P 0

n generated by the sets

D

(

T

(

ni

;1)

T

(

ni

)] :=f(

!t

) :

!

2

DT

(

ni

;1)(

!

)

< t

T

(

ni

)(

!

)g, where

i

1

and

D

2 FT

(ni;1). By virtue of Proposition 2.2, we have

0n =

Q

X(

jP 0

n). Recall that

here

Q

X is a nite measure.

The sequence (

0n) is bounded in

IL

2(

Q

X), and thus is in a compact set for the weak

topology in

IL

2(

Q

X). So there exists a subsequence, again denoted by

0n for simplicity,

which converges weakly to a variable

0 in

IL

2(

Q

X).

Let us rst show that

0 =

Q

X-a.s. Take

= 1D]st], where

D

is

Fs-measurable.

Then

Q

X(

0n

)

!

Q

X(

0

). Consider the two stopping times

S

n = inff

T

(

ni

) :

i

2

INT

(

ni

)

s

g and

T

n= inff

T

(

ni

) :

i

2

INT

(

ni

)

t

g. Then

Q

X(

0n

) =

Q

X(

0n1

D(S nTn

]) +

Q

X(

0n1

D(sS n

])

;

Q

X(

0n1

D(tT n

])

Q

X(

) =

Q

X(

1D(S nT

n]) +

Q

X(

1D(sSn])

;

Q

X(

1D

(tTn])

:

9 =

(3.2)

If

A

(

ns"

) =f

S

n

> s

+

"

g we have

Q

X(

D

(

sS

n])

Q

X(

A

(

ns"

)

IR

+) +

Q

X(

(

ss

+

"

])

:

Since

Q

X(

:

IR

+) is absolutely continuous w.r.t.

P

and since (3.1) implies

P

(

A

(

ns"

)) !

0 as

n

! 1, we also have

Q

X(

A

(

ns"

)

IR

+)

other hand,

Q

X((

ss

+

"

])!0 as

"

!0 because

Q

X is a nite measure, so we deduce

that

Q

X(

D

(

sS

n])! 0. The variables

0n being

Q

X-uniformly integrable, we deduce

that

Q

X(

0n1

X(

). Then by a monotone class argument, this relation holds for all

bounded predictable

, which yields

=

0

Q

indeed strong convergence. Thus the

IL

2(

Q

X)-convergence of the original sequence

0n to

follows. 2

Remark 3.2

When the subdivisions (

n) are ner and ner, the sequence of

-elds P

square-integrable martingale and the convergence to

readily follows from the fact that

P 0

n increases toP up to

P

-null sets (by (3.1)). 2

3.2 A general continuous time convergence theorem

We have again a stochastic basis (

F

(Ft)t

0

P

) with F =

W

Ft, supporting locally

square-integrable martingales

X

and

X

n _{and square-integrable variables}

U

n _and

U

_{. We}

have the following (unique) decompositions, as in (1.2):

U

=

+R

n_{) is a square-integrable}

mar-tingale orthogonal to

X

(resp. to

X

n_).

Below, we consider again the measure

Q

X associated with

X

by (1.3). It is not

necessarily nite, so we recall that

n !Q

X

means that

n

!

in

R

-measure for one

(hence for all) nite measure

R

equivalent to

Q

X. Our main result is the following:

Theorem 3.3

Assume that

U

n !

U

in

IL

Proof

. 1) To begin with, we introduce the following orthogonal decompositions for the locally square-integrable martingales

X

n_{and the square-integrable martingales}

N

n _(recall

(3.3)) below the processes

L

n_{are locally square-integrable martingales and}

T

n_are

square-integrable martingales (recall also that the orthogonality between local martingales is denoted by ?):

X

_nt =

X

n₊Z t

N

nt = Z t 0

ns

dX

s+

T

nt

T

n?

