De…nition 1 We de…ne for a function f : R ! R f (t ) := lim

1 Introduction

In the next sections we shall describe the mathematical model of a set of particles which move freely as independent Brownian motions but may proliferate at random times with certain random time-dependent rates of proliferation.

Let us anticipate a few notations.

De…nition 1 We de…ne for a function f : R ! R f (t ) := lim

s%t

f (s); the left limit of f at t;

Jf (t) := f (t) f (t ); the jump size of f at t:

Further, let (X

t

)

t 0

be a stochastic process. We de…ne its quadratic variation, when the limit exists and is independent of the partitions,

[X]

_t

= P lim

k!1 n 1

X

j=0

X

_tk

j+1^t

X

_tk j^t

2

;

where the maximal distance of two consequent sites in the partition f0 = t

^k0

< t

^k₁

< <

t

^k_n

= T g converges to 0 as k ! 1. We denote the continuous part of the quadratic variation by [X]

^c

, i.e.

[X]

^c_t

= [X]

_t

X

s t

JX

_s²

:

2 The model

Assume we have a …ltered probability space ( ; F; F

t

; P ), stopping times T

₀^a;N

< T

₁^a;N

, independent Brownian motions B

^a_t

,

I

^a;N

:= [T

₀^a;N

; T

₁^a;N

) X

_t^a;N

=

( if t = 2 I

^a;N

X

^{(a; );N}

T₁^{(a; );N}

+ B

_t^a

if t 2 I

^a;N

and X

^a;N

T₁^a;N

= lim

t"T1^a;N

X

_t^a;N

. Or

X

_t^a;N

=

( if t = 2 I

^a;N

X

^a;N

T₀^a;N

+ B

_t^a

if t 2 I

^a;N

where X

^a;N

T₀^a;N

= X

^{(a; );N}

T₁^{(a; );N}

= lim

_t"T^a;N

1

X

_t^a;N

.

It remains to specify T

₁^a;N

, given T

₀^a;N

and the other objects. Assume we have, on ( ; F; F

t

; P ) independent standard Poisson processs N

_t^0;a

t 0

and denote their …rst jump times by

^0;a

(N

_t^0;a

is equal to zero for t <

^0;a

and equal to 1 at t =

^0;a

; recall that

0;a

> 0 with probability one). Assume we have continuous adapted non-decreasing process

a;N

t t 0

, equal to zero for t T

₀^a;N

, with the following property: either

^a;N_t

<

^0;a

for all t, and in this case we set T

₁^a;N

= +1, or there exists T

1^a;N

such that

^a;N_t

<

^0;a

for t < T

₁^a;N

and

^a;N_t

=

^0;a

for t T

₁^a;N

. De…ne

N

_t^a;N

= N

^0;aa;N t

:

The stopping time T

₁^a;N

is the …rst (and unique) jump time of N

_t^a;N

. More precisely, we have

N

_t^a;N

= 1

_[Ta;N

1 ;1)

(t) : Indeed, if T

₁^a;N

= +1, where 1

_[T^a;N

1 ;1)

(t) is the function identically equal to zero, we have

a;N

t

<

^0;a

for all t, hence N

^0;aa;N t

is also identically equal to zero; if T

₁^a;N

< 1 and t < T

1^a;N

, where 1

_[T^a;N

1 ;1)

(t) = 0, we have

^a;N_t

<

^0;a

and thus N

^0;aa;N t

= 0; …nally, if T

₁^a;N

< 1 and t T

₁^a;N

, where 1

_[Ta;N

1 ;1)

(t) = 1, we have

^a;N_t

=

^0;a

and thus N

^0;aa;N t

= N

^0;a0;a

= 1.

3 Preliminaries on Itô formula for processes de…ned on ran- dom time intervals

3.1 SDEs on random time intervals In the sequel, let us always assume T

₁

> T

₀

.

