Itô’s Formula

in probability. On the other hand, writing

pXn1 iD0

Xtⁿ_iC1.Xtⁿ_iC1Xtⁿ_i/D

pXn1 iD0

Xt_iⁿ.Xtⁿ_iC1Xtⁿ_i/C

pXn1 iD0

.Xtⁿ_iC1Xtⁿ_i/²;

and using Proposition4.21, we get

n!1lim

pXn1 iD0

Xtⁿ_iC1.Xt_iⁿ_C1Xtⁿ_i/D Z t

0 XsdXsC hX;Xit; in probability. The resulting limit is different fromRt

0XsdXsunless the martingale part of X is degenerate. Note that, if we add the previous two convergences, we arrive at the formula

.Xt/².X0/²D2 Z t

0 XsdXsC hX;Xit

which is a special case of Itô’s formula of the next section.

5.2 Itô’s Formula

Itô’s formula is the cornerstone of stochastic calculus. It shows that, if we apply a twice continuously differentiable function to ap-tuple of continuous semimartin- gales, the resulting process is still a continuous semimartingale, and there is an explicit formula for the canonical decomposition of this semimartingale.

Theorem 5.10 (Itô’s formula) Let X¹; : : : ;X^pbe p continuous semimartingales, and let F be a twice continuously differentiable real function onR^p. Then, for every t0,

F.X¹t; : : : ;Xt^p/DF.X₀¹; : : : ;X₀^p/C Xp

iD1

Z t 0

@F

@xⁱ.Xs¹; : : : ;X_s^p/dX_sⁱ C1

2 Xp i;jD1

Z t 0

@²F

@xⁱ@x^j.Xs¹; : : : ;X_s^p/dhXⁱ;X^jis:

Proof We first deal with the casep D 1and we writeX D X¹for simplicity. Fix t> 0and consider an increasing sequence0 Dt₀ⁿ < <tⁿ_pn D tof subdivisions ofŒ0;twhose mesh tends to0. Then, for everyn,

F.Xt/DF.X₀/C

pXn1 iD0

.F.Xtⁿ_iC1/F.Xtⁿ_i//:

114 5 Stochastic Integration For every i 2 f0; 1; : : : ;pn 1g, we apply the Taylor–Lagrange formula to the functionŒ0; 13 7!F.Xtⁿ_i C.Xtⁿ_iC1Xtⁿ_i//, between D0and D1, and we get that

F.Xtⁿ_iC1/F.Xtⁿ_i/DF⁰.Xtⁿ_i/.Xtⁿ_iC1Xtⁿ_i/C1

2fn;i.Xtⁿ_iC1Xt_iⁿ/²;

where the quantityfn;ican be written asF⁰⁰.Xtⁿ_i Cc.Xt_iⁿ_C1Xtⁿ_i//for somec2Œ0; 1. By Proposition5.9withHsDF⁰.Xs/, we have

n!1lim

pXn1 iD0

F⁰.Xtⁿ_i/.Xtⁿ_iC1Xtⁿ_i/D Z t

0 F⁰.Xs/dXs;

in probability. To complete the proof of the casepD1of the theorem, it is therefore enough to verify that

n!1lim

pXn1 iD0

fn;i.Xtⁿ_iC1Xtⁿ_i/²D Z t

0 F⁰⁰.Xs/dhX;Xis; (5.15) in probability. We observe that

sup

0ipn1jfn;iF⁰⁰.Xtⁿ_i/j sup

0ipn1 sup

x2ŒX_tn i^X_tn

iC1;X_tn i_X_tn

iC1jF⁰⁰.x/F⁰⁰.Xtⁿ_i/j

! :

The right-hand side of the preceding display tends to0a.s. asn ! 1, as a simple consequence of the uniform continuity ofF⁰⁰(and of the sample paths ofX) over a compact interval.

