DIFFERENTIAL SUBORDINATION UNDER CHANGE OF LAW

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DIFFERENTIAL SUBORDINATION UNDER CHANGE OF LAW

Komla Domelevo, Stefanie Petermichl

To cite this version:

Komla Domelevo, Stefanie Petermichl. DIFFERENTIAL SUBORDINATION UNDER CHANGE OF

LAW. Annals of Probability, Institute of Mathematical Statistics, 2018. �hal-01977592�

arXiv:arXiv:1604.01606

DIFFERENTIAL SUBORDINATION UNDER CHANGE OF LAW

By Komla Domelevo

^‡

and Stefanie Petermichl

^†,∗,‡

Universit´ e Paul Sabatier

^‡

We prove optimal

L²

bounds for a pair of Hilbert space valued differentially subordinate martingales under a change of law. The change of law is given by a process called a weight and sharpness in this context refers to the optimal growth with respect to the charac- teristic of the weight. The pair of martingales is adapted, uniformly integrable, and c` adl` ag. Differential subordination is in the sense of Burkholder, defined through the use of the square bracket. In the scalar dyadic setting with underlying Lebesgue measure, this was proved by

Wittwer

[34], where homogeneity was heavily used. Re- cent progress by

Thiele

–

Treil

–

Volberg

[30] and

Lacey

[20], inde- pendently, resolved the so–called non–homogenous case using discrete in time filtrations, where one martingale is a predictable multiplier of the other. The general case for continuous–in–time filtrations and pairs of martingales that are not necessarily predictable multipliers, remained open and is adressed here. As a very useful second main result, we give the explicit expression of a Bellman function of four variables for the weighted estimate of subordinate martingales with jumps. This construction includes an analysis of the regularity of this function as well as a very precise convexity, needed to deal with the jump part.

1. Introduction.

The paper by Nazarov – Treil – Volberg [23] has set the groundwork for the early advances in modern weighted theory in harmonic analysis and probability that started around twenty years ago. In their paper the authors show necessary and sufficient conditions for a dyadic martingale transform to be bounded in the L

²

two–weight setting. The methodology of their proof could be used to get the first sharp result in the real valued one–weight setting, for the dyadic martingale transform [34]. Sharpness in this setting means best control on growth with respect to the A

₂

characteristics

Q

^F₂

(w) = sup

τ

ess . sup

ω

E (w|F

_τ

) E (w

⁻¹

|F

_τ

),

∗

Supported by ERC grant CHRiSHarMa 682402.

†

The second author is a member of IUF

MSC 2010 subject classifications: Primary 60G44; secondary 60G46 Keywords and phrases: weight, differential subordination, Bellman function

1

DOMELEVO AND PETERMICHL

where the underlying filtration F is dyadic and the supremum is taken over all adapted stopping times τ . Equivalently, without the use of stopping times, the A

₂

characteristic can be defined by

Q

^F₂

(w) = sup

I

1

|I | Z

w 1

|I | Z

w

⁻¹

, where the supremum runs over all dyadic intervals.

The area of sharp weighted estimates has seen substantial progress with new, beautiful proofs of Wittwer ’s result and its extensions to the time shifted martingales referred to as ‘dyadic shift’ [21][31]. Related, important questions in harmonic analysis, such as boundedness of the Beurling–Ahlfors transform [27], Hilbert transform [25], general Calderon-Zygmund operators [18][22][20] and beyond [5][18] have been solved, beautifully advancing pro- found understanding of the objects at hand.

During the early days of weighted theory in harmonic analysis, before optimal weighted estimates were within reach, say, for the maximal operator or the Hilbert transform [17] similar questions were asked in probability theory, concerning stochastic processes with continuous–in–time filtrations [6][19]. The difficulty that arises in the non–homogenous setting, typically seen when these processes have jumps, were already observed back then and this restriction was made in one form or another in these papers. Certain basic facts about weights do not hold true for all jump processes, such as the classical self improvement of the A

₂

characteristic of the weight [6].

Another obstacle typical for working with weights is the non–convexity of the set inspired by the A

2

characteristic: {r, s ∈ R

+

: 1 6 rs 6 Q} with Q > 1. Such continuity–in–space assumptions still appear regularly for these or other reasons when adressing weights, see [3][24].

Wittwer ’s proof also uses the homogeneity that arises from the dyadic filtration where the underlying measure is Lebesgue in a subtle but cru- cial way. This homogeneity assumption has only recently been removed in the papers [30] and [20]. These authors work with discrete–in–time gen- eral filtrations with arbitrary underlying measure, where one martingale is a predictable multiplier of the other. A direct passage using the results for discrete–in–time filtrations to the continuous–in–time filtration case where one uses Ba˜ nuelos’ and Wang’s definition

Y differentially subordinate to X (1.1)

:⇔ [X, X]

t

− [Y, Y ]

t

nonnegative and nondecreasing

is only possible in very special cases, such as predictable multipliers of

stochastic integrals - this passage is standard and explained in one of Burkholder’s

CHANGE OF LAW

early works on L

^p

estimates for pairs of differentially subordinate martin- gales [7]. (In full generality, this unweighted L

^p

problem was only much later resolved in [33].)

In this article, we tackle the sharp weighted estimate in full generality, using the notion of differentail subordination of Burkholder (1.1) and the martingale A

₂

characteristic,

Q

^F₂

[w] = sup

τ

ess . sup

ω

(w)

τ

(w

⁻¹

)

τ

.

We prove that for L

²

integrable Hilbert space valued martingales X, Y with Y differentially subordinate to X there holds

kY k

_L2(w)

. Q

^F₂

[w]kXk

_L2(w)

,

where the implied constant is numeric and does not depend upon the di- mension, the pair of martingales or the weight. The linear growth in the quantity Q

^F₂

(w) is sharp.

The proof in this paper is different from the proofs in [20] and [30]. In [20] so–called sparse operators are used while in [30] the authors reduce the estimate through the use of the so–called outer measure space theory.

Our approach is the following. We derive an explicit Bellman function of four variables adapted to the problem. It has certain conditions on its range, a continuous sub–convexity as well as discrete one–leg convexity, such as seen in [30] for two smaller Bellman functions (their functions make up a part of ours). We heavily use the explicit form of the Bellman function and its regularity properties in several parts in our proof to handle the delicacy of the continuous–in–time processes with values in Hilbert space. The resulting function is in the ‘dualized’ or ‘weak form’, which is in a contrast to the

‘strong form’ of a Burkholder type functional often seen when using the strong subordination condition (1.1). (The explicit form of a Burkholder type functional for this weighted question is still open). Indeed, the form of the strong differential subordination condition is adapted to work well for Burkholder type functionals and arises naturally in this setting. The passage to its use in the weak form is accomplished through the use of the so–called Ellipse Lemma and requires a Bellman function solving the entire problem at once as opposed to splitting the problem into pieces. This is the first use of this strategy for problems in probability and should allow generalisations of numerous existing results as well as an alternative (allbeit more complicated) proof of Wang’s extension to Burkholder’s famous estimates using [32] or [2].

Note that for these L

^p

problems, fewer difficulties arise, even in the presence

DOMELEVO AND PETERMICHL

of jumps. This is thanks to the convexity of the domain in the L

^p

problem.

The one–leg–convexity required to control the jumps is almost free in the L

^p

case, when using a trick from [11]. This trick is not available here because of the non–convex domain.

Our result gives through the formula in [4] a probabilistic proof of the weighted estimate for the Beurling–Ahlfors transform with its implication, a famous borderline regularity problem for the Beltrami equation, solved in [27]. Other applications are discussed in the last section. They include a dimensionless weighted bound for discrete and semi–discrete second order Riesz transforms. The Bellman function constructed here has already been used to obtain a dimensionless weighted bound of the Bakry Riesz vector on Riemannian manifolds under a condition on curvature.

1.1. Differentially subordinate martingales.

Consider first discrete–in–time martingales. For that let (Ω, F

_∞

, P) be a probability space with a nondecreasing sequence F = (F

_n

)

_n>0

of sub σ–fields of F

_∞

such that F

₀

contains all F

_∞

–null sets. We are interested in H –valued martingales, where H is a separable Hilbert space with norm |· |

H

and scalar product h·, ·i

_H

: if f = {f

_n

}

_n∈N

is a H –valued martingale adapted to F, we note f

_n

= P

n

k=0

df

_k

, with the convention df

₀

:= f

₀

, and df

_k

:= f

_k

− f

k−1

, for k > 1. Similarily, if g is another adapted H–valued martingale, we note g

n

= P

n

k=0

dg

_k

with the same conventions. One says that g is differentially subordinate to f if one has almost surely for k > 0, |dg

_k

|

_H

6 |df

_k

|

_H

.

In this paper we consider continuous–in–time filtrations. Let again (Ω, F

_∞

, P ) a probability space with a nondecreasing right continuous family F = (F

_t

)

_t>0

of sub σ–fields of F

∞

such that F

₀

contains all F

∞

–null sets. We are inter- ested in H –valued c` adl` ag martingales, where H is a separable Hilbert space.

In order to clearly define differential subordination in this setting, we make use of the square bracket or quadratic variation process.

Recall that the quadratic variation process of a real–valued semimartin- gale X is the process denoted by [X, X] := ([X, X]

_t

)

_t>0

and defined as (see e.g. Protter [28])

[X, X]

t

= X

_t²

− 2 Z

t

0

X

s−

dX

s

where we have set X

0−

= 0. Similarily, the quadratic covariation of two

semimartingales X and Y is the following process also known as the bracket

CHANGE OF LAW

process

[X, Y ]

_t

:= X

_t

Y

_t

− Z

t

0

X

s−

dY

_s

− Z

Y

s−

dX

_s

.

In the Hilbert–space–valued setting, one identifies H with `

2

and defines [X, X] = P

∞

n=1

[X

ⁿ

, Xn], where X

ⁿ

is the nth coordinate of X.

Definition 1 (differential subordination) . Let X and Y two adapted c` adl` ag semimartingales taking values in a separable Hilbert space. We say that Y is differentially subordinate by quadratic variation to X iff

[X, X]

t

− [Y, Y ]

t

is a nondecreasing and nonnegative function of t > 0.

Let us denote by X

^c

the unique continuous part of X with [X, X]

_t

= |X

₀

|

²

+ [X

^c

, X

^c

]

_t

+ X

0<s6t

|∆X

_s

|

²

,

where ∆X

_t

:= X

_t

− X

t−

. There holds [X, X]

^c_t

= [X

^c

, X

^c

]

_t

and ∆[X, X]

_t

=

|∆X

_t

|

²

. We have the following characterisation distinguishing the continuous and jump parts proved by Wang in [33]

Lemma 1 . If X and Y are semimartingales, then Y is differentially subordinate to X if and only if (i) [X, X]

^c_t

− [Y, Y ]

^c_t

is a nonnegative and nondecreasing function of t, (ii) the inequality |∆Y

_t

| 6 |∆X

_t

| holds for all t > 0 and (iii) |Y

₀

| 6 |X

₀

|.

1.2. Martingales in non–homogeneous weighted spaces.

Let again (Ω, F

_∞

, P ) be a probability space with a nondecreasing right continuous family F := (F

_t

)

_t>0

of sub σ–fields of F

_∞

such that F

₀

contains all F

_∞

–null sets. If X and Y are adapted c` adl` ag square integrable H –valued martingales and Y is differentially subordinate to X, then it is obvious that

(1.2) kY k

_L2

6 kXk

_L2

.

Recall here that kXk

_L2

:= sup

_t

kX

_t

k

_L2

, where (1.3) kX

_t

k

²_L2

:= E |X

_t

|

²

=

Z

Ω

|X

_t

(ω)|

²

dP(ω).

DOMELEVO AND PETERMICHL

Assume again that Y is differentially subordinate to X. We might insist on the underlying probability space at hand by saying in short that X and Y are P –martingales and that Y is P –differentially–subordinate to X. The main concern of this paper is to obtain sharp inequalities similar to (1.2) under a change of law in the definition of the L

²

–norm according to [9].

Let w be a positive, uniformly integrable martingale (that we often identify with its closure w

∞

) that we call a weight. Let d Q := d P

^w

:= wd P and (Ω, F

_∞

, Q ) be a probability space with the same assumptions as (Ω, F

_∞

, P ) but with a change of the probability law.

Question 1 . Let P and Q such that (Ω, F

_∞

, P ) and (Ω, F

_∞

, Q ) are two filtered probability spaces as described above. Does there exist a constant C

_P,Q

> 0 depending only on ( P , Q ) such that if X and Y are uniformly integrable P –martingales adapted to F and Y is P –differentially–subordinate to X, then

kY k

_2,Q

6 C

_P,Q

kXk

_2,Q

?

We look for C

_P,Q

:= C

w

allowing to compare kY k

_2,Q

:= kY k

_2,w

and kXk

_2,Q

:= kXk

_2,w

. We will need also u = w

⁻¹

the inverse weight and we as- sume u uniformly integrable. We will finally note d P

^u

:= ud P . It follows that P

^w

and P

^u

are probability measures on Ω up to the obvious normalisations.

The necessary condition on the weight is classical:

Definition 2 (A

2

class) . Let (Ω, F, (F

_t

)

_t>0

, P ) be a filtered probabil- ity space. We say that the weight w > 0 is in the A

₂

–class, iff the A

₂

– characteristic of the weight w, noted Q

^F₂

[w] and defined as

Q

^F₂

[w] := sup

τ

ess . sup

ω

(w)

τ

(w

⁻¹

)

τ

with the first supremum running over all adapted stopping times, is finite.

We often write Q

^F₂

[w] := sup

_τ

ess . sup

_ω

w

τ

u

τ

where u := w

⁻¹

is the inverse weight.

2. Statement of the main results.

Theorem 1 (differential subordination under change of law) . Let X and Y be two adapted uniformly integrable c` adl` ag H –valued martingales such that Y is differentially subordinate to X. Let w be a weight in the A

2

class. Then

kY k

_L2(w)

. Q

^F₂

[w]kXk

_L2(w)

and the linear growth in Q

^F₂

[w] is sharp.

CHANGE OF LAW

The upper estimate of this result will be a consequence of the following bilinear estimate:

Proposition 1 (bilinear estimate) . Let X and Y be two adapted uni- formly integrable c` adl` ag H –valued martingales such that Y is differentially subordinate to X. Let w an admissible weight in the A

₂

class. Then

E Z

∞

0

|d[Y, Z]

_t

| . Q

^F₂

[w]kXk

_L2(w)

kZk

_L2(u)

.

The proof of the above proposition will be based on the Bellman function method. Let us fix some notation. Let us denote by V the quadruplet

V := (x, y, r, s) ∈ H × H × R

^∗+

× R

^∗+

=: S .

The variables (x, y) will be associated to H –valued martingales whereas the variables (r, s) to R–valued martingales for the weights. We introduce the domain

D

_Q

:= {V ∈ S : 1 6 rs 6 Q} .

We will often restrict our attention to truncated weights, that is given 0 <

ε < 1, we consider weights w such that ε 6 w 6 ε

⁻¹

. This means the variables r and s are bounded below and above and the domain becomes

D

^ε_Q

:=

V ∈ D

_Q

: ε 6 r 6 ε

⁻¹

, ε 6 s 6 ε

⁻¹

.

Lemma 2 (existence and properties of the Bellman function) . There exists a function B (V ) = B

_Q

that is C

¹

on D

^ε_Q

, and piecewise C

²

, with the estimate

B (V ) . |x|

²

r + |y|

²

s and on each subdomain where it is C

²

there holds

d

²

B > 2

Q |dx||dy|.

Whenever V and V

₀

are in the domain, the function has the property B (V ) − B(V

0

) − dB(V

0

)(V − V

0

) > 2

Q |x − x

0

||y − y

0

|.

Moreover, we have the estimates

|(∂

_x²

Bdx, dx)| . ε

⁻¹

|dx|

²

, |(∂

_y²

Bdy, dy)| . ε

⁻¹

|dy|

²

with the implied constants independent of V and (dx, dy).

We have an explicit expression of the function described in Lemma 2.

This is, aside from Theorem 1, one of the main results of this paper.

DOMELEVO AND PETERMICHL

3. Existence and properties of the Bellman function.

Proof of Lemma 2 (existence and properties of the Bellman function) We give an explicit expression for such a function. Let V = (x, y, r, s) and W = (r, s). We first consider

B

1

(x, y, r, s) = hx, xi

r + hy, yi s . Then trivially 0 6 B

₁

6

^hx,xi_r

+

^hy,yi_s

and

(d

²

B

₁

dV, dV ) = 2

r hdx, dxi + 2hx, xi

r

³

(dr)

²

− 4 hx, dxi r

²

dr + 2

s hdy, dyi + 2hy, yi

s

³

(ds)

²

− 4 hy, dyi s

²

ds

= 2

r D

dx − x

r dr, dx − x r dr

E

+ 2 s

D dy − y

s ds, dy − y s ds

E

> 0.

Letting V

₀

= (x

₀

, y

₀

, r

₀

, s

₀

) and V = (x, y, r, s) we also calculate

−(B

₁

(V

0

) − B

1

(V ) + dB

1

(V

0

)(V − V

0

))

= −

x

²₀

r

0

− x

²

r + 2x

0

r

0

(x − x

0

) − x

²₀

r

₀²

(r − r

0

)

− y

²₀

s

0

− y

²

s + 2y

₀

s

0

(y − y

0

) − y

²₀

s

²₀

(s − s

0

)

= r x

r − x

₀

r

0

, x r − x

₀

r

0

+ s

y s − y

₀

s

0

, y s − y

₀

s

0

. We now consider the two functions from [30];

K(r, s) =

√ rs

√ Q

1 −

√ rs 8 √

Q

,

N (r, s) =

√ rs

√ Q 1 − (rs)

^3/2

128Q

^3/2

!

in the domain 1 6 rs 6 Q. We have 0 6 K 6

1 − 1 8 √

Q √

√ rs Q <

√ rs

√ Q 6 1,

CHANGE OF LAW

0 6 N 6

1 − 1

128Q

^3/2

√

√ rs Q <

√ rs

√ Q 6 1, so in particular rs > rs − K

²

> rs

1 −

_Q¹

. One calculates that

−(d

²

KdW, dW ) > 1

8Q |dr||ds|,

−(d

²

N dW, dW ) & 1

Q

²

s

²

(dr)

²

,

−(d

²

N dW, dW ) & 1

Q

²

r

²

(ds)

²

.

Furthermore, if W and W

₀

belong to the domain of K or N , then one also has

K(W

0

) − K(W ) + dK(W

0

)(W − W

0

) & 1

Q |r − r

0

||s − s

0

|, N (W

0

) − N (W ) + dN (W

0

)(W − W

0

) & 1

Q

²

s

0

s|r − r

0

|

²

, N (W

0

) − N (W ) + dN (W

0

)(W − W

0

) & 1

Q

²

r

0

r|s − s

0

|

²

. These remarkable one-leg concavity properties were proven in [30].

Let now

B

₂

= hx, xi

2r −

_s(N(r,s)+1)¹

+ hy, yi

s = hx, xi

r + M (r, s) + hy, yi s , where

M(r, s) = r − 1

s(N (r, s) + 1) .

We will need an appropriate lower bound for the Hessian of B

₂

. One easily checks that the functions

F (x, r, M ) = hx, xi r + M , G(r, s, N ) = 1

s(N + 1)

are convex, directly from the analysis of their Hessians. In order to estimate the Hessian of B

₂

from below, one merely requires estimates of derivatives;

−∂

_M

F = hx, xi

(r + M )

²

> hx, xi

4r

²

and −∂

_N

G = 1

s(N + 1)

²

> 1

4s .

DOMELEVO AND PETERMICHL

Since 0 6 r −

s(N(r,s)+1)¹

6 r, we know 0 6 B

₂

6

^|x|_r²

+

^|y|_s²

. Therefore (d

²

B

₂

dV, dV )

& hx, xi 4r

²

1 s(N + 1)

²

1

Q

²

|dr|

²

s

²

+ 2 s

D dy − y

s ds, dy − y s ds

E

& |x|

²

s

Q

²

r

²

|dr|

²

+ 2 s

D dy − y

s ds, dy − y s ds

E

& |x|

Qr |dr|

dy − y

s ds .

The function B

2

has the additional property

−(B

₂

(V

₀

) − B

₂

(V ) + dB

₂

(V

₀

)(V − V

₀

))

& hx

₀

, x

₀

i

Q

²

r

₀²

s(r − r

₀

)

²

+ s y

s − y

₀

s

₀

, y

s − y

₀

s

₀

. Indeed, write

hx, xi 2r −

s(N(r,s)+1)¹

= H(x, r, s, N(r, s)) with H(x, r, s, N ) = hx, xi 2r −

_s(N¹₊₁₎

. Observe that H is convex and that

−∂

_N

H & hx, xi Q

²

r

²

s .

Setting P

0

= (x

0

, r

0

, s

0

, N

0

) and P = (x, r, s, N), we have by convexity of H that H(P ) > H(P

₀

) + dH(P )(P − P

₀

). Equivalently

H(P ) − H(P

₀

)

−∂

_x

H(P

₀

)(x − x

₀

) − ∂

_r

H(P

₀

)(r − r

₀

) − ∂

_s

H(P

₀

)(s − s

₀

)

> −∂

_N

H(P

₀

)(N

₀

− N ).

Recall the one–leg concavity property of N ;

N (r

0

, s

0

) − N (r, s) + ∂

r

N (r

0

, s

0

)(r − r

0

) + ∂

s

N (r

0

, s

0

)(s − s

0

)

& 1

Q

²

s

0

s|r − r

0

|

²

.

Setting N

0

= N (r

0

, s

0

), N = N (r, s) and using the lower derivative estimate of H and the chain rule, we obtain the following one–leg convexity for B

₂

:

B

2

(V ) − B

2

(V

0

) − dB

2

(V

0

)(V − V

0

)

& hx

₀

, x

₀

i

Q

²

r

₀²

s|r − r

₀

|

²

+ s y

s − y

₀

s

0

, y s − y

₀

s

0

.

CHANGE OF LAW

Analogously

B

3

= hx, xi

r + hy, yi

2s −

_r(N(r,s)+1)¹

has the same size estimates as well as

(d

²

B

₃

dV, dV ) & |y|

Qs |ds|

dx − x

r dr and one–leg convexity

B

3

(V ) − B

3

(V

0

) − dB

3

(V

0

)(V − V

0

)

& hy

₀

, y

₀

i

Q

²

s

²₀

r|s − s

₀

|

²

+ r x

r − x

₀

r

0

, x r − x

₀

r

0

. Let us now consider

H

4

(x, y, r, s, K) = sup

a>0

β(a, x, y, r, s, K ) = sup

a>0

hx, xi

r + aK + hy, yi s + a

⁻¹

K

. Testing for critical points gives

∂

_a

β = − hx, xiK

(r + aK)

²

+ hy, yiK (as + K)

²

. So ∂

_a

β = 0 if and only if

a = a

⁰

= |y|r − |x|K

|x|s − |y|K .

Since only a > 0 are admissible, we require that |y|r − |x|K and |x|s − |y|K have the same sign. To determine sign change of ∂

a

β at a

⁰

, consider

− |x|

r + aK + |y|

as + K = (|y|r − |x|K) − a(|x|s − |y|K) (r + aK)(as + K) .

If the signs are negative, then the sign change is from negative to positive otherwise from positive the negative. For a maximum to be attained at a

⁰

> 0 we require that both numerator and denominator be positive. Then, if K is relatively small, meaning |y|r − |x|K and |x|s − |y|K are positive, we have

H

₄

(x, y, r, s, K) = β(a

⁰

, x, y, r, s)

= hx, xi(|x|s − |y|K) r(|x|s − |y|K ) + (|y|r − |x|K)K

+ hy, yi(|y|r − |x|K) s(|y|r − |x|K) + (|x|s − |y|K)K

= hx, xis − 2|x||y|K + hy, yir

rs − K

²

.

DOMELEVO AND PETERMICHL

Observe that by the above considerations on K, the denominator is never 0. The case |x| = 0 or |y| = 0 corresponds to other parts of the domain, so when K is small in the above sense, this function is in C

²

.

When |y|r − |x|K 6 0 or |x|s − |y|K 6 0, the supremum is attained at the boundary and H

₄

=

^hy,yi_s

or H

₄

=

^hx,xi_r

. Thanks to the size restrictions on K, we never have both |x|s − |y|K 6 0 and |y|r − |x|K 6 0 unless x, y = 0.

Indeed

|x|(|x|s − |y|K) + |y|(|y|r − |x|K)

= |x|

²

r − 2 |x||y|

rs K + |y|

²

s

= |x|

√ r − |y|

√ s

2

+ 2|x||y|

√ rs

1 − K

√ rs

.

With 1 −

^√K_rs

> 0 we see that the above is never negative and the vanishing of the quantity implies x = y = 0. If |x|s − |y|K 6 0 and |y|r − |x|K > 0 then

^hx,xi_r

<

^hy,yi_s

and H

₄

=

^hy,yi_s

, if |y|r − |x|K 6 0 and |x|s − |y|K > 0 then H

4

=

^hx,xi_r

.

Notice that when

^hx,xi_r

=

^hy,yi_s

and x, y 6= 0 then |y|r − |x|K > 0 and

|x|s − |y|K > 0. Indeed, we have seen

^|x|_r²

− 2

^|x||y|_rs

K +

^|y|_s²

> 0. Thus

hx,xi

r

=

^hy,yi_s

>

^|x||y|_rs

K and |y|r − |x|K > 0 and |x|s − |y|K > 0.

Thus H

₄

∈ C

²

for these parts of the domain. We also see from these considerations that in order to see H

4

∈ C

¹

we only need to check the cuts |x|s − |y|K = 0 and |y|r − |x|K > 0 as well as |y|r − |x|K = 0 and

|x|s − |y|K > 0.

When |y|r − |x|K > 0 and |x|s − |y|K > 0 (we call this part of the domain R

₁

),

(∂

x

H

4

, dx) = 2 hdx, xi

|x|

|x|s − |y|K rs − K

²

,

∂

_r

H

₄

= − (|x|s − |y|K)

²

(rs − K

²

)

²

,

∂

_K

H

₄

= −2 (|x|s − |y|K)(|y|r − |x|K) (rs − K

²

)

²

.

We first prove that ∂

x

H

4

is continuous throughout. Recall that we have to treat three regions: R

₂

, where |y|r − |x|K > 0 and |x|s− |y|K 6 0, R

₃

, where

|x|s − |y|K > 0 and |y|r − |x|K 6 0 and R

1

. Inside R

2

we have H

4

=

^hy,yi_s

and thus ∂

_x

H

₄

= 0. Inside R

₃

we have H

₄

=

^hx,xi_r

and thus ∂

_x

H

₄

=

^2hx,dxi_r

.

CHANGE OF LAW

Inside R

1

we have

∂

_x

H

₄

= 2 hdx, xi

|x|

|x|s − |y|K rs − K

²

= 2hx, dxi |x|s − |y|K

r(|x|s − |y|K) + (|y|r − |x|K)K .

Consider three cases, first, let us first approach a boundary point of R

₁

from within R

₁

such that |y|r −|x|K = a > 0 and |x|s−|y|K = 0. Let 0 < ε < a/2 and assume a− ε < |y|r − |x|K < a + ε and 0 < |x|s − |y|K < ε. There holds

|h∂

_x

H

₄

, dxi| 6 2|dx|

_rs−K^ε ₂

. ε|dx| since rs − K

²

is bounded below. Letting ε → 0 shows continuity at this point. Second, let us approach a boundary point such that |x|s − |y|K = a > 0 and |y|r − |x|K = 0 from within R

1

. Assume therefore again 0 < ε < a/2, a − ε < |x|s − |y|K < a + ε and 0 < |y|r − |x|K < ε. We show there holds (∂

_x

H

₄

, dx) .

_a^ε

|dx|. Since

1

r − (|y|r − |x|K)K |x|s − |y|K r

²

(|x|s − |y|K)

²

6 (|x|s − |y|K)

r(|x|s − |y|K) + (|y|r − |x|K)K

6 1

r , we have

2hx, dxi(|x|s − |y|K)

r(|x|s − |y|K) + (|y|r − |x|K )K − 2hx, dxi r

6 2|hx, dxi| (|y|r − |x|K)K

r

²

(|x|s − |y|K) . |x||dx| ε

a .

Since 0 < |y|r − |x|K < ε and s, r, K controlled, one can deduce from

|x|s − |y|K ∼ a that |x| ∼ a. Last, let us approach |y|r − |x|K = 0 and

|x|s − |y|K = 0. To this end, one can see that if 0 < |y|r − |x|K < ε and 0 < |x|s − |y|K < ε then |x|, |y| . ε, establishing continuity in the third case.

The derivative ∂

_r

H

₄

is similar since the term

^|x|s−|y|K_rs−K2

reappears as a square and in R

3

notice that H

4

=

^hx,xi_r

so ∂

r

H

4

= −

^hx,xi_r2

. It is easy to see that the derivative ∂

_K

H

₄

is zero in R

₂

and R

₃

as well as when approaching the boundary of R

1

.

By symmetry in the variables, the function is in C

¹

. As a consequence,

B

₄

(x, y, r, s) = H

₄

(x, y, r, s, K(r, s)) ∈ C

¹

.

DOMELEVO AND PETERMICHL

Function H

4

is a supremum of convex functions and therefore convex. It has been shown indirectly in [23] that −∂

_K

H

₄

> 0 everywhere and that in R

⁰₁

⊂ R

1

where |y|r−2|x|K > 0 and |x|s−2|y|K > 0 we have −∂

_K

H

4

&

^|x||y|_rs

. We present an easier argument. Recall that

−∂

_K

H

4

= 2 (|x|s − |y|K)(|y|r − |x|K) (rs − K

²

)

²

= 2 (|x|s − |y|K)(|y|r − |x|K)|x||y|

(r(|x|s − |y|K) + K(|y|r − |x|K))(s(|y|r − |x|K) + K(|x|s − |y|K)) . So −∂

_K

H

4

> c

^|x||y|_rs

if

rs

c > K

²

+ rs + Kr(|x|s − |y|K )

|y|r − |x|K + Ks(|y|r − |x|K)

|x|s − |y|K .

Now K

²

6 1 6 rs and when |y|r − 2|x|K > 0 then |y|r − |x|K > |x|K.

Similarly |x|s − |y|K > |y|K. So the last two terms are bounded by

^Kr|x|s_|x|K

+

Ks|y|r

|y|K

= 2rs. So c = 1/4 works. In R

⁰₁

, (d

²

B

4

dV, dV ) > 4 |x||y|

8rsQ |dr||ds| = |x||y|

2rsQ |dr||ds|.

We need to add more functions with the good concavity for other K. Let B

5

= hx, xi

2r −

s(K(r,s)+1)¹

+ hy, yi s .

Since 0 6 r −

s(K(r,s)+1)¹

6 r we know 0 6 B

5

6

^|x|_r²

+

^|y|_s²

. Now the Hessian estimate becomes

(d

²

B

5

dV, dV ) > hx, xi 4r

²

1 s(K + 1)

²

1

8Q |dr||ds| > |x|

²

128Qsr

²

|dr||ds|.

So the function B

₅

is convex and when 2|x|K > |y|r, then (d

²

B

5

dV, dV ) > |x||y|

256KQsr |dr||ds| > |x||y|

256Qsr |dr||ds|.

If we set

B

₆

= hx, xi

r + hy, yi

2s −

r(K(r,s)+1)¹

,

CHANGE OF LAW

then B

₆

is convex, 0 6 B

₆

6

^|x|_r²

+

^|y|_s²

and when 2|y|K > |x|s then (d

²

B

₆

dV, dV ) > |x||y|

256Qsr |dr||ds|.

Putting the above facts together, we see that the function B

7

= B

4

+B

5

+B

6

satisfies

(d

²

B

₇

dV, dV ) & |x||y|

Qsr |dr||ds|.

Through similar considerations as above, we have discrete one-leg convexity B

7

(V ) − B

7

(V

0

) − dB

7

(V

0

)(V − V

0

) & |x

₀

||y

₀

|

Qs

₀

r

₀

|r − r

0

||s − s

0

|.

Letting for appropriate fixed c

i

,

(3.1) B = c

₁

B

₁

+ c

₂

B

₂

+ c

₃

B

₃

+ c

₇

B

₇

,

we obtain 0 6 B .

^|x|_r²

+

^|y|_s²

and d

²

B >

_Q²

|dx||dy| in the regions where B ∈ C

²

. Indeed,

(d

²

B

1

dV, dV ) > 4

Q |dx||dy| + 4|x||y|

Qrs |dr||ds| − 4|y|

Qs |dx||ds| − 4|x|

Qr |dy||dr|, (d

²

B

2

dV, dV ) >

√ 3|x|

2Qr |dy||dr| −

√ 3|x||y|

2Qrs |dr||ds|, (d

²

B

3

dV, dV ) >

√ 3|y|

2Qs |dx||ds| −

√ 3|x||y|

2Qrs |dr||ds|, (d

²

B

₇

dV, dV ) > |x||y|

256Qrs |dr||ds|,

where the last inequality holds in the regions where the function B

₄

∈ C

²

. The weighted sum of these inequalities according to (3.1) yields the desired inequality on convexity. Now,

B

₁

(V ) − B

₁

(V

₀

) − dB

₁

(V

₀

)(V − V

₀

)

& rs Q x r − x

₀

r

₀

y s − y

₀

s

₀

> rs

Q

x r − x

₀

r

₀

, y s − y

₀

s

₀

,

B

₂

(V ) − B

₂

(V

₀

) − dB

₂

(V

₀

)(V − V

₀

)

& s Q

|x

₀

|

r

₀

|r − r

₀

| y s − y

0

s

₀

> s

Q

x

0

r

₀

, y s − y

0

s

₀

|r − r

₀

|,

DOMELEVO AND PETERMICHL

B

3

(V ) − B

3

(V

0

) − dB

3

(V

0

)(V − V

0

)

& r Q

|y

₀

| s

0

|s − s

0

| x r − x

0

r

0

> r

Q

x r − x

0

r

0

, y

0

s

0

|s − s

0

|,

B

7

(V ) − B

7

(V

0

) − dB

7

(V

0

)(V − V

0

)

& 1

Q |x

₀

||y

₀

||r − r

₀

||s − s

₀

| > 1

Q |hx

₀

, y

₀

i||r − r

₀

||s − s

₀

|.

Notice that the last inequalities also remain true when we replace x by Θx and x

0

by Θx

0

where the rotation Θ is chosen so that Θ(x − x

0

) and y − y

0

have the same direction and thus we may assume that hx − x

₀

, y − y

₀

i =

|x − x

0

||y − y

0

|.

Summing the above inequalities gives

Q(B (V ) − B(V

₀

) − dB (V

₀

)(V − V

₀

))

&

x r − x

0

r

₀

r, y

s − y

0

s

₀

s + y

0

s

₀

(s − s

0

)

+ x

₀

r

₀

(r − r

₀

), y

s − y

₀

s

₀

s + y

₀

s

₀

(s − s

₀

)

= x

r − x

₀

r

0

r, y − y

₀

+ x

₀

r

0

(r − r

₀

), y − y

₀

= hx − x

₀

, y − y

₀

i = |x − x

₀

||y − y

₀

|

and we have proved the one-leg convexity. It remains to bound the second derivatives in x and y. Let ε be the cut off of the weights so that ε 6 r, s 6 ε

⁻¹

. We calculate

∂

_x²

hx, xi r dx, dx

= 2hdx, dxi

r . ε

⁻¹

hdx, dxi;

∂

_x²

hx, xi

r + M(r, s) dx, dx

= 2hdx, dxi

r + M(r, s) 6 2hdx, dxi

r . ε

⁻¹

hdx, dxi;

∂

_x²

hx, xis − 2|x||y|K + hy, yir rs − K

²

dx, dx

= 2hdx, dxis

rs − K

²

− 2|y|K rs − K

²

hdx, dxi

|x| − hx, dxi

²

|x|

³

. ε

⁻¹

hdx, dxi, where the last implied constant uses the lower bound for rs−K

²

> rs

1 −

_Q¹

>

1 −

_Q¹

. We used hx, dxi

²

6 hx, xihdx, dxi which yields

^hdx,dxi_|x|

−

^hx,dxi_|x|3 ²

> 0.

CHANGE OF LAW

These imply that for V ∈ D

_Q,ε

we have

(3.2) (∂

²_x

B(V )dx, dx) . ε

⁻¹

|dx|

²

.

This concludes the proof of Lemma 2.

Convexities of the form d

²

B(V ) > 2|dx||dy| can be self improved using the following interesting lemma:

Lemma 3 . (ellipse lemma, Dragicevic – Treil – Volberg [14]) Let H be a Hilbert space with A, B two positive definite operators on H . Let T be a self-adjoint operator on H such that

(T h, h) > 2(Ah, h)

^1/2

(Bh, h)

^1/2

for all h ∈ H. Then there exists τ > 0 satisfying

(T h, h) > τ (Ah, h) + τ

⁻¹

(Bh, h) for all h ∈ H .

For our specific Bellman function, we will need a quantitative version:

Lemma 4 (quantitative ellipse lemma for B) . Let V ∈ D

^ε_Q

. Assume moreover that B is C

²

at V . Then there exists τ (V ) > 0 such that

Qd

²_V

B (V ) > τ (V )|dx|

²

+ (τ (V ))

⁻¹

|dy|

²

. Moreover, we have the bound

Q

⁻¹

ε . τ (V ) . Qε

⁻¹

.

Proof of Lemma 4 (quantitative ellipse lemma for B) Let V ∈ D

_Q^ε

. We have already seen in Lemma 2 that

d

²_V

B(V ) > 2

Q |dx||dy|.

The ellipse lemma [14] implies the existence of τ (V ) such that for all vectors dx and dy there holds

Qd

²_V

B (V ) > τ (V )|dx|

²

+ (τ (V ))

⁻¹

|dy|

²

.

We can estimate τ (V ) by testing the Hessian on any dV of the form dV = (dx, 0, 0, 0),

τ (V )|dx|

²

6 Q(d

²_V

B(V )dV, dV ) = Q(∂

_x²

B(V )dx, dx) . Qε

⁻¹

|dx|

²

,

DOMELEVO AND PETERMICHL

where the last inequality follows from (3.2). Hence τ (V ) . Qε

⁻¹

as claimed.

The same bound holds for (τ (V )

⁻¹

) by testing against dV = (0, dy, 0, 0).

Finally, we have proved that for all V ∈ D

^ε_Q

, Q

⁻¹

ε . τ (V ) . Qε

⁻¹

.

We now address the lack of smoothness of B. All functions aside from H

₄

that appear are at least in C

²

. We apply a standard mollifying procedure via convolution with ϕ

_`

directly on H

4

(x, y, r, s, K), now only taking real variables with x, y positive, 1 < rs < Q and 0 < K < 1. Here ϕ denotes a standard mollifying kernel in the five real variables (x, y, r, s, K) ∈ R

⁵

with support in the corresponding unit ball, whereas ϕ

_`

(·) := `

⁻⁵

ϕ(·/`) denotes its scaled version with support of size `. By slightly changing the constructions, the upper and lower estimate on the product rs can be mod- ified at the cost of a multiplicative constant in the final estimate of the Bellman function. Also take into account that the weights are cut, there- fore bounded above and below. Further, we will assume that the positive variables x and y be bounded below. These considerations give us enough room to smooth the function H

₄

. It is important that H

₄

is at least in C

¹

and its second order partial derivatives exist almost everywhere. So we have d

²

(H

4

∗ ϕ

`

) = (d

²

H

4

) ∗ ϕ

`

. Last, we are observing that as long as the norms of vectors |x| and |y| are bounded away from 0, our function H

₄

∗ ϕ

_`

, mol- lified in R

⁵

remains smooth when taking vector variables (observe that the final Bellman function only depends upon |x| and |y|). It is important that the smoothing happens before the function is composed with K, we there- fore preserve fine convexity properties, in particular also the much needed one-leg convexity. Size estimates change slightly, but are recovered when the mollifying parameter goes to 0. These details are either standard and have appeared in numerous articles on Bellman functions, or an easy consequence of the above construction.

Lemma 5 (regularised Bellman function and its properties) . Let ε > 0 given. Let 0 < ` 6 ε/2. There exists a function B

_`

(x, y, r, s) defined on the domain

D

_Q^ε,`

:=

V ∈ D

^ε_Q

; |x| > `, |y| > ` ⊂ D

_Q^ε

such that for all V

0

, V ∈ D

^ε,`_Q

, we have

B

_`

. (1 + `) |x|

²

r + |y|

²

s

,

CHANGE OF LAW

(3.3) d

²_V

B

_`

(V ) > 2

Q |dx||dy|, (3.4) B

_`

(V ) −B

_`

(V

₀

)−d

_V

B

_`

(V

₀

)(V −V

₀

) > 1

Q |∆x||∆y| = 1

Q |x −x

₀

||y −y

₀

|.

Moreover the quantitative ellipse lemma now holds in the form Qd

²_V

B

_`

(V ) > τ

_`

(V )|dx|

²

+ (τ

_`

(V ))

⁻¹

|dy|

²

,

where τ

`

:= τ

`

(V ) is a continuous function of its arguments, and where Q

⁻¹

ε . τ

_`

(V ) . Qε

⁻¹

.

4. Dissipation estimates.

Let V := (X, Z, u, w) a c` adl` ag adapted martingale with values in D

^ε_Q

. In order to bound the H –valued martingale X := (X

¹

, X

²

, . . .) away from 0, it is classical to introduce the R × H –valued martingales X

^a

:= (a, X

¹

, X

²

, . . .) where a > 0. It follows that kX

^a

k

²

= kXk

²

+a

²

and kX

^a

k > a, and the same construction holds for Z. We note V

^a

:= (X

^a

, Z

^a

, u, w). Given a smoothing parameter ` > 0 a smoothing parameter, take a > ` then it follows that

V ∈ D

^ε_Q

⇒ V

^a

∈ D

_Q^ε,`

.

The main result of this section is the following dissipation estimate:

Proposition 2 (dissipation estimate) . Let ε > 0, ` > 0 as defined above. Let V a c` adl` ag adapted martingale with V ∈ D

_Q^ε

. Let F

_t

:= E (|X

∞

|

²

w

_∞^ε

|F

_t

) and G

t

:= E (|Z

_∞

|

²

u

^ε_∞

|F

_t

). Let finally a > `. We have

Q(1 + `)( E F

_t

+ E G

_t

+ 2a

²

ε

⁻¹

)

& 1 2 E

Z

t 0

τ

_`

(V

s−

)d[X, X]

^c_s

+ (τ

_`

(V

s−

))

⁻¹

d[Z, Z]

^c_s

+ E X

0<s6t

|∆X

_s

||∆Z

_s

|.

We need the preliminary lemma

Lemma 6 (comparison of quadratic forms in stochastic integrals) . Let Q denote the set of quadratic forms from R

^m

× R

^m

→ R . Let A := (A

_αβ

)

_16α,β6m

and B := (B

_αβ

)

_16α,β6m

two Q–valued c` adl` ag processes. Assume for all t > 0 and a.s. that A(t) > B(t) (resp. A(t) > |B(t)|), in the sense that

(AdV, dV ) > (BdV, dV ) (resp .(AdV, dV ) > |(BdV, dV )|)

DOMELEVO AND PETERMICHL

for all dV ∈ R

^m

. Abbreviating A

s−

: d[V, V ]

s

= P

α,β

(A

_αβ

)

s−

d[V

α

, V

_β

]

s

we have for all t > 0,

E Z

t

0

A

s−

: d[V, V ]

_s

> E Z

t

0

B

s−

: d[V, V ]

_s

,

resp . E Z

t

0

A

s−

: d[V, V ]

s

> E Z

t

0

|B

_s−

: d[V, V ]

s

|

.

Proof of Lemma 6 (comparison of quadratic forms in stochastic integrals) Under the hypotheses above, let us consider the case A(t) > B(t), the case A(t) > |B(t)| being treated in the same manner. Given t > 0, assume that

Z

_t

0

A

s−

: d[V, V ]

_s

= X

α,β

Z

_t

0

(A

_αβ

)

s−

d[V

_α

, V

_β

]

_s

< ∞,

otherwise the claim is proved. Given the process V , let σ

_n

:= (0 6 T

₀ⁿ

6 T

₁ⁿ

6 . . . 6 T

_iⁿ

6 . . . 6 T

_kⁿ_n

6 t) denote a random partition of stopping times tending to the identity as n tends to infinity. Given α and β, we have that A

_αβ

is a R –valued c` adl` ag process. It follows (see e.g. Protter [28]) that the stochastic integral

(4.1)

Z

t 0

A

_αβ

(s−)d[V

_α

, V

_β

]

_s

is the limit in ucp (uniform convergence in probability) as n tends to infinity of sums

S

_αβ^A

:=

kn−1

X

i=0

A

_αβ

(T

_iⁿ

)(V

^T

n

αi+1

− V

^T

n

αi

)(V

^T

n i+1

β

− V

^T

n i

β

)

involving the stopping times defined above. Since A > B, summing w.r.t.

α, β yields, for any s ∈ [0, t],

CHANGE OF LAW



 X

α,β

S

_αβ^A



 (s) := X

α,β kn−1

X

i=0

A

_αβ

(T

_iⁿ

)(V

^T

n

α,si+1

− V

_α,s^Tin

)(V

^T

n i+1

β,s

− V

_β,s^Tin

)

=

kn−1

X

i=0

X

α,β

A

_αβ

(T

_iⁿ

)(V

^T

n

α,si+1

− V

α,s^Tin

)(V

^T

n i+1

β,s

− V

_β,s^Tin

)

>

kn−1

X

i=0

X

α,β

B

αβ

(T

_iⁿ

)(V

^T

n

α,si+1

− V

^T

n

α,si

)(V

^T

n i+1

β,s

− V

^T

n i

β,s

)

>



 X

α,β

S

_αβ^B



 (s),

with an obvious definition for S

_αβ^B

. Passing to the limit in the sums P

α,β

gives the result.

Proof of Proposition 2 (dissipation estimates)

Step 1. We first pas to a finite dimensional case. Let V a c` adl` ag adapted martingale with V ∈ D

^ε_Q

. Then V

^a

∈ D

_Q^ε,`

. We note X

^a,m

the projection of X

^a

∈ R × H onto R × R

^m

, and introduce accordingly Z

^a,m

and V

^a,m

. Notice that [X

^a

, X

^a

] = a

²

+ [X, X] and similarly [X

^a,m

, X

^a,m

] = a

²

+ [X

^m

, X

^m

].

Since V

^a,m

∈ D

_Q^ε,`

where B

_`

is C

²

and we can apply Itˆ o’s formula and obtain, for all t > 0, almost sure paths,

B

`

(V

_t^a,m

)

= B

`

(V

₀^a,m

) + Z

t

0+

d

V

B(V

_s−^a,m

)dV

_s^m

+ 1 2

Z

t 0+

d

²_V

B

`

(V

_s−^a,m

) : d[V

^m

, V

^m

]

^c_s

+ X

0<s6t

{B

_`

(V

_s^a,m

) − B

_`

(V

_s−^a,m

) − d

_V

B

_`

(V

_s−^a,m

)∆V

_s^m

}.

Thanks to Lemma 4 and Lemma 6, the concavity properties (3.3) of B

_`

imply for the continuous part

1 2

Z

t 0+

d

²_V

B

_`

(V

_s−^a,m

) : d[V

^m

, V

^m

]

^c_s

> 1 2Q

Z

t 0+

τ

_`

(V

_s−^a,m

)d[X

^m

, X

^m

]

^c_s

+ (τ

_`

(V

_s−^a,m

))

⁻¹

d[Z

^m

, Z

^m

]

^c_s

.

DOMELEVO AND PETERMICHL

Also, the concavity properties (3.4) of B

_`

for the jump part B

_`

(V

_s^a,m

) − B

_`

(V

_s−^a,m

) − d

_V

B

_`

(V

_s−^a,m

)∆V

_s^m

> 1

Q |∆X

_s^m

||∆Z

_s^m

|.

Plugging the continuous and jump dissipation estimates into Itˆ o’s formula yields for all times, almost sure paths,

B

`

(V

_t^a,m

) > B

`

(V

₀^a,m

) + Z

t

0+

d

V

B

`

(V

_s−^a,m

)dV

_s^m

+ 1

2Q Z

t

0

τ

_`

(V

_s−^a,m

)d[X

^m

, X

^m

]

^c_s

+ (τ

_`

(V

_s−^m

))

⁻¹

d[Z

^m

, Z

^m

]

^c_s

+ 1

Q X

0<s6t

|∆X

_s^m

||∆Z

_s^m

|.

Step 2. For technical reasons in the proof, we work with bounded martin- gales that we obtain through a usual stopping procedure. Recall that V is a c` adl` ag adapted martingale with V ∈ D

^ε_Q

and V

^a

∈ D

^ε,`_Q

. For all M ∈ N, de- fine the stopping time T

_M

:= inf {t > 0; |V

^a

|

²_t

+ [V

^a

, V

^a

]

_t

> M

²

}, so that T

_M

is a stopping time that tends to infinity as M goes to infinity. It follows that V

^a,TM

is a local martingale, and that V

^a,TM−

and [V

^a

, V

^a

]

^TM−

are bounded semimartingales. Let m ∈ N

^?

and V

^a,m

the projection of V

^a

onto R

^m

⊂ H . For each M , there exists a sequence {T

_M,k

}

_k>1

of stopping times such that T

_M,k

↑ T

M

as k ↑ ∞, and such that (V

^a,m

)

^TM,k

is a martingale. Since

|V

^a,m

| 6 |V

^a

|, it follows that (V

^a,m

)

^TM,k−

is a bounded semimartingale, to

which we can apply the dissipation estimate of Step 1 above and obtain

CHANGE OF LAW

B

`

(V

_t∧T^a,m

M,k−

)

> B

_`

(V

₀^a,m

) +

Z

t∧T_M,k− 0+

d

_V

B

_`

(V

_s−^a,m

)dV

_s^m

+ 1

2Q

Z

t∧T_M,k− 0

τ

_`

(V

_s−^a,m

)d[X

^m

, X

^m

]

^c_s

+ (τ

_`

(V

_s−^a,m

))

⁻¹

d[Z

^m

, Z

^m

]

^c_s

+ 1

Q

X

0<s<t∧T_M,k

|∆X

_s^m

||∆Z

_s^m

|

= B

`

(V

₀^a,m

) +

Z

t∧T_M,k 0+

d

V

B

`

(V

_s−^a,m

)dV

_s^m

+ 1

2Q

Z

t∧T_M,k− 0

τ

_`

(V

_s−^a,m

)d[X

^m

, X

^m

]

^c_s

+ (τ

_`

(V

_s−^a,m

))

⁻¹

d[Z

^m

, Z

^m

]

^c_s

+ 1

Q

X

0<s<t∧T_M,k

|∆X

_s^m

||∆Z

_s^m

| − d

V

B

_`

(V

_t∧T^a,m

M,k−

)∆V

_t∧T^m_M,k

.

Taking expectation and then letting k → ∞, the dominated convergence theorem yields

EB

`

(V

_t∧T^a,m

M−

) (4.2)

> E B

_`

(V

₀^a,m

) + 1

2Q E Z

t∧TM

0

τ

`

(V

_s−^a,m

)d[X

^m

, X

^m

]

^c_s

+ (τ

`

(V

_s−^a,m

))

⁻¹

d[Z

^m

, Z

^m

]

^c_s

+ 1

Q E X

0<s<t∧T_M

|∆X

_s^m

||∆Z

_s^m

| − E {d

_V

B

_`

(V

_t∧T^a,m

M−

)∆V

_t∧T^m_M

}.

Observe that we used size properties of B

_`

, the definition of the stopping time T

_M,k

, the estimate of the τ

_`

provided by Lemma 5 and the size control of the weights.

Step 3. Now, we wish to return to the infinite dimensional case. First recall that 0 6 B

`

(V ) . (1 + `)(X

²

/u + Y

²

/w). Let F

t

:= E(|X

∞

|

²

w

∞

|F

_t

), G

t

:=

E (|Z

_∞

|

²

u

∞

|F

_t

) and F

_t^a

:= E (|X

_∞^a

|

²

w

∞

|F

_t

), G

^a_t

:= E (|Z

_∞^a

|

²

u

∞

|F

_t

). Notice that F

_t^a

= E ((|X

_∞

|

²

+ a

²

)w

∞

|F

_t

) 6 F

_t

+ E (a

²

w

^ε_∞

|F

_t

) 6 F

_t

+ a

²

ε

⁻¹

. It follows, thanks to Jensen inequality, that

B

_`

(V

_t^a

) 6 C

₀

(1 + `)(F

_t^a

+ G

^a_t

) . (1 + `)(F

_t

+ G

_t

+ 2a

²

ε

⁻¹

).