Martingale representation for degenerate diffusions

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Contents lists available atScienceDirect

Journal

of

Functional

Analysis

www.elsevier.com/locate/jfa

Martingale

representation

for

degenerate

diﬀusions

A.S. Üstünel

Bilkent University,Math.Dept.,Ankara,Turkey

a r t i c l e i n f o a bs t r a c t

Article history:

Received 10 May 2018 Accepted 14 December 2018 Available online 17 December 2018 Communicated by L. Gross Keywords: Entropy Degenerate diﬀusions Martingale representation Relative entropy Innovation process

Causal Monge–Ampère equation

Let(W,H,μ) betheclassicalWienerspaceonIRd_._Assume

thatX = (Xt) isadiﬀusionprocesssatisfyingthestochastic

diﬀerential equation dXt = σ(t,X)dBt+ b(t,X)dt, where

σ : [0,1]×C([0,1],IRn₎_{→ IR}n_⊗IRd_,_b_{: [0,}_1]_×C([0,_1],_IRn₎_→

IRn_,_{B is}_an _IRd_-valued_Brownian _motion._We_suppose_that

the weak uniqueness of this equation holds for any initial condition.WeprovethatanysquareintegrablemartingaleM w.r.t.totheﬁltration(Ft(X),t∈ [0,1]) canberepresentedas

Mt= E[M0] +

t

0

Ps(X)αs(X).dBs

whereα(X) isanIRd_-valued_process_adapted_to₍_F t(X),t∈

[0,1]),satisfyingE₀t(a(Xs)αs(X),αs(X))ds<∞,a= σσ

and Ps(X) denotes a measurable version of the orthogonal

projection from IRd _to _σ(X

s)(IRn). In particular, for any

h∈ H,wehave E[ρ(δh)|F1(X)] = exp ⎛ ⎝ 1 0 (Ps(X) ˙hs, dBs) −1 2 1 0 |Ps(X) ˙hs|2ds ⎞ ⎠ , (0.1)

where ρ(δh) = exp(₀1( ˙hs,dBs)− 1₂ |H|2H). In the case

the process X is adapted to the Brownian ﬁltration, this E-mailaddress:ustunel@fen.bilkent.edu.tr.

https://doi.org/10.1016/j.jfa.2018.12.004

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result gives a new development as an inﬁnite series of the L2_-functionals _of _the _degenerate _diﬀusions. _We _also

give an adequate notion of “innovation process” associated to a degenerate diﬀusion which corresponds to the strong solutionwhentheBrownianmotionisreplacedbyanadapted perturbationofidentity.Thislatterresultgivesthesolution ofthecausalMonge–Ampèreequation.

1. Introduction

Therepresentationof randomvariables with thestochasticintegralswith respect to somebasicprocesseshasalonghistoryandalsoithasveryimportantapplications,e.g., insignal theory, filtering, optimal control, finance, stochastic differential equations, in physics,etc.Letusrecallthequestion:assumethatwearegivenacertainsemimartingale

X indexedby[0,1] forexample.LetF(X)= (Ft(X),t∈ [0,1]) denoteitsﬁltration.The

question is under which conditions can we represent any martingale adapted to the filtrationofX as astochastic integralw.r.t. afixed martingaleof thefiltrationF(X)? For the case of Wiener process, this questionhas a very longhistory and it is almost impossibletogiveanexactaccountofthecontributingworks.Toourknowledge,ithas been answeredforthefirst timeintheworkof K.Itô,cf. [10,15]. In[2], C.Dellacherie hasgiven adifferent point of viewto provetherepresentation theorem for theWiener andPoissonprocesses,basedontheuniquenessoftheirlaws.Thecaseofnondegenerate diffusionprocesseshasbeenelucidatedin[12],alsostudiedin[3,11] (cf.alsothereferences there) with also some remarks about the degenerate case.The general case, using the notionofmultiplicityisgivenin[1].