When we write

dX

_t

= b

_t

dt +

_t

dB

_t

on t 2 [T

0

; T

₁

] where T

₀

; T

₁

are stopping times, we mean that

X

_t

= X

_T0

+ Z

t

T0

b

_s

ds + Z

t

T0

s

dB

_s

for t 2 [T

0

; T

₁

] : When we write

dX

_t

= b

_t

dt +

_t

dB

_t

on t 2 (T

0

; T

₁

) where T

0

; T

1

are stopping times, we mean that

dX

_t

= b

_t

dt +

_t

dB

_t

on t 2 T

0⁰

; T

₁⁰

for every pair of stopping times T

₀⁰

T

₁⁰

2 (T

0

; T

₁

). Similar de…nitions apply to other cases,

like [T

0

; T

1

).

3.2 Itô formula on random time intervals If 2 C

^1;2

and

X

_t

= X

_T0

+ Z

t

T0

b

_s

ds + Z

t

T0

s

dB

_s

for t 2 [T

0

; T

₁

) then

(t; X

_t

) = (T

₀

; X

_T0

) + Z

t

T0

(@

_s

(s; X

_s

) ds + r (s; X

s

) b

_s

) ds + Z

t

T0

r (s; X

s

)

_s

dB

_s

+ Z

t

T0

1

2 T r

_s ^T_s

D

²

(s; X

_s

) ds for t 2 [T

0

; T

₁

):

3.3 SDEs for processes with jumps Assume, for the process Y

_t

2 R, we have

Y

_t

= Y

_T0

+ Z

t

T0

b

_s

ds + Z

t

T0

s

dB

_s

for t 2 [T

0

; T

₁

) Y

_t

= 0 otherwise.

Then Y

t

satis…es, for all t 0, the following identity:

Y

t

= Z

t

0

1

_s2[T0;T1)

b

s

ds + Z

t

0

1

_{s2[T0;T1) s}

dB

s

+ Y

_T0

1

_{t T0}

Y

_T1

1

_{t T1}

for t 0:

The proof of this identity is simply made by checking the validity for t < T

₀

, then for t 2 [T

0

; T

₁

) and …nally for t T

₁

.

If needed, we may formally write

d (Y

_t

) = (

_t=T0

1

_{T0>0 t=T1}

) Y

_t

+ 1

_t2[T0;T1)

b

_t

dt + 1

_{t2[T0;T1) t}

dB

_t

:

4 Back to our case

Proposition 2 Let be a function from R

^d

f g ! R, of class C

²

on R

^d

, equal to zero at . Then

X

_t^a;N

= X

^a;N

T₀^a;N

1

_{t T}^a;N

0

X

^a;N

T₁^a;N

1

_{t T}^a;N

1

+

Z

t 0

1

_s2I^a;N

r X

_s^a;N

dB

_s^a

+ 1 2

Z

t 0

1

_s2I^a;N

X

_s^a;N

ds:

Proof. We have

X

_t^a;N

= 0 if t = 2 I

^a;N

X

_t^a;N

= X

^a;N

T₀^a;N

+ Z

t

T₀^a;N

r X

_s^a;N

dB

_s^a

+ 1 2

Z

t T₀^a;N

X

_s^a;N

ds if t 2 I

^a;N

:

Hence it is su¢ cient to apply the result of Subsection 3.3.

Set

S

_t^N

:= 1 N

X

a2 t

X_t^a;N

= 1 N

X

a2

1

_t2Ia;N

X_t^a;N

so that, for as above (namely ( ) = 0) S

_t^N

; = 1

N X

a2

1

_t2Ia;N

X

_t^a;N

= 1 N

X

a2

X

_t^a;N

:

Proposition 3 Let be a function from R

^d

f g ! R, of class C

²

on R

^d

, equal to zero at . Then

S

_t^N

; = S

₀^N

; + 1 N

X

a2

X

^a;N

T₁^a;N

1

_{t T}a;N 1

+ M

_t^{1; ;N}

+ 1 2

Z

t 0

S

_s^N

; ds where

M

_t^{1; ;N}

:= 1 N

X

a2

Z

t 0

1

_s2Ia;N

r X

_s^a;N

dB

_s^a

:

Proof. We have S

_t^N

; = 1

N X

a2

X

^a;N

T₀^a;N

1

_{t T}a;N 0

1 N

X

a2

X

^a;N

T₁^a;N

1

_{t T}a;N 1

+ 1 N

X

a2

Z

t 0

1

_s2Ia;N

r X

_s^a;N

dB

_s^a

+ 1 N

X

a2

1 2

Z

t 0

1

_s2Ia;N

X

_s^a;N

ds

= 1 N

X

a2

X

^a;N

T₀^a;N

1

_{t T}^a;N

0

1 N

X

a2

X

^a;N

T₁^a;N

1

_{t T}^a;N

1

+ M

_t^{1; ;N}

+ 1 2

Z

t 0

S

^N_s

; ds:

Now,

1 N

X

a2

X

^a;N

T₀^a;N

1

_{t T}^a;N

0

1 N

X

a2

X

^a;N

T₁^a;N

1

_{t T}^a;N

1

= S

₀^N

; + 1 N

X

a2

X

^a;N

T₁^a;N

1

_{t T}^a;N

1

: Indeed, in the …rst sum,

_N¹

P

a2

X

^a;N

T₀^a;N

1

_{t T}^a;N

0

, there are terms with T

₀^a

= 0, which sum-up to S

₀^N

; , and terms with T

₀^a

> 0, which mean that two new particles arise at time T

₀^a

and one particle dies; the particle which dies is included in the sum

_N¹

P

a2

X

^a;N

T₁^a;N

1

_{t T}a;N 1

. The sum of two new particles minus the old particle is equal to one new particle, and a way

to describe the sum of only one new particle is

_N¹

P

a2

X

^a;N

T₁^a;N

1

_{t T}^a;N

1

. Recall that, in our scheme, we always have T

₁^a;N

> 0 (since T

₁^a;N

> T

₀^a;N

), hence there is no danger to add also particles already existing at time zero, in the sum

_N¹

P

a2

X

^a;N

T₁^a;N

1

_{t T}^a;N

1

. Proposition 4 Let be bounded (in addition to the previous properties). Then the process

M

_t^{2; ;N}

:= 1 N

X

a2

X

^a;N

T₁^a;N

1

_{t T}a;N 1

1 N

X

a2

Z

t 0

X

_s^a;N

d

^a;N_s

is a martingale, with respect to the …ltration G

t

:= F

t

.

Proof. We have

X

^a;N

T₁^a;N

1

_{t T}a;N 1

= Z

t

0

X

_s^a;N

dN

_s^a;N

: It is known that, since N

_t^a;N

= N

^0;aa;N

t

, the process N

_t^{a;N a;N}_t

is a G

^t

-martingale; namely,

a;N

t

is the compensator of N

_t^a;N

(notice: these facts have an elementary proof in the case when

^a;N_t

is deterministic; but the result is true also in the random case; a reference available on line is [1], lemma 3.2.

By the theory of stochastic integration with respect to point processes, the compensator of R

t

0

X

_s^a;N

dN

s^a;N

is the Lebesgue-Stilties integral Z

t

0

X

_s^a;N

d

^a;N_s

:

The di¤erence of the two processes, M

_t^{2; ;N}

, is a martingale (not only a local martingale,

due to the boundedness of ). The result remains true under summation (this require some

technical work, omitted).

Corollary 5 If

^a;N_t

has the form

a;N

t

=

Z

t 0

1

_s2I^a;N

X

_s^a;N

; S

_s^N

ds then

S

_t^N

; = S

₀^N

; + 1 2

Z

t 0

S

_s^N

; ds + Z

t

0

; S

_s^N

S

_s^N

; ds + M

_t^{1; ;N}

+ M

_t^{2; ;N}

:

Proof. We observe that 1

N X

a2

Z

t 0

X

_s^a;N

d

^a;N_s

= 1 N

X

a2

Z

t 0

X

_s^a;N

1

_s2Ia;N

X

_s^a;N

; S

_s^N

ds = Z

t

0

S

_s^N

; ; S

_s^N

ds:

Let us now take, as a particular case of (x), the function

N

(x

0

x). We write h

^N_t

:=

_N

S

_t^N

so, for instance, S

_t^N

;

_N

(x

₀

) = h

^N_t

(x

₀

). Renaming, at the end of the computation, x

0

by x, we get:

Corollary 6

h

^N_t

(x) = h

^N₀

(x) + 1 2

Z

t 0

h

^N_s

(x) ds + Z

t

0

N

; S

_s^N

S

_s^N

(x) ds + M

_t^1;N

(x) + M

_t^2;N

(x)

where

N

; S

^N_s

S

_s^N

(x) denotes R

N

(x y) y; S

_s^N

S

_s^N

(dy) and M

_t^1;N

(x) := 1

N X

a2

Z

t 0

1

_s2Ia;N

r

N

x X

_s^a;N

dB

_s^a

:

M

_t^2;N

(x) := 1 N

X

a2

N

x X

^a;N

T₁^a;N

1

_{t T}a;N 1

Z

t 0

N

; S

_s^N

S

_s^N

(x) ds:

5 Estimates on the martingale terms

Let us recall a few general facts. Given a semi-martingale M

_t

(di che regolarità? ) we associate to it the quadratic variation [M; M ]

_t

and the predictable quadratic variation hM; Mi

t

, which is the predictable compensator of [M; M ]

_t

. If M is a (local?) martingale, it is also the unique predictable process such that M

_t²

hM; Mi

t

is a martingale (local?).

It follows

E M

_t²

= E [hM; Mi

t

] = E [[M; M ]

_t

] :

Example 7 If M

_t

= B

_t

, then [M; M ]

_t

= t, hM; Mi

t

= t, E M

_t²

= t.

Example 8 If N

_t

is an increasing jump process with discrete jumps, then [N; N ]

_t

= P

s t

(J N

s

)

²

.

Example 9 If the jumps have size 1, then [N; N ]

_t

= N

t

and hN; Ni

t

is thus the compen- sator A

t

of N

t

itself. Set M

t

= N

t

A

t

and assume A

t

continuous (by a theorem, it is so if the jumps are not predicatble). Then [M; M ]

_t

= [N; N ]

_t

= N

_t

, hM; Mi

t

= A

_t

and

E M

_t²

= E h

(N

_t

A

_t

)

²

i

= E [M

_t

] = E [A

_t

] :

Example 10 In particular, if N

t

= N

⁰_t

, N

_t⁰

standard Poisson, the previous facts hold with A

_t

=

_t

.

Example 11 If d

_s

=

_s

ds, H is predictable and bounded, and M

t

:=

Z

t 0

H

s

d N

⁰_s s

= Z

t

0

H

s

dN

⁰_s

Z

t

0

H

s s

ds then [M; M ]

_t

= R

0

H

s

dN

⁰_{s t}

. But, in case N

⁰_t

= 1

_[T;1)

(t), Z

t

0

H

_s

dN

⁰_s

= H

_T

1

_[T;1)

(t) = H

_T

N

⁰_t

and thus Z

0

H

_s

dN

⁰_s

t

= H

_T

N

⁰ _t

= H

_T²

N

⁰_t

= Z

t

0

H

_s²

dN

⁰_s

whose compensator (hence the compensator of [M; M ]

_t

) is

hM; Mi

t

= Z

t

0

H

_s²

d

s

: Summarizing,

E M

_t²

= E H

_T²

N

⁰_t

= E Z

t

0

H

_s²

dN

⁰_s

= E Z

t

0

H

_s²

d

_s

:

Given two semi-martingales M

_t⁽ⁱ⁾

(di che regolarità? ), i = 1; 2, we associate to them the joint or mutual variation M