SincePpn1

iD0 .Xtⁿ_iC1Xtⁿ_i/²converges in probability (Proposition4.21), it follows from the last display that

ˇˇˇˇ ˇ

pXn1 iD0

fn;i.Xt_iⁿ_C1Xt_iⁿ/²

pXn1 iD0

F⁰⁰.Xtⁿ_i/.Xtⁿ_iC1Xtⁿ_i/²ˇˇ ˇˇˇ !

n!10

in probability. So the convergence (5.15) will follow if we can verify that

n!1lim

pXn1 iD0

F⁰⁰.Xtⁿ_i/.Xtⁿ_iC1Xtⁿ_i/² D Z t

0 F⁰⁰.Xs/dhX;Xis; (5.16) in probability. In fact, we will show that (5.16) holds a.s. along a suitable sequence of values ofn(this suffices for our needs, because we can replace the initial sequence

5.2 Itô’s Formula 115

of subdivisions by a subsequence). To this end, we note that

pXn1 iD0

F⁰⁰.Xtⁿ_i/.Xt_iⁿ_C1Xt_iⁿ/²D Z

Œ0;tF⁰⁰.Xs/ n.ds/;

wherenis the random measure onŒ0;tdefined by n.dr/WD

pXn1 iD0

.Xtⁿ_iC1Xt_iⁿ/²ıtⁿ_i.dr/:

WriteDfor the dense subset ofŒ0;tthat consists of allt_iⁿforn1and0ipn. As a consequence of Proposition4.21, we get for everyr2D,

n.Œ0;r/ !

n!1hX;X˛

r

in probability. Using a diagonal extraction, we can thus find a subsequence of values ofnsuch that, along this subsequence, we have for everyr2D,

n.Œ0;r/ !^a:s:

n!1hX;X˛

r;

which implies that the sequencen converges a.s. to the measure1_Œ0;t.r/dhX;Xir, in the sense of weak convergence of finite measures. We conclude that we have

Z

Œ0;tF⁰⁰.Xs/ n.ds/ !^a:s:

n!1

Z t

0 F⁰⁰.Xs/dhX;Xis

along the chosen subsequence. This completes the proof of the casepD1.

In the general case, the Taylor–Lagrange formula, applied for everyn 1and everyi2 f0; 1; : : : ;pn1gto the function

Œ0; 137!F.X¹_tⁿ

i C.X¹_tⁿ

iC1X¹_tⁿ

i/; : : : ;X_t^pn

i C.X_t^pn iC1X_t^pn

i// ; gives

F.X_t¹ⁿ

iC1; : : : ;X_t^pn

iC1/F.X¹_tⁿ

i; : : : ;X_t^pn i/D

Xp kD1

@F

@x^k.X¹_tⁿ

i; : : : ;X^p_tn i/ .X_t^kn

iC1X_t^kn

i/ C

Xp k;lD1

f_n^k_;^;_i^l 2 .X_t^kn

iC1X_t^kn

i/.X_t^ln

iC1X^l_tⁿ

i/

116 5 Stochastic Integration

where, for everyk;l2 f1; : : : ;pg, f_n^k_;^;_i^lD @²F

@xk@xl.Xtⁿ_i Cc.Xtⁿ_iC1Xtⁿ_i//;

for somec2Œ0; 1(here we use the notationXtD.X_t¹; : : : ;X^p_t/).

Proposition5.9can again be used to handle the terms involving first derivatives.

Moreover, a slight modification of the arguments of the casep D 1shows that, at least along a suitable sequence of values ofn, we have for everyk;l2 f1; : : : ;pg,

n!1lim

pXn1 iD0

f_n^k_;^;_i^l.X^k_tⁿ

iC1X_t^kn

i/.X_t^ln

iC1X_t^ln

i/D Z t

0

@²F

@xk@xl.X_s¹; : : : ;X^p_s/dhX^k;X^lis

in probability. This completes the proof of the theorem. ut An important special case of Itô’s formula is the formula of integration by parts, which is obtained by takingp D 2andF.x;y/ D xy: ifX andY are two continuous semimartingales, we have

XtYtDX₀Y₀C Z t

0 XsdYsC Z t

0 YsdXsC hX;Yi_t: In particular, ifY DX,

X_t²DX²₀C2 Z t

0 XsdXsC hX;Xit:

WhenXDMis a continuous local martingale, we know from the definition of the quadratic variation thatM² hM;Miis a continuous local martingale. The previous formula shows that this continuous local martingale is

M₀²C2 Z t

0 MsdMs:

We could have seen this directly from the construction ofhM;Miin Chap.4 (this construction involved approximations of the stochastic integralRt

0MsdMs).

LetB be an .Ft/-real Brownian motion (recall from Definition3.11 that this means thatBis a Brownian motion, which is adapted to the filtration.Ft/and such that, for every0s<t, the variableBtBsis independent of the-fieldFs). An .Ft/-Brownian motion is a continuous local martingale (a martingale ifB₀ 2 L¹) and we already noticed that its quadratic variation ishB;BitDt.