X:

(3.6)

In what follows we prove a bit more than is strictly necessary for the present theorem, but the following facts will also be used in the subsequent results. The orthogonality of

X

n _and

N

n _yields Z t

0

ns

ns

d

h

XX

is+h

L

n

T

n

it = 0

8

t

a.s. (3.7)

We also have

h

X

n;

XX

n;

X

it = Z t

0

(

ns;1) 2

d

h

XX

is+h

L

n

L

nit

(3.8)

Q

X((

n

n+

n;

) 2)

E

((

U

n;

U

) 2)

! 0

(3.9)

E

(h

T

n

T

ni 1)

E

(h

N

n

N

ni 1)

E

(R 1 0 (

ns)

2

d

h

L

n

L

nis)

E

(( R

1

0

ns

dX

ns) 2)

9 =

E

((

U

n) 2)

K

(3.10)

for some constant

K

, and where we have used that

U

n !

U

in

IL

2(

P

) for the last two

properties.

2) After these preliminaries, we can go the proof of our claim. First, we can write the (pathwise) Lebesgue decomposition of the process h

L

n

T

ni, which is of locally bounded

variation, w.r.t. the increasing process h

XX

i as h

L

n

T

nit= Rt

0

ns

d

h

XX

is+

A

nt, where

A

n _{is a function of locally bounded variation which is singular w.r.t. to} h

XX

i. Then

(3.7) yields

n

n₊

n _{= 0}

Q

_X ;a.s. (3.11)

But it is well known by Kunita-Watanabe inequality that the variation of the process

h

L

n

T

ni over 0

t

] is smaller than or equal to p

h

L

n

L

nit p

h

T

n

T

nit, while by the above

Lebesgue decomposition it is bigger than Rt 0

j

nsj

d

h

XX

is. Then we readily deduce from

(3.4), (3.8) and (3.10) that Rt 0

j

nsj

d

h

XX

is !P 0 for all

t

, so in view of (3.11) we get

n

n !Q

X 0

:

(3.12)

Next, (3.4) and (3.8) on the one hand, (3.9) on the other hand, give us:

n

n₊

n !Q

X

n !Q

X 1

:

(3.13)

Now, combining (3.12) and (3.13) readily gives us

n!Q X

.

2

Associated with this theorem, we have a result about the rate of convergence:

Theorem 3.4

Assume that

U

n !

U

in

IL

2(

P

) and that

X

and

X

n are locally

square-integrable martingales satisfying (3.4). Assume further that there is a sequence (

a

n) in

IR

+ going to +

1 such that the sequence (

a

n(

U

n;

U

) :

n

1) is bounded in

IL

2(

P

) and

that for each

t

the sequence of variables (

a

2

nh

X

n;

XX

n;

X

it :

n

2

IN

) is uniformly

tight. Then the sequence (

a

n(

n;

) :

n

2

IN

) is uniformly tight with respect to any nite

Proof

. 1) We choose a nite measure

R

equivalent to

Q

X. Let us rst recall that

if (

u

n_{) is a sequence of processes such that for all}

t

_{the sequence of random variables}

(Rt 0

j

u

nsj

d

h

XX

is :

n

1) is tight, then the sequence (

u

n) is

R

-tight.

Applying this to (3.8) and (3.9) multiplied by

a

2

ngives that

the two sequences

a

n(

n;1)

a

n(

n

n₊

n

;

) are

R

-tight. (3.14)

We also deduce from (3.8) that for each

t

the sequence (h

L

n

L

nit:

n

1) is tight. Exactly

as in the last step of the previous proof, we deduce that the sequences (Rt 0

a

n

j

nsj

d

h

XX

is:

n

1) are tight, hence the sequence (

a

n

n) is

R

-tight. In view of (3.11) we deduce that

the sequence

a

n

n

n is

R

-tight. (3.15)

Now, we can write

a

n(

n;

) =

a

n

n₍₁

;

n_{)(1 +}

n_{) +}

a

_n

n₍

n

n₊

n

;

);

a

n

n

n₊

a

_n

₍

n ;1)

:

We also know that the sequences (

n_{) and (}

n_{) are}

R

_{-tight (by (3.13) and the previous}

theorem). Then the result readily follows from (3.14) and (3.15). 2

3.3 A discrete version of Section (3.2)

Now we consider for each

n

a subdivision

n= (

T

(

ni

) :

i

2

IN

) of stopping times on the

basis (

F

(Ft)

P

) with F = W

Ft, satisfying (2.4), and we suppose that the sequence

(

n) satises (3.1). For each

n

we have a square-integrable martingale

X

n and a

square-integrable variable

U

n_{. Analoguous to (2.5), we set}

X

_{ni =}

X

_nT

(ni). As in (2.6) we have

(3.3), as well as the decomposition

U

n ₌

n₊ 1 X i=1

ni!

X

ni+

N

n

1

:

(3.16)

Then, as in (2.7) we set

0nt =

ni if

T

(

ni

;1)

< t

T

(

ni

)

:

(3.17)

Theorem 3.5

Assume that

U

n!

U

in

IL

2(

P

) and that

X

n and

X

are square-integrable

martingales and that

E

(h

X

n

;

XX

n

;

X

i 1)

! 0

:

(3.18)

Then the sequence

0n converges to

=

(

XU

) in

Q

X-measure.

Proof

. In view of Proposition 2.2 we have

0n=

Q

Xn(

n jP

0

n), whereP 0

n is the

-eld on

~ dened in the proof of Theorem 3.1. Let also P be the predictable

-eld on ~. We

We can nd a probability measure

R

on (~

P) which dominates all the nite measures

Q

X and

Q

Ln, and such that

Q

X

aR

for some constant

a

. We can thus nd nonnegative

R

-integrable and predictable functions

VV

n _{such that}

V

a

and

Q

X =

V

R

Q

L

Now, (3.18) and (3.8), then (3.9), then (3.10), yield

n

3.3, and in view of (3.19), (3.21) and (3.11), we obtain

n

n !IL

1(

R

). Putting all these results together yields

W

n !IL

On the other hand, Bayes' rule yields

0n =

Q

Now let us apply the proof of Theorem 3.1 to

R

instead of

Q

X: rst with

V

instead

of

, which, since

V

is bounded, yields

R

(

V

jP

2(

R

). Combining this with (3.23) yields

R

(

W

njP

In the previous theorem, we would like to replace (3.18) by (3.4), with

X

and

X

n

3.4 Application to the Euler scheme

We apply the previous results to the Euler approximation scheme for a stochastic dier-ential equation. The setting, similar to that of Subsection 2.4, is as follows: we have a locally square-integrable martingale

Z

on a space (

F

(Ft)t

0

P

) with F =

W

Ftand a

locally Lipschitz continuous function with linear growth

g

, and

X

is the (unique) solution of the following stochastic equation (where

X

0 is a given

F

0-measurable square-integrable

variable):

X

t =

X

0+ Z t

0

g

(

X

s;)

dZ

s

:

(3.25)

In comparison with Subsection 2.4, we relax the assumptions on

g

and

Z

and allow an arbitrary initial condition

X

0. We also consider subdivisions

n= (

T

(

ni

) :

i

2

IN

) of

stopping times satisfying (2.4), such that (3.1) holds. With

n

0 = 0 and

nt=

T

(

ni

;1)

for

T

(

ni

;1)

< t

T

(

ni

), we have the \continuous" Euler approximation at stage

n

,

which is the solution of

X

nt =

X

0+ Z t

0

g

(

X

nn

s)

dZ

s

:

(3.26)

Let

U

and

U

n _{be square-integrable variables such that}

U

n !

U

in

IL

2: typically

U

=

f

(

X

t) and

U

n =

f

(

X

nt) for some

t

, where

f

is a bounded continuous function. In

this case, since by a well known result (see e.g. 11])

X

n _{goes in probability to}

X

_{, locally}

uniformly in time, we do indeed have

U

n!

U

in

IL

2.