⁽¹⁾

; M

⁽²⁾ _t

and the predictable joint variation M

⁽¹⁾

; M

⁽²⁾ _t

, which is the predictable compensator of M

⁽¹⁾

; M

⁽²⁾ _t

. If M

⁽¹⁾

; M

⁽²⁾

are (local) mar- tingales, is it true that M

⁽¹⁾

; M

⁽²⁾ _t

is also the unique predictable process such that M

_t⁽¹⁾

M

_t⁽²⁾

M

⁽¹⁾

; M

⁽²⁾ _t

is a martingale (local?)? It would follow

E h

M

_t⁽¹⁾

M

_t⁽²⁾

i

= E hD

M

⁽¹⁾

; M

⁽²⁾

E

t

i

= E hh

M

⁽¹⁾

; M

⁽²⁾

i

t

i :

Notice that, if M

⁽¹⁾

; M

⁽²⁾

are jump processes that do not jump simultaneously, then M

⁽¹⁾

; M

⁽²⁾ _t

= 0, hence E h

M

_t⁽¹⁾

M

_t⁽²⁾

i

= 0.

Example 12 If a = 1; 2, N

_t^a

= N

^0;aa

t

, N

_t^0;a

independent standard Poisson, d

^a_s

=

^a_s

ds, H

^a

predictable and bounded, and

M

_t^(a)

:=

Z

t 0

H

_s^a

d N

^0;aa s

as

= Z

t

0

H

_s^a

dN

^0;aa s

Z

t 0

H

_s^{a a}_s

ds

= H

_T^aa

N

^0;aa t

Z

t 0

H

_s^{a a}_s

ds

then h

M

⁽¹⁾

; M

⁽²⁾

i

t

= h

H

_T¹1

N

^0;11

; H

_T²2

N

^0;22

i

t

= 0 since they do not jump simultaneously. Hence E h

M

_t⁽¹⁾

M

_t⁽²⁾

i

= 0.

5.1 First martingale term

By taking a …nite sum approximation and then passing to the limit, one can check that E M

_t^{1; ;N 2}

= 1

N

²

X

a2

E

" Z

t 0

1

_s2Ia;N

r X

_s^a;N

dB

_s^a

2

# :

We have used the fact that E

Z

t 0

1

_s2I^a;N

r X

_s^a;N

dB

_s^a

Z

t

0

1

_s2Ia0;N

r X

_s^a0;N

dB

^a_s⁰

= 0

for a 6= a

⁰

, a known property for Itô integrals w.r.t. indepedent Brownian motions, when the integrands are of class M

²

(here it is true since r

N

is bounded). Then by Itô isometry,

E M

_t^{1; ;N 2}

= 1 N

²

X

a2

E Z

t

0

1

_s2I^a;N

r X

_s^{a;N 2}

ds

= 1 N E

"Z

t 0

1 N

X

a2

1

_s2I^a;N

r X

_s^{a;N 2}

ds

#

= 1 N E

Z

t 0

D

jr j

²

; S

_s^N

E ds 1

N kr k

²₁

Z

t

0

E 1; S

^N_s

ds:

It follows

N !1

lim E M

_t^{1; ;N 2}

= 0

by an estimate on E 1; S

_s^N

discussed afterwards.

Similarly,

E M

_t^1;N

(x)

²

= 1 N

²

X

a2

E

" Z

t 0

1

_s2I^a;N

r

^N

x X

_s^a;N

dB

_s^a

2

#

= 1 N

²

X

a2

E Z

t

0

1

_s2Ia;N

r

N

x X

_s^{a;N 2}

ds

= 1 N E

"Z

t 0

1 N

X

a2

1

_s2Ia;N

r

N

x X

_s^{a;N 2}

ds

#

= 1 N E

Z

t 0

D

jr

N

(x )j

²

; S

_s^N

E ds :

Here we have to use the structure of

N

. Precisely, if we assume that

N

(x) =

_N^d _N¹

x , we have D

jr

^N

(x )j

²

; S

_s^N

E

=

_N^{d 2}

Z

d

N

r

N¹

(x y)

²

S

^N_s

(dy) : (1) To control such term it is necessary to assume

sup

N 2N d 2 N

N < 1:

When the second moments E M

_t^{1; ;N 2}

and E M

_t^{1;N 2}

have been estimated, one can prove similar estimates with the supremum in time inside the expectation, using the martingale property.

