The Quadratic Variation of a Continuous Local Martingale

80 4 Continuous Semimartingales Proof We start by proving the first assertion. Uniqueness is an easy consequence of Theorem 4.8. Indeed, let A and A⁰ be two increasing processes satisfying the condition given in the statement. Then the processAtA⁰_tD.M_t²A⁰_t/.M_t²At/ is both a continuous local martingale and a finite variation process. It follows that AA⁰D0.

In order to prove existence, consider first the case whereM₀ D 0 andM is bounded (henceMis a true martingale, by Proposition4.7(ii)). FixK > 0and an increasing sequence0 D tⁿ₀ < t₁ⁿ < < tⁿ_pn D K of subdivisions ofŒ0;Kwith mesh tending to0.

We observe that, for every0 r < s and for every boundedFr-measurable variableZ, the process

NtDZ.Ms^tMr^t/

is a martingale (the reader is invited to write down the easy proof!). It follows that, for everyn, the process

Xⁿ_t D

p_n

X

iD1

Mtⁿ_i1.Mtⁿ_i^tMtⁿ_i1^t/

is a (bounded) martingale. The reason for considering these martingales comes from the following identity, which results from a simple calculation: for everyn, for every j2 f0; 1; : : : ;png,

M²_tⁿ

j 2Xⁿt_jⁿD Xj

iD1

.Mtⁿ_i Mtⁿ_i1/²: (4.4)

Lemma 4.10 We have

n;m!1lim EŒ.XKⁿX^m_K/²D0:

Proof of the lemma Let us fix n m and evaluate the product EŒXKⁿX_K^m. This product is equal to

p_n

X

iD1 p_m

X

jD1

EŒMtⁿ_i1.Mtⁿ_i Mt_iⁿ₁/Mt^m_j1.Mt^m_j Mt^m_j1/:

In this double sum, the only terms that may be nonzero are those corresponding to indicesi andj such that the interval.t^m_j1;t^m_j is contained in .t_iⁿ ₁;t_iⁿ. Indeed, suppose thattⁿ_i t^m_j1(the symmetric caset^m_j tⁿ_i1is treated in an analogous way).

4.3 The Quadratic Variation of a Continuous Local Martingale 81

Then, conditioning on the-fieldFt^m_j1, we have EŒMtⁿ_i1.Mtⁿ_i Mt_iⁿ₁/Mt^m_j1.Mt^m_j Mt^m_j1/

DEŒMtⁿ_i1.Mt_iⁿMtⁿ_i1/Mt^m_j1EŒMt_j^mMt^m_j1 jFt_j^m₁D0:

For everyj D 1; : : : ;pm, writein;m.j/for the unique indexisuch that.t^m_j1;t^m_j .tⁿ_i1;t_iⁿ. It follows from the previous considerations that

EŒXKⁿX^m_KD X

1jpm;iDi_n;m.j/

EŒMtⁿ_i1.Mt_iⁿMtⁿ_i1/Mt^m_j1.Mt^m_j Mt^m_j1/:

In each termEŒMtⁿ_i1.Mtⁿ_i Mtⁿ_i1/Mt^m_j1.Mt^m_j Mt^m_j1/, we can now decompose Mtⁿ_i Mtⁿ_i1D X

kWin;m.k/Di

.Mt^m_k Mt_k1^m /

and we observe that, ifkis such thatin;m.k/Dibutk6Dj, EŒMtⁿ_i1.Mt_k^mMt^m_k1/Mt_j^m₁.Mt^m_j Mt^m_j1/D0

(condition onFt^m_k1 ifk > jand onFt^m_j1 ifk< j). The only case that remains is kDj, and we have thus obtained

EŒXⁿKX_K^mD X

1jpm;iDin;m.j/

EŒMtⁿ_i1Mt^m_j1.Mt^m_j Mt^m_j1/²:

As a special case of this relation, we have EŒ.XK^m/²D X

1jpm

EŒM²t^m_j1.Mt^m_j Mt^m_j1/²:

Furthermore,

EŒ.XKⁿ/²D X

1ipn

EŒM²tⁿ_i1.Mtⁿ_i Mtⁿ_i1/² D X

1ipn

EŒM²tⁿ_i1EŒ.Mtⁿ_i Mtⁿ_i1/²jFtⁿ_i1 D X

1ipn

E h

M²_tⁿ

i1

X

jWin;m.j/Di

EŒ.Mt^m_j Mt^m_j1/²jFtⁿ_i1i

D X

1jpm;iDin;m.j/

EŒMt²ⁿ_i1.Mt^m_j Mt^m_j1/²;

where we have used Proposition3.14in the third equality.