Note that

X

and

X

n_{are locally square-integrable martingales. Recall also (1.3). Then}

as a corollary of Theorem 3.3 we get:

Theorem 3.6

Let

U

n !

U

in

IL

2(

P

), and let

=

(

XU

) and

n =

(

X

n

U

n). Then

n!Q X

.

Proof

. It is enough to prove that (3.4) holds. Note

h

X

n;

XX

n;

X

it = Z t

0

(

g

(

X

_nn s)

;

g

(

X

s ;))

2

d

h

ZZ

is 2

Z t 0

(

g

(

X

_nn s)

;

g

(

X

n s))

2

d

h

ZZ

is+ 2 Z t

0

(

g

(

X

n s)

;

g

(

X

s ;))

2

d

h

ZZ

is

:

We have already mentioned that

X

n _{goes to}

X

_{uniformly in time, in}

P

_{-measure. Thus}

the sequence sups t(j

X

sj+j

X

nsj) is bounded in probability and, since

g

is continuous and

locally bounded, it follows that the rst term in the right side of the above inequality goes to 0 in probability for each

t

. On the other hand

ns !

s

and

ns

< s

for all

s >

0: thus

for all

!

and all

s >

0 we have

X

n s(

!

)

!

X

s

;(

!

). Thus, by the continuity of

g

again,

the second term in the right side of the above inequality goes to 0 for all

!

: hence (3.4) is proved. 2

Let us now pass to the \discrete" Euler approximation:

Here we have some problems of integrability, because in order to apply the previous results we need each

X

n _{to be a discrete-time locally square-integrable martingale. On the other}

hand we do not wish to assume that

X

,

X

n _and

Z

_{are square-integrable up to innity.}

In order to resolve this problem, we suppose that

Z

is a martingale square-integrable on compacts, and also that there is a constant

K

such that for all

in

:

T

(

ni

);

T

(

ni

;1)

K:

(3.28)

Then the process

Z

stopped at any time

T

(

ni

) (which is bounded by (3.28)), is a square-integrable martingale, and

Z

ni =

Z

T(ni) is a martingale square-integrable on compacts

w.r.t. (FT (ni))i

0. Due to the linear growth of

g

, and similarly to (5.2) of the Appendix,

one also checks easily that

X

ni=

X

_nT(ni)is also a martingale square-integrable on compacts

w.r.t. the ltration (FT (ni))i

0.

As soon as

U

n _{is square-integrable, analogous to (3.16), we may thus write}

U

n ₌

and orthogonal to the discrete time locally square-integrable martingale (

X

_ni)i0 (resp.

(

Z

ni)i0), and

ni and

ni are FT

(ni;1)-measurable. Further, we set

0nt =

ni

Theorem 3.7

Assume (3.28) and that

Z

is a martingale square-integrable on compacts. If the variables

U

n _and

U

_are FT-measurable for some

T

2

IR

same if we replace

Z

by the stopped process

Z

T0

in (3.29), and also

=

(

Z

T0

U

). So we can assume that

Z

=

Z

T _{is square-integrable. Applying Theorem 3.5 with}

X

n₌

X

₌

Z

then yields that

0n

The last three terms above are orthogonal martingales, and thus by identication with the previous expression we get that a.s.:

ni

g

(

X

ni;1) =

ni1fg( Xn

i;1

This yields

0ns

g

(

X

n n s) =

0ns1 fg(X

n n s

)6=0g

Q

Z-a.s. (3.32)

A similar argument shows that

s

g

(

X

s;) =

s1 fg(X

s;

)6=0g

Q

Z-a.s. (3.33)

As seen in the proof of Theorem 3.6,

g

(

X

_nn s)

!

g

(

X

s

;) in probability for all

s

. Then

one deduces from the fact that

0n

!

in

Q

Z-measure and from (3.32) and (3.33) that

0n

!

in

Q

Z-measure on the set f(

!t

) : j

g

(

X

t ;(

!