5.2 Second martingale term

Consider now

M

_t^{2; ;N}

= 1 N

X

a2

X

^a;N

T₁^a;N

1

_{t T}a;N 1

1 N

X

a2

Z

t 0

X

_s^a;N

d

^a;N_s

= 1 N

X

a2

Z

t 0

X

_s^a;N

d N

_s^{a;N a;N}_s

which is a martingale with respect to the …ltration G

t

:= F

t

. Since the jumps of N

s^a;N

and N

s^a0;N

, for a 6= a

⁰

never occur at the same time, we have (see 12) E

Z

t 0

X

_s^a;N

d N

_s^{a;N a;N}_s

Z

t 0

X

_s^a0;N

d N

_s^{a0;N a}_s^0;N

= 0:

Hence

E M

_t^{2; ;N 2}

= 1 N

²

X

a2

E

" Z

t 0

X

_s^a;N

d N

_s^{a;N a;N}_s

2

# : It is known, see 11, that

E

" Z

t 0

X

_s^a;N

d N

_s^{a;N a;N}_s

2

#

= E

"

X

^a;N

T₁^a;N 2

1

_{t T}a;N 1

#

or also E

" Z

t 0

X

_s^a;N

d N

_s^{a;N a;N}_s

2

#

= E Z

t

0

X

_s^{a;N 2}

d

^a;N_s

: Then, using the …rst expression,

E M

_t^{2; ;N 2}

= 1 N

²

X

a2

E

"

X

^a;N

T₁^a;N 2

1

_{t T}a;N 1

#

= k k

²₁

1 N

X

a2

E

"

1 N

X

a2

1

_{t T}a;N 1

#

k k

²₁

1

N E 1; S

_t^N

:

Alternatively, using the second expression, if

^a;N_t

has the form

a;N

t

=

Z

t 0

1

_s2Ia;N

X

_s^a;N

; S

_s^N

ds we get

E M

_t^{2; ;N 2}

= 1 N

²

X

a2

E Z

t

0

X

_s^{a;N 2}

d

^a;N_s

= 1 N

²

X

a2

E Z

t

0

X

_s^{a;N 2}

1

_s2Ia;N

X

_s^a;N

; S

^N_s

ds

= 1 N

²

X

a2

E Z

t

0

X

_s^{a;N 2}

1

_s2I^a;N

X

_s^a;N

; S

_s^N

ds

= 1 N E

Z

t 0

Z

j (x)j

²

x; S

_s^N

S

_s^N

(dx) ds

which here gives rise to a similar estimate as above.

Similarly,

E M

_t^2;N

(x)

²

= 1 N

²

X

a2

E

" Z

t 0

N

x X

_s^a;N

d N

_s^{a;N a;N}_s

2

#

= 1 N

²

X

a2

E Z

t

0

N

x X

_s^{a;N 2}

d

^a;N_s

= 1 N

²

X

a2

E

"

N

x X

^a;N

T₁^a;N 2

1

_{t T}a;N 1

# :

To control the last term we need to integrate in dx. The previous term is more suitable:

1 N

²

X

a2

E Z

t

0

N

x X

_s^{a;N 2}

d

^a;N_s

= 1 N

²

X

a2

E Z

t

0

N

x X

_s^{a;N 2}

1

_s2Ia;N

X

_s^a;N

; S

_s^N

ds

= 1 N E

Z

t 0

Z

j

N

(x y)j

²

y; S

_s^N

S

_s^N

(dy) ds : If we assume that

N

(x) =

_N^d _N¹

x , we have

Z

j

N

(x y)j

²

y; S

_s^N

S

_s^N

(dy) =

_N^d

Z

d N

1

N

(x y)

²

y; S

_s^N

S

_s^N

(dy) : (2)

6 Estimates on h

^N_t

Recall that

h

^N_t

(x) = h

^N₀

(x) + 1 2

Z

t 0

h

^N_s

(x) ds + Z

t

0

N

; S

_s^N

S

_s^N

(x) ds + M

_t^1;N

(x) + M

_t^2;N

(x) : We want an identity for h

^N_t

(x)

²

. We need Itô formula for processes with jumps. Recall the following result, also known as generalized Itô formula.

Lemma 13 Let X be a one-dimensional semimartingale such that X

t

; X

t

take values in an open set U R and f : U ! R twice continuously di¤erentiable. Then f(X) is a semimartingale and

f (X

_t

) =f (X

₀

) + Z

t

0

f

⁰

(X

_s

)dX

_s

+ 1 2

Z

t 0

f

⁰⁰

(X

_s

)d [X]

^c_s

+ X

s t

Jf (X

_s

) f

⁰

(X

_s

)JX

_s

Lemma 14

h

^N_t

(x)

²

= h

^N₀

(x)

²

+ Z

t

0

h

^N_s

(x) h

^N_s

(x) ds + Z

t

0

2h

^N_s

(x)

_N

; S

_s^N

S

_s^N

(x) ds +

Z

t 0

2h

^N_s

(x) dM

_s^1;N

(x) + Z

t

0

2h

^N_s

(x) dM

_s^2;N

(x) + 1

N Z

t

0

D

jr

^N

(x )j

²

; S

_s^N

E

ds + 1

N

²

X

a2 ^N

2N

(x X

^a;N

T₁^a;N

)1

_{t T}a;N 1

:

Notice also that 1 N

²

X

a2 ^N 2

N

(x X

^a;N

T₁^a;N

)1

_{t T}a;N

1

= 1

N

²

X

a2 ^N

Z

t 0

2

N

(x X

_s^a;N

)dN

_s^a;N

:

Proof. We apply generalized Itô formula to the function f (x) = x

²

and the process X

t

= h

^N_t

(x) (N and x given). We …rst get

h

^N_t

(x)

²

= h

^N₀

(x)

²

+ Z

t

0

2h

^N_s

(x) dh

^N_s

(x) + [X]

^c_t

+ X

s t

J h

^N_s

(x)

²

2h

^N_s

(x) Jh

^N_s

(x)

where, after substitution of dh

^N_s

(x), the term R

t

0

2h

^N_s

(x) dh

^N_s

(x) give rise to the sum of the terms (we analyze each one of them on a separate line)

Z

t 0

2h

^N_s

(x) 1

2 h

^N_s

(x) ds = Z

t

0

h

^N_s

(x) h

^N_s

(x) ds

(the number of jumps of h

^N_s

(x) is …nite on [0; t], hence we may change the value of the Riemann integral at a …nite numebr of points)

Z

t 0

2h

^N_s

(x)

_N

; S

_s^N

S

_s^N

(x) ds = Z

t

0

2h

^N_s

(x)

_N

; S

_s^N

S

_s^N

(x) ds Z

t

0

2h

^N_s

(x) dM

_s^1;N

(x) = Z

t

0

2h

^N_s

(x) dM

_s^1;N

(x)

(the argument about the …nite number of jumps of h

^N_s

(x) in [0; t] applies also to Itô integrals w.r.t. Brownian motion)

Z

t 0

2h

^N_s

(x) dM

_s^2;N

(x) :

The quadratic variation [X]

_t

is equal to

[X]

_t

= M

^1;N

(x)

_t

+ M

^2;N

(x)

_t

+ 2 M

^1;N

(x) ; M

^2;N

(x)

_t

:

The quadratic variation M

^2;N

(x)

_t

has only jumps; the covariation M

^1;N

(x) ; M

^2;N

(x)

_t

is equal to zero since, at the jumps of M

^2;N

(x), M

^1;N

(x) is continuous. Hence term [X]

^c_t

is equal to

[X]

^c_t

= M

^1;N

(x)

_t

= 1 N

²

X

a2

Z

t 0

1

_s2Ia;N

r

N

x X

_s^a;N

dB

_s^a

= 1 N

²

X

a2

Z

t 0

1

_s2I^a;N

r

^N

x X

_s^{a;N 2}

ds

= 1 N

Z

t 0

D

jr

^N

(x )j

²

; S

_s^N

E

ds:

It remains to understand the term P

s t

J h

^N_s

(x)

²

2h

^N_s

(x) Jh

^N_s

(x) . At every jump time r, we have

J h

^N_r

(x)

²

2h

^N_r

(x)Jh

^N_r

(x)

= h

^N_r

(x)

²

h

^N_r

(x)

²

2h

^N_r

(x) h

^N_r

(x) h

^N_r

(x)

= h

^N_r

(x)

²

+ h

^N_r

(x)

²

2h

^N_r

(x) h

^N_r

(x)

= h

^N_r

(x) h

^N_r

(x)

²

= Jh

^N_r

(x)

²

: Moreover,

Jh

^N

T₁^a;N

(x) = 1

N

^N

(x X

^a;N

T₁^a;N

) hence

X

r t

Jh

^N_r

(x)

²

= X

a2 ^N

Jh

^N

T₁^a;N

(x)

²

1

_{t T}a;N 1

= 1 N

²

X

a2 ^N

2N

(x X

^a;N

T₁^a;N

)1

_{t T}a;N 1

:

Lemma 15 One has E

" Z

t 0

2h

^N_s

(x) dM

_s^1;N

(x)

2

#

4

_N^{d 2}

N E

Z

t 0

h

^N_s

(x)

²

Z

d

N

r

N¹

(x y)

²

S

_s^N

(dy) ds :

Proof. Since Z

t

0

2h

^N_s

(x) dM

_s^1;N

(x) = 1 N

X

a2

Z

t 0

2h

^N_s

(x) 1

_s2Ia;N

r

N

x X

_s^a;N

dB

^a_s

We have, as above,

E

" Z

t 0

2h

^N_s

(x) dM

_s^1;N

(x)

2

#

= 1 N

²

X

a2

E

" Z

t 0

2h

^N_s

(x) 1

_s2I^a;N

r

^N

x X

_s^a;N

dB

_s^a

2

#

= 1 N

²

X

a2

E Z

t

0

4 h

^N_s

(x)

²

1

_s2Ia;N

r

^N

x X

_s^{a;N 2}

ds

= 4 N E

"Z

t 0

h

^N_s

(x)

²

1 N

X

a2

1

_s2I^a;N

r

^N

x X

_s^{a;N 2}

ds

#

= 4 N E

Z

t 0

h

^N_s

(x)

²

D

jr

^N

(x )j

²

; S

_s^N

E ds : Using (1), we get

E

" Z

t 0

2h

^N_s

(x) dM

_s^1;N

(x)

2

#

4

_N^{d 2}

N E

Z

t 0

h

^N_s

(x)

²

Z

d

N

r

N¹

(x y)

²

S

_s^N

(dy) ds :

Lemma 16 One has E

" Z

t 0

2h

^N_s

(x) dM

_s^2;N

(x)

2

#

4

_N^d

N E

Z

t 0

h

^N_s

(x)

²

Z

d N

2 1

N

(x y) y; S

_s^N

S

_s^N

(dy) ds : Proof. Since

Z

t 0

2h

^N_s

(x) dM

_s^2;N

(x) = 1 N

X

a2

Z

t 0

2h

^N_s

(x)

_N

x X

_s^a;N

d N

_s^{a;N a;N}_s

as above we have E

" Z

t 0

2h

^N_s

(x) dM

_s^2;N

(x)

2

#

= 1 N

²

X

a2

E

" Z

t 0

2h

^N_s

(x)

N

x X

_s^a;N

d N

_s^{a;N a;N}_s

2

#

= 1 N

²

X

a2

E Z

t

0

4 h

^N_s

(x)

^{2 2}_N

x X

_s^a;N

X

_s^a;N

; S

_s^N

ds

= 4 N E

Z

t 0

h

^N_s

(x)

²

Z

2N

(x y) y; S

_s^N

S

_s^N

(dy) ds :