82 4 Continuous Semimartingales

If we combine the last three displays, we get EŒ.Xⁿ_KX^m_K/²DE

h X

1jpm;iDin;m.j/

.Mtⁿ_i1Mt_j^m₁/².Mt^m_j Mt_j^m₁/²i :

Using the Cauchy–Schwarz inequality, we then have EŒ.XKⁿ X_K^m/²E

h

sup

1jpm;iDin;m.j/.Mtⁿ_i1Mt_j^m₁/⁴i₁₌₂ Eh X

1jpm

.Mt^m_j Mt^m_j1/²₂i₁₌₂ :

By the continuity of sample paths (together with the fact that the mesh of our subdivisions tends to0) and dominated convergence, we have

n;m!1lim;nmE h

sup

1jpm;iDin;m.j/.Mt_iⁿ₁Mt^m_j1/⁴i D0:

To complete the proof of the lemma, it is then enough to prove the existence of a finite constantCsuch that, for everym,

Eh X

1jpm

.Mt^m_j Mt^m_j1/²₂i

C: (4.5)

LetAbe a constant such thatjMtj Afor everyt 0. Expanding the square and using Proposition3.14twice, we have

Eh X

1jpm

.Mt_j^mMt^m_j1/²₂i

DEh X

1jpm

.Mt^m_j Mt_j^m₁/⁴i

C2Eh X

1j<kpm

.Mt^m_j Mt^m_j1/².Mt^m_k Mt^m_k1/²i 4A²Eh X

1jpm

.Mt^m_j Mt^m_j1/²i

C2

pXm1 jD1

E

h.Mt^m_j Mt^m_j1/²E h X^pm

kDjC1

.Mt^m_k Mt^m_k1/²ˇˇˇFt^m_j

ii

D4A²Eh X

1jpm

.Mt^m_j Mt^m_j1/²i

C2

pXm1 jD1

E

h.Mt^m_j Mt^m_j1/²EŒ.MKMt^m_j/²jFt^m_j i

4.3 The Quadratic Variation of a Continuous Local Martingale 83

12A²Eh X

1jpm

.Mt^m_j Mt^m_j1/²i D12A²EŒ.MKM₀/²

48A⁴

which gives the bound (4.5) withCD48A⁴. This completes the proof. ut We now return to the proof of the theorem. Thanks to Doob’s inequality inL² (Proposition3.15(ii)), and to Lemma4.10, we have

n;m!1lim E h

sup

tK.X_tⁿX^m_t /²i

D0: (4.6)

In particular, for everyt 2 Œ0;K, .X_tⁿ/n1 is a Cauchy sequence inL² and thus converges inL². We want to argue that the limit yields a processYindexed byŒ0;K with continuous sample paths. To see this, we note that (4.6) allows us find a strictly increasing sequence.nk/k1of positive integers such that, for everyk1,

E h

suptK.Xt^nkC1X_t^nk/²i 2^k:

This implies that

E hX¹

kD1

sup

tKjXt^nkC1X_t^nkji

<1

and thus

X1 kD1

suptKjXⁿ_t^kC1X_t^nkj<1; a.s.