)

j

"

g, for every

" >

0. Hence the

same convergence holds also on the set

A

= f(

!t

) :

g

(

X

t ;(

!

))

6

= 0g, and since

Q

X is

absolutely continuous w.r.t.

Q

Z and does not charge the complement of

A

, we deduce

that

0n

!

in

Q

X-measure. 2

Remark 3.8

The same proof as above would also work for Theorem 3.6: we have

n₌

(

ZU

n₎!

=

(

ZU

) by (1.4), while the relation (3.32) holds between

n and

n.

4 Weak convergence results

In this section we consider the weak convergence of integrands: we have a sequence

X

n _{of locally square-integrable martingales, each dened on its own probability space}

(n

Fn

(Fnt)

P

n), and for each

n

a square-integrable variable

U

n on the relevant space.

The aim is to prove that if (

X

n

U

n_{) converges in law to (}

XU

_{), with}

X

_{a locally}

square-integrable martingale and

U

a square-integrable variable on the space (

F

(Ft)

P

), then

(

X

n

₍

X

n

U

n_{)) converges in law to (}

X

₍

XU

_{)) in some sense.}

It seems impossible to solve such a general problem, so we will concentrate on some particular cases.

4.1 Application of the Clark-Haussmann formula

Here we consider a sequence of processes of the form studied in Subsection 2.4. More precisely, we have

Z

,

g

and

X

x _{as in this subsection, given on (}

F

(Ft)

P

). For each

n

,

we also have a Levy process

Z

n_{which is a martingale square-integrable on compacts on a}

space (n

Fn

(Fnt)

P

n), satisfying (2.15) with

c

n and

F

n, and as before, we assume that

the numbers

~

c

=

c

+Z

F

(

dz

)

z

2

~

c

n =

c

n+ Z

F

n(

dz

)

z

2

are nite and strictly positive.

Then we have dierentiable functions

g

n, and we consider the equations (2.17) and

(2.18) w.r.t.

Z

n _and

g

_{n, and whose solutions are denoted by}

X

nx _and

X

0nx. We make

the following assumptions. First on

g

n and

g

:

j

g

n(0)j

K

j

g

0

n(

x

)j

K

j

g

0

n(

x

);

g

0

n(

y

)j

K

j

x

;

y

j

(4.1)

g

n

g

g

0

Next on

Z

n _and

Z

_{: we basically assume that}

Z

n _{converges in law to}

Z

_{, plus a slightly}

stronger assumption which is reminiscent of the Lindeberg condition more precisely we assume that

~

c

n!

c

~ R

F

n(

dz

)

h

(

z

) ! R

F

(

dz

)

h

(

z

)

for

h

continuous, bounded and vanishing in a neighborhood of 0,

9 =

(4.3)

A

(

x

) = sup_n Z

F

n(

dz

)

z

21 fjzjxg

! 0 as

x

!1

:

(4.4)

These two conditions imply that the second convergence in (4.3) also holds when

h

is con-tinuous, and

h

(

x

) =

O

(

x

2) at innity, and

h

(

x

) =

o

(

x

2) at 0. They imply the convergence

in law of

Z

n _to

Z

_{(see e.g. 9]).}

Then we can state:

Theorem 4.1

Assume (4.1), (4.2), (4.3) and (4.4). Let

f

be a di erentiable function with a bounded and Lipschitz derivative and

T >

0. The processes

=

(

X

x

f

₍

X

_{xT)) and}

n₌

₍

X

nx

f

₍

X

nx

T )) have versions which are left continuous with right limits, and if we

set

(+)s= limt#st>s

t and

(+)ns= limt#st>s

nt, the processes (

X

nx

n_{(+)) converge in}

law for the Skorokhod topology on

IR

2 to(

X

x

(+)).