Although the nondegenerate caseis completely settled, thedegenerate case hasnot been solved in a deﬁnitive way, in the sense that a unique minimal martingale with respect to which the above mentioned property of representation holds, has not been discovered. In this workwe are doing exactlythis: we prove thatfor adegenerate dif-fusion whose law is unique, there exists a minimal martingale with respect to which everysquareintegrable,F1(X)-measurablefunctionalcanberepresentedasastochastic

integral of an F(X)-adapted process. To do this we ﬁrst prove the density of a class of F1(X)-measurable stochastic integralsin L2(F1(X)) using the methodlaunched by

C. Dellacherie, then we show that a sequence from this class which is approximating any element of L2₍_F

1(X)) deﬁnes an L2-convergingsequence of processes with values

intherange oftheadjointofthediﬀusioncoeﬃcient,i.e., withvaluesinσ_(t,_X)(IRn

). Because of the degeneracy of σ, we can not determine the limit of this sequence but itsimage underthe orthogonalprojection Pt(X): IRd → σ(t,X)(IRn) anditslimit is

perfectlywell-deﬁnedandinthisway weseethattheminimalmartingaleforthe repre-sentationofthefunctionalsofthediﬀusion(Itôprocess)isnothingbutdmt= Pt(X)dBs

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Note thateventhe adaptabilityof (mt,t ∈ [0,1]) tothe ﬁltrationF(X) is notevident.

This result of representation gives also existence of non-orthogonal chaos representa-tion of the elements of L2₍_F

1(X)) as the multiple ordered integrals with respect to

the martingale (mt,t ∈ [0,1]) of elements of {L2(Cn,dt1× · · · × dtn),n ≥ 1}, where Cn ={(t1,. . . ,tn)∈ [0,1]n: t1> . . . > tn}.

Inthethirdsectionwedefinethenotionofinnovationprocessassociated toa degen-eratediffusionprocess1.Inthiscasethedefinitionofinnovationprocessisdifferentfrom theclassicalcaseoftheperturbationofidentitysincewehavetotakeintoaccountalso the actionofthe projectionoperator-valuedprocess(Pt(X),t∈ [0,1]).InSection3we

extendtheinnovationrepresentationtheoremofFujisaki–Kallianpur–Kunita,[8],under the hypothesis of strongexistence and uniquenessto the caseof degenerate diffusions. Thisresultconfirmsthevalidityofthechoicethatwehavedonetodefinetheinnovation process. Inparticular,usingthisinnovationprocesswecancalculatetheconditional ex-pectation ofthe Girsanovexponential of an adapteddrift u with respect to the sigma algebraF1(XU),whereXU isthesolutionofthediffusionstochasticdifferentialequation

where therandominputisequalto U = B + u.

TheresultsofthethirdsectionisthenappliedtothesolutionoftheadaptedMonge– Ampère equation inthe caseof thedegenerate diﬀusions,which extendsthe resultsof [7,13,14], cf. also[4–6].Inparticularwe calculatetherelative entropyofthelawof XU

with respecttothelawofX by theuseofprecedingresults.

Letusnotetoﬁnishthisintroductionthatmostoftheseresultsareeasilyextendibleto moregeneralsituations,forexamplethestrongexistenceanduniquenesshypothesiscan beweakenedintheentropiccalculations,wehavetriedtofollowmaximumhomogeneity inthehypothesisandthesepossible extensionsmaybetreatedinseparate works. 2. Stochasticintegral representationoffunctionalsofdiﬀusions

Let X = (Xt,t ∈ [0,1]) be a weak solution of the following stochastic diﬀerential

equation:

dXt= b(t, X)dt + σ(t, X)dBt, X0= x, (2.2)

where B = (Bt,t ∈ [0,1]) is an IRd-valued Brownian motion and σ : [0,1] × C([0,1],IRn) → L(IRd,IRn) and b : [0,1]× C([0,1],IRn)→ IRn are measurable maps, adapted to the naturalﬁltration of C([0,1],IRn) and of linear growth.Recall that the classical theorem of Yamada–Watanabe says that the strong uniqueness implies the uniqueness inlaw of theaboveSDE, cf. [16,9]. Hence weakuniquenessis easier to ob-tain in the applications. In this section we shall assume the weak uniqueness of the equation andprovethemartingalerepresentationpropertyforthecasewhereσ maybe degenerate.

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Moreprecisely, let(Ft(X),t ∈ [0,1]) be theﬁltration of X andlet us denote by K

thesetofIRn-valued,(Ft(X),t∈ [0,1])-adaptedprocessesα(X),s.t.

E 1 0 (a(s, X)αs(X), αs(X))ds <∞ , wherea(s,w)= σ(s,w)σ(s,w),s∈ [0,1],w∈ C([0,1],IRn). Theorem1.ThesetΓ={N ∈ L2₍_F

1(X)): N = E[N ]+

1

0(αs(X),σ(s,X)dBs),α∈ K}

isdensein L2₍_F 1(X)).

Proof. Suppose that there is some M ∈ L2₍_F

1(X)) which is orthogonal (in L2) to Γ.

Usingthe usualstoppingtechnique,we canassumethatthecorresponding (Ft(X),t∈

[0,1])-martingaleisboundedandpositivewhoseexpectationisequaltoone.The orthog-onalityimpliesthat(Mt(f (Xt)−

t

0Lf (s,X)ds),t∈ [0,1]) isagaina(local)martingale

foranysmoothfunctionf : IRn→ IR,whereL isdeﬁnedas

Lf (t, X) = 1 2 i,j ai,j(t, X)∂i,jf (Xt) + i bi(t, X)∂if (Xt) . (2.3)

Thisimplies,with PropositionIV.2.1of[9],that,underthemeasureM· P ,X isagain aweak solution of 2.2. Bythe uniqueness inlaw, we get X(M · P )= X(P ), since M

is F1(X)-measurable, we should have M = 1. Hence the functionals of the diﬀusion

whichareorthogonaltotheabovesetofstochasticintegralsarealmost surelyconstant. Consequently,thesetΓ istotalinL2₍_F

1(X)). 2

Thefollowing isthe extensionofthemartingalerepresentationtheoremtothe func-tionalsofdegeneratediﬀusions:

Theorem 2.Denote by Ps(X) a measurable version of the orthogonal projection from

IRd ontoσ(s,X)_(IRn₎_{⊂ IR}d _and_let_F _{∈ L}2₍_F

1(X)) be any randomvariable with zero

expectation.Thenthereexistsaprocessξ(X)∈ L2

a(dt× dP ;IRd),adapted to(Ft(X),t∈ [0,1]), suchthat F (X) = 1 0 (Ps(X)ξs(X), dBs)IRd= 1 0 (ξs(X), Ps(X)dBs)IRn a.s.

Conversely,any stochasticintegralof theform

1

0

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where ξ(X) is an (Ft(X),t ∈ [0,1])-adapted, measurable process with E₀1|Ps(X)ξs(X)|2ds<∞,givesrise toanF1(X)-measurablerandomvariable.

Proof. From Theorem 1, there exists a sequence (Fn(X),n ≥ 1) ⊂ Γ, where Γ is

de-ﬁned inthe statementof Theorem 1, convergingto F (X) in L2_. _We_can _suppose _that

E[Fn(X)]= 0,foranyn≥ 1.Hence

Fn(X) = 1 0 (γ_sn(X), σ(s, X)dBs)IRd= 1 0 (σ(s, X)γ_sn(X), dBs)IRd.

As explained above γn is an (Ft(X),t ∈ [0,1])-adapted, IRn-valued process satisfying E₀1(a(Xs)γsn(X),γsn(X))ds<∞. SinceFn(X)→ F (X) inL2, lim n,m→∞E 1 0 |σ_{(s, X)γ}n s(X)− σ(s, X)γsm(X)|2ds = 0 . Let αn s(X) = σ(Xs)γsn(X), as Ps(X)αns(S) = Ps(X)σ(Xs)γsn(X) = σ(Xs)γsn(X), (Ps(X)αns(X),n ≥ 1) converges to some ξs(X) in L2a(ds× dP ;IR d ). As Ps(X) is an

orthogonal projection, (Ps(X)αns(X),n≥ 1) converges to Ps(X)ξs(X) alsoin L2a(ds× dP ;IRd).Therefore 1 0 Ps(X)ξs(X).dBs= lim n 1 0 Ps(X)αns(X).dBs = lim n 1 0 (γn_s(X), σ(s, X)dBs) = lim n Fn(X) = F (X) .

Let now G ∈ L2_{(P ) be} _given _by _G₌ 1

0(Ps(X)ηs(X),dBs) and assumethat it is not

F1(X)-measurable.Then G− E[G|F1(X)] is orthogonal toL2(F1(X)).It follows from

theﬁrstpartofthetheoremthatwecanrepresentE[G|F1(X)] as

1

0(Ps(X)ξs(X),dBs).

Letus deﬁneh= η− ξ,then theorthogonalitymentionedaboveimpliesthat

E ⎡ ⎣ 1 0 (Ps(X)hs, dBs). 1 0 (αs(X), σ(s, X)dBs) ⎤ ⎦ = 0 for any(Ft(X),t∈ [0,1])-adapted,measurable α suchthatE

1

0(a(s,X)αs,αs)ds<∞.

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Remark.Letη beanadaptedprocesssuchthatηsbelongstotheorthogonalcomplement

of σ(IRn) in IRd ds× dP -a.s. Then η + ξ can also be used to representF (X). Hence

ξ(X) isnotuniquebutP (X)ξ(X) isalwaysunique.

Theorem3.Let ˙u∈ L2(dt× dP,IRd) beadaptedtotheBrownianﬁltration,thenwehave

E ⎡ ⎣ 1 0 ( ˙us, dBs)|F1(X) ⎤ ⎦ = 1 0 (E[Ps(X) ˙us|Fs(X)], dBs) (2.4) almostsurely.

Proof. Wehavetoproveﬁrstthattherighthandsideof(2.4) isF1(X)-measurable.We

knowfromTheorem 2thattheleftsideof(2.4) canbe representedas

E ⎡ ⎣ 1 0 ( ˙us, dBs)|F1(X) ⎤ ⎦ = 1 0 Ps(X)ξs(X).dBs for some ξ ∈ L2 a(dt× dP ;IRd). Let F (X) = 1 0 Ps(X)αs(X).dBs be any element of L2(F1(X)) withα(X)∈ L2a(dt× dP ;IRd).Wehave E ⎡ ⎣ ⎛ ⎝ 1 0 ˙us.dBs ⎞ ⎠ F (X) ⎤ ⎦ = E 1 0 ( ˙us, Ps(X)αs(X))ds = E 1 0 (Ps(X)E[ ˙us|Fs(X)], Ps(X)αs(X))ds = E ⎡ ⎣ ⎛ ⎝ 1 0 Ps(X)ξs(X).dBs ⎞ ⎠ F (X) ⎤ ⎦ = E 1 0 (Ps(X)ξs(X), Ps(X)αs(X))ds ,

hencePs(X)ξs(X)= Ps(X)E[ ˙us|Fs(X)] ds× dP -a.s.,inparticular

1

0

(E[Ps(X) ˙us|Fs(X)], dBs)

is F1(X)-measurable. Theaboveidentiﬁcation assures then thevalidity ofthe relation

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Corollary 1.Leth∈ H1([0,1],IRd) (i.e., the Cameron–Martin space),denote by ρ(δh) theWick exponential exp(₀1( ˙hs,dBs)−1₂

• Assume that Ft(X)⊂ Ft(B) forany t ∈ [0,1], where (Ft(B),t ∈ [0,1]) represents theﬁltration of theBrownian motion. Deﬁnethe martingalem= (mt,t∈ [0,1]) as mt=

t

0Ps(X)dBs,then theset

K ={ρ(δm(h)) : h∈ H} is total in L2(F1(X)), where ρ(δm(h)) = exp

1 0( ˙hs, dms)− 1 2 1 0 |Ps(X) ˙hs| 2_ds_.

In particular, any element F of L2₍_F

1(X)) canbe written in aunique way as the

sum F = E[F ] + ∞ n=1 Cn (fn(s1, . . . , sn), dms1⊗ . . . ⊗ dmsn) (2.5)

where Cn isthen-dimensionalsimplexin [0,1]n and fn∈ L2(Cn,ds⊗n)⊗ (IRd)⊗n.

• More generally, without the hypothesis Ft(X) ⊂ Ft(B), forany F ∈ L2(F1(X))∩

L2₍_F

1(B)), theconclusionsof theﬁrstpartof thetheorem holdtrue.

Proof. Let F ∈ L2₍_F

1(X)), assume thatF is orthogonal to K,i.e. E[F ρ(δm(h))] = 0

forany h∈ H.FromCorollary1ρ(δm(h))= E[ρ(δh)|F1(X)],hence

E[F ρ(δh)] = E