Consequently, except on the negligible setN where the series in the last display diverges, the sequence of random functions.Xt^nk; 0 t K/converges uniformly onŒ0;Kas k ! 1, and the limiting random function is continuous by uniform convergence. We can thus setYt.!/ D limXt^nk.!/, for every t 2 Œ0;K, if ! 2

˝nN, and Yt.!/ D 0, for everyt 2 Œ0;K, if ! 2 N. The process.Yt/₀tK

has continuous sample paths andYtisFt-measurable for everyt2 Œ0;K(here we use the fact that the filtration is complete, which ensures thatN 2 Ft for every t0). Furthermore, since theL²-limit of.Xtⁿ/n1must coincide with the a.s. limit of a subsequence,Ytis also the limit ofX_tⁿinL², for everyt 2 Œ0;K, and we can pass to the limit in the martingale property forXⁿ, to obtain thatEŒYt j Fs D Ys

for every0 st K. It follows that.Yt^K/t0is a martingale with continuous sample paths.

84 4 Continuous Semimartingales On the other hand, the identity (4.4) shows that the sample paths of the process M_t²2X_tⁿare nondecreasing along the finite sequence.tⁿ_i; 0ipn/. By passing to the limitn! 1along the sequence.nk/k1, we get that the sample paths ofM_t² 2Ytare nondecreasing onŒ0;K, except maybe on the negligible setN. For every t 2Œ0;K, we setA^.t^K/ DM²_t 2Yton˝nN , andA^.t^K/ D 0onN . ThenA^.₀^K/ D 0,A^.t^K/isFt-measurable for everyt 2 Œ0;K,A^.K/ has nondecreasing continuous sample paths, and.M_t^K² A^._t^K^K//t0is a martingale.

We apply the preceding considerations withK D `, for every integer ` 1, and we get a process.A^{.`/</sup>t /<sub>0</sub>t`. We then observe that, for every` 1,A<sup>.`}_t^^C<sub>`</sub><sup>1/</sup> D A<sup>.`/</sup><sub>t^`</sub>for everyt 0, a.s., by the uniqueness argument explained at the beginning of the proof. It follows that we can define an increasing processhM;Misuch that hM;MitDA<sup>.`/</sup>t for everyt2Œ0; `and every`1, a.s., and clearlyM_t² hM;Mit

is a martingale.

In order to get (4.3), we observe that, ifK> 0and the sequence of subdivisions 0 D t₀ⁿ < tⁿ₁ < < tⁿ_pn D K are fixed as in the beginning of the proof, the processA^._t^K^K/ must be indistinguishable fromhM;Mit^K, again by the uniqueness argument (we know that bothM_t^K² A^._t^K^K/ andM²_t^K hM;Mit^Kare martingales).

In particular, we havehM;MiK D A^._K^K/a.s. Then, from (4.4) withj D pn, and the fact thatX_Kⁿ converges inL²toYK D ¹₂.MK² A^._K^K//, we get that

n!1lim

pn

X

jD1

.Mtⁿ_j Mt_jⁿ₁/²D hM;MiK

inL². This completes the proof of the theorem in the case whenM₀ D0andMis bounded.

Let us consider the general case. Writing Mt D M₀ C Nt, so that M²_t D M₀² C2M₀Nt CN_t², and noting that M₀Nt is a continuous local martingale (see Exercise4.22), we see that we may assume thatM₀ D0. We then set

TnDinfft0W jMtj ng

and we can apply the bounded case to the stopped martingalesM^Tn. SetA^Œn D hM^Tn;M^Tni. The uniqueness part of the theorem shows that the processesA^Œ_t^T^nC1_n and A^Œtⁿare indistinguishable. It follows that there exists an increasing processAsuch that, for everyn, the processesAt^TnandA^Œtⁿare indistinguishable. By construction, M_t^T² _nAt^Tn is a martingale for everyn, which precisely implies thatM_t²Atis a continuous local martingale. We takehM;Mit D At, which completes the proof of the existence part of the theorem.

Finally, to get (4.3), it suffices to consider the caseM₀ D 0. The bounded case then shows that (4.3) holds ifMandhM;Mitare replaced respectively byM^Tn and hM;Mit^Tn (even with convergence inL²). Then it is enough to observe that, for everyt> 0,P.tTn/converges to1whenn! 1. ut

4.3 The Quadratic Variation of a Continuous Local Martingale 85 Proposition 4.11 Let M be a continuous local martingale and let T be a stopping time. Then we have a.s. for every t0,

hM^T;M^TitD hM;Mit^T:

This follows from the fact thatM_t^T² hM;Mi_t^Tis a continuous local martingale (cf. property (c) of continuous local martingales).

Proposition 4.12 Let M be a continuous local martingale such that M₀D0. Then we havehM;Mi D0if and only if MD0.

Proof Suppose that hM;Mi D 0. Then M_t² is a nonnegative continuous local martingale and, by Proposition4.7 (i),M_t² is a supermartingale, henceEŒMt² EŒM²₀D0, so thatMtD0for everyt. The converse is obvious. ut The next theorem shows that properties of a continuous local martingale are closely related to those of its quadratic variation. IfAis an increasing process,A1

denotes the increasing limit ofAtast! 1(this limit always exists inŒ0;1).

Theorem 4.13 Let M be a continuous local martingale with M₀ 2L². (i) The following are equivalent:

(a) M is a (true) martingale bounded in L². (b) EŒhM;Mi₁ <1.

Furthermore, if these properties hold, the process M_t² hM;Mitis a uniformly integrable martingale, and in particular EŒM²₁DEŒM₀²CEŒhM;Mi₁. (ii) The following are equivalent:

(a) M is a (true) martingale and Mt2L²for every t0.

(b) EŒhM;Mit <1for every t0.

Furthermore, if these properties hold, the process M²_t hM;Mitis a martingale.

Remark In property (a) of (i) (or of (ii)), it is essential to suppose thatM is a martingale, and not only a continuous local martingale. Doob’s inequality used in the following proof is not valid in general for a continuous local martingale!

Proof

(i) ReplacingM byMM₀, we may assume thatM₀ D 0 in the proof. Let us first assume that M is a martingale bounded in L². Doob’s inequality in L² (Proposition3.15(ii)) shows that, for everyT> 0,

E h

0tTsup M_t²

i4EŒM²T:

By lettingTgo to1, we have E

h sup

t0M_t²

i4sup

t0EŒM²tDWC<1:

86 4 Continuous Semimartingales Set Sn D infft 0 W hM;Mit ng. Then the continuous local martingale M²_t^Sn hM;Mi_t^Sn is dominated by the variable

sup

s0M²_s Cn;

which is integrable. From Proposition4.7(ii), we get that this continuous local martingale is a uniformly integrable martingale, hence

EŒhM;Mi_t^SnDEŒM_t^S² _nE h

sups0M²_s iC:

By lettingn, and thenttend to infinity, and using monotone convergence, we get EŒhM;Mi1C<1.

Conversely, assume thatEŒhM;Mi1 <1. SetTn Dinfft0W jMtj ng.

Then the continuous local martingaleM²_t^Tn hM;Mit^Tn is dominated by the variable

n²C hM;Mi1;

which is integrable. From Proposition 4.7 (ii) again, this continuous local martingale is a uniformly integrable martingale, hence, for everyt0,

EŒM²t^TnDEŒhM;Mit^TnEŒhM;Mi1DWC⁰<1:

By lettingn ! 1and using Fatou’s lemma, we getEŒMt² C⁰, so that the collection.Mt/t0 is bounded inL². We have not yet verified that.Mt/t0 is a martingale. However, the previous bound onEŒM²_t^Tnshows that the sequence .Mt^Tn/n1is uniformly integrable, and therefore converges both a.s. and inL¹to Mt, for everyt0. Recalling thatM^Tnis a martingale (Proposition4.7(iii)), the L¹-convergence allows us to pass to the limitn! 1in the martingale property EŒMt^Tn jFsDMs^Tn, for0s<t, and to get thatMis a martingale.

Finally, if properties (a) and (b) hold, the continuous local martingaleM² hM;Miis dominated by the integrable variable

supt0M_t²C hM;Mi₁

and is therefore (by Proposition4.7(ii)) a uniformly integrable martingale.

(ii) It suffices to apply (i) to.Mt^a/t0for every choice ofa0. ut