Proof

. 1) A version of

is given by (2.20), with

given by (2.21). We wish to prove here that this version is left continuous with right limits. We can rewrite

as

k

(

syz

) = R 1 0(

Q

T

;s

f

0(

y

+

uzg

(

y

)) ;

Q

T

;s

f

0(

y

))

du

(

sy

) =

Q

T;s

f

0(

y

) +1 ~ cR

F

(

dz

)

z

2

k

(

syz

)

:

9 =

(4.5)

In view of (5.9) of the appendix and of the properties of

f

, we have for 0

s

t

T

: j

k

(

syz

)j

C

j

k

(

syz

);

k

(

tyz

)j

C

(1 +j

y

j(1 +j

z

j)) p

t

;

s

9 =

(4.6)

for a constant

C

. Now, (4.3) and (4.4) yield that R

F

(

dz

)

z

21 fjzjxg

A

(

x

), so the above

estimates and (5.9) again yield that for all

N >

1_

T

;1=4 and for two other constants

C

0,

C

00:

j

(

sy

);

(

ty

)j

C

0(1+

N

j

y

j) p

t

;

s

+

CA

(

N

)

C

00(1+

j

y

j)(

t

;

s

)

1=4+

A

((

t

;

s

)

;1=4)

(4.7) (take

N

= (

t

;

s

)

;1=4 to get the last estimate). On the other hand, as in the proof of

Theorem 2.6 we have (2.25) with

Q

t instead of

Q

nt, hence it is clear from (4.5) and (4.7)

and another application of (5.9) that (

sy

) 7!

(

sy

) is continuous: hence (2.20) readily

yields that

is left continuous with right limits.

2) Similarly, for each

n

we associate with

Z

n_,

F

_{n, ~}

c

_n,

g

_{n the functions}

k

_{n and}

_{n given}

by (4.5). Exactly as before, we obtain that

n_{, as given by (2.20) with}

_nand

X

nx_instead

of (4.1), (4.3) and (4.4), it is clear that the estimates (4.6) and (4.7) hold for all

k

n and

n with constants

C

,

C

0 independent of

n

.

3) Now we apply again the stability results of 10]: by (4.1), (4.2) and (4.3), for any sequence

y

n!

y

, the processes (

X

ny

n

X

0ny

n) converge in law to (

X

y

X

0y), and further

the estimate (5.2) of the Appendix yields that each sequence (

X

0ny n

t )n1 is uniformly

integrable. Hence if

Q

_nt is associated with (

X

nx

X

0nx) by (2.19) we readily deduce that

(2.25) holds.

We will deduce that if

y

n!

y

and

s

n!

s

we have

n(

s

n

y

n) !

(

sy

)

:

(4.8)

Indeed, by (2.25) and (5.9) of the Appendix, we have

Q

_nT;s n

f

0(

y

n) !

Q

T ;s

f

0(

y

), hence

also

k

n(

s

n

y

n

z

n) !

k

(

syz

) as soon as

z

n !

z

because of (4.2). Hence for (4.8) it

remains to prove that if

h

n(

z

) =

k

n(

s

n

y

n

z

) and

h

(

z

) =

k

(

syz

)

1 ~

c

n Z

F

n(

dz

)

z

2

h

n(

z

) !

1 ~

c

Z

F

(

dz

)

z

2

h

(

z

)

(4.9)

knowing that

h

n(

z

n)!

h

(

z

) if

z

n!

z

and

h

is continuous andj

h

nj

C

for a constant

C

.

Now, consider the probability measures

G

n(

dz

) = 1 ~

cn(

F

n(

dz

)

z

2+

c

n

"

0(

dz

)) and

G

(

dz

) = 1

~

c(

F

(

dz

)

z

2+

c"

0(

dz

). Since

h

n(0) =

h

(0) = 0, (4.9) reads as

G

n(

h

n)

!

G

(

h

). Furthermore

(4.3) and (4.4) imply that

G

nconverges weakly to

G

.

By the Skorokhod representation theorem we can nd random variables

V

n,

V

on a

suitable probability space, such that

V

n and

V

have laws

G

n and

G

, and that

V

n !