In this particular case, Itô’s formula reads F.Bt/DF.B₀/C

Z t

0 F⁰.Bs/dBsC 1 2

Z t

0 F⁰⁰.Bs/ds:

5.2 Itô’s Formula 117 TakingX_t¹ D t,X_t² D Bt, we also get for every twice continuously differentiable functionF.t;x/onRCR,

F.t;Bt/DF.0;B₀/C Z t

0

@F

@x.s;Bs/dBsC Z t

0.@F

@t C 1 2

@²F

@x²/.s;Bs/ds:

LetBtD.B¹_t; : : : ;B^d_t/be ad-dimensional.Ft/-Brownian motion. Note that the componentsB¹; : : : ;B^dare.Ft/-Brownian motions. By Proposition4.16,hBⁱ;B^ji D 0wheni 6D j(by subtracting the initial value, which does not change the bracket hBⁱ;B^ji, we are reduced to the case whereB¹; : : : ;B^dare independent). Itô’s formula then shows that, for every twice continuously differentiable functionFonR^d,

F.B¹_t; : : : ;B^d_t/ DF.B¹₀; : : : ;B^d₀/C

Xd iD1

Z t 0

@F

@xi.B¹_s; : : : ;B^d_s/dBⁱ_sC 1 2

Z t

0 F.B¹_s; : : : ;B^d_s/ds: The latter formula is often written in the shorter form

F.Bt/DF.B₀/C Z t

0 rF.Bs/dBsC 1 2

Z t

0 F.Bs/ds;

whererFstands for the vector of first partial derivatives ofF. There is again an analogous formula forF.t;Bt/.

Important remark It frequently occurs that one needs to apply Itô’s formula to a functionF which is only defined (and twice continuously differentiable) on an open subset U ofR^p. In that case, we can argue in the following way. Suppose that there exists another open setV, such that.X₀¹; : : : ;X₀^p/ 2 V a.s. andVN U (hereVN denotes the closure ofV). TypicallyV will be the set of all points whose distance fromU^c is strictly greater than ", for some " > 0. SetTV WD infft 0 W .Xt¹; : : : ;X^pt/ … Vg, which is a stopping time by Proposition 3.9. Simple analytic arguments allow us to find a function G which is twice continuously differentiable onR^pand coincides withFonV. We can now apply Itô’s formula toN obtain the canonical decomposition of the semimartingaleG.X_t^T¹ _V; : : : ;X_t^T^p _V/ D F.X_t^T¹ _V; : : : ;X_t^T^p _V/, and this decomposition only involves the first and second derivatives ofFonV. If in addition we know that the process.Xt¹; : : : ;X^pt/a.s. does not exitU, we can let the open setVincrease toU, and we get that Itô’s formula for F.Xt¹; : : : ;Xt^p/remains valid exactly in the same form as in Theorem5.10. These considerations can be applied, for instance, to the functionF.x/ D logxand to a semimartingaleX taking strictly positive values: see the proof of Proposition5.21 below.

118 5 Stochastic Integration We now use Itô’s formula to exhibit a remarkable class of (local) martingales, which extends the exponential martingales associated with processes with indepen- dent increments. A random process with values in the complex planeCis called a complex continuous local martingale if both its real part and its imaginary part are continuous local martingales.

Proposition 5.11 Let M be a continuous local martingale and, for every2C, let E.M/tDexp

Mt² 2 hM;Mit

:

The processE.M/is a complex continuous local martingale, which can be written in the form

E.M/tDe^M0C Z t

0 E.M/sdMs:

Remark The stochastic integral in the right-hand side of the last display is defined by dealing separately with the real and the imaginary part.

Proof IfF.r;x/is a twice continuously differentiable function onR², Itô’s formula gives

F.hM;Mit;Mt/DF.0;M₀/C Z t

0

@F

@x.hM;Mis;Ms/dMs

C Z t

0

@F

@r C 1 2

@²F

@x²

.hM;Mis;Ms/dhM;Mis:

Hence,F.hM;Mit;Mt/is a continuous local martingale as soon asF satisfies the equation

@F

@r C1 2

@²F

@x² D0:

This equation holds forF.r;x/D exp.x ₂²r/(more precisely for both the real and the imaginary part of this function). Moreover, for this choice ofF we have

@F

@x DF, which leads to the formula of the statement. ut