V

everywhere. Then

G

n(

h

n) =

E

(

h

n(

V

n)) and

G

(

h

) =

E

(

h

(

V

)), and the fact that

h

n(

z

n)!

h

(

z

) if

z

n!

z

yields that

h

n(

V

n)!

h

(

V

) everywhere. Since furtherj

h

nj

C

, it follows

that

G

n(

h

n)!

G

(

h

): hence (4.9) and (4.8) are proved.

4) Observe that

(+)ns =

n(

sX

snx)10T)(

s

) and

(+)s =

(

sX

xs)10T)(

s

). Further,

(4.8) implies that

n!

locally uniformly. Since

X

nx converges in law to

X

x and since

X

x _{has no xed time of discontinuity, an application of the continuous mapping theorem}

yields that (

X

nx

₍₊₎n_{) converges in law for the Skorokhod topology to (}

X

x

_(+)). 2

Remark 4.2

Suppose now that

f

is a continuously dierentiable function on

IR

k _with

all partial derivatives bounded and Lipschitz, and let 0

< T

1

< ::: < T

k. Set

=

(

X

x

f

₍

X

_xT

1

:::X

xT

k)) and

n₌

₍

X

nx

f

₍

X

_Tnx 1

:::X

nx

Tk )), as in Remark 2.9. Then the

statement of Theorem 4.1 holds, with exactly the same proof.

4.2 A discrete time version

Here we consider a \discrete time" version of the previous results. The setting is as follows, and will also be the same in the next subsection.

For each

n

we have a sequence (

Y

_ni)i1of i.i.d. variables on a given space (

n

Fn

P

n),

with

E

n₍

Y

_{ni) = 0}

E

n₍₍

Y

_ni)2

) = 1

n

E

n₍₍

Y

_ni)4

where

"

n!0. These conditions imply that the partial sums processes

Z

_nt = nt] X i=1

Y

_ni (4.11)

converge weakly to a standard Wiener process

Z

=

W

, dened on a (possibly dierent) ltered space (

F

(Ft)t

0

P

). We also have a function

g

on

IR

which is dierentiable

with a bounded Lipschitz derivative, and we consider the dierence equation

X

0nx =

x

X

nx

i =

X

inx;1+

g

(

X

nx

i;1)

Y

ni

(4.12)

whose solution is a square-integrable martingale w.r.t. the discrete-time ltration Fni =

(

Y

nj :

j

i

). We also consider the associated continuous-time martingale w.r.t. the

ltration (Fn nt])t

0:

X

tnx =

X

nxnt]

:

(4.13)

This process

X

nx _{can be viewed as the solution of the stochastic dierential equation}

X

tnx =

x

+ Z t

0

g

(

X

snx;)

dZ

ns

(4.14)

and by stability theorems (see 10]) it converges weakly to the unique strong solution of the following equation:

X

xt =

x

+Z t 0

g

(

X

xs)

dZ

s

:

(4.15)

We even have that the pair (

Z

n

X

nx_{) weakly converges to (}

ZX

x_{). Further}

X

nx_and

X

nx

are also related by (2.5) with

T

i =

i=n

, and

X

nxis a locally square-integrable martingale.

Now we let

T >

0 and

f

be a dierential function with a bounded and Lipschitz derivative. Then

U

n₌

f

₍

X

_Tnx_{) is square-integrable. We can consider the decomposition}

(3.16), which gives

ni, and we associate

0n as in (3.17) with

T

(

ni

) =

i=n

. On the other

hand

U

=

f

(

X

_xT) is also square-integrable, and we set

=

(

X

x

U

_).

Here again, by construction

0nis left continuous with right limits, and we set

(+)0ns=

limt#st>s

0ns. On the other hand, the version of

given by Theorem 2.6 is not only left

continuous, but even continuous except at time

T

: this is because the function

of (2.21) is continuous, and the process

X

x _{also is continuous: then the process}

_s=

₍

sX

_xs)1

0T)(

s

)

is another version of

, which is right continous with left limits (and also continuous except at

T

) and diers from the rst version at time

T

only.

Theorem 4.3

Assume (4.10), (4.11), (4.12), (4.14) and (4.15) with

g

di erentiable with a bounded and Lipschitz derivative. Let

f

be a di erentiable function with a bounded and Lipschitz derivative and let

T >

0. Then the processes (

X

nx

₍₊₎0n) converge in law for

the Skorokhod topology on

IR

2 to (

X

x

).

Proof

. The explicit form of

is given by (2.20), with

taking the simple form

(

sy

) =

Q

T;s

f

0(

y

). Now, if

P

nt

f

(

x

) =

E

(

f

(

X

nx

t )), we readily deduce from Proposition 2.1 and

from (4.10) and (4.12) and (4.13) that a version of

ni is given by

ni =

8 < :

n g(

Xnx

i;1 )

R

_ni(

dy

)

yP

n T;

i n

f

(

X

_inx ;1+

g

(

X

nx

i;1)

y

) if in

< T g

(

X

nx i;1)

6

= 0

where

_nidenotes the law of

Y

_ni. In view of the properties of

g

, we readily deduce by induc-tion on

i

that

x

7!

X

nx

i is dierentiable (for all

!

), hence

x

7!

X

nx

t is also dierentiable

and its derivative satises

X

0nx

t = 1 +

Z t 0

g

0(

X

nx s;)

X

0nx s;

dZ

ns

and we set

Q

nt

f

(

x

) =

E

(

f

(

X

_tnx)

X

0nx

t ). Then by virtue of (4.10) and of the properties of

g

again, one easily checks that

Q

nt

f

0(

y

) is bounded in (

ny

) and continuous in

y

, and that @@y

P

nt

f

(

y

) =

Q

nt

f

0(

y

) since further the

Y

ni's are centered, we get

Z

ni(

dy

)

yP

n T;

i n

f

(

X

_inx;1+

g

(

X

nx i;1)

y

) =

g

(

X

inx;1)

n Q

nT; i n

f

0(

X

nx i;1) +

"

ni

where supij

"

nij!0. Therefore if

n(

s

) =

i=n

when

i=n < s

(

i

+ 1)

=n

, we deduce that

a version of

(+)0n is given by

(+)0ns

=

Q

_nT;n(s)

f

0

(

X

nxn(s))1 0

n

(T)](

s

) +

00n s

where supsj

00n

s j ! 0. By the same argument as in Theorem 4.1 one has

Q

ns n

f

0(

y

n) !

Q

s

f

0(

y

) when

s

n !

s

and

y

n !

y

. Since

X

nx converges in law to

X

x, the result then

follows as in Theorem 4.1 again. 2

Remark 4.4

Exactly as in Remark 4.2, the same result holds when instead of

f

(

X

_xT) and

f

(

X

_Tnx) we consider the variables

f

(

X

_xT1

:::X

xTk) and

f

(

X

nx T1

:::X

nx

Tk ), where

f

is a continuously dierentiable function on

IR

k _{with all partial derivatives bounded and}

Lipschitz, and 0

< T

1

< ::: < T

k.

4.3 Another discrete time version

Here we consider exactly the same setting as in the previous subsection: we have (4.10), (4.11), (4.12), (4.13), (4.14) and (4.15).

The only two dierences are that we only assume

g

to be locally Lipschitz with at most linear growth, and that we will prove a convergence theorem for more general variables than

f

(

X

_xT), but in a much weaker sense.

More precisely, we consider a function on the Skorohod space

ID

of all right contin-uous with left limits functions on

IR

+, which is bounded, continuous for the local uniform

topology, and measurable w.r.t. the

-eldDT generated by the coordinates on

ID

up to

some time

T >

0 (recall that if is continuous for the Skorokhod topology, it is a fortiori continuous for the local uniform topology). Then we take

U

= (

X

x_{) and}

U

n_{= (}

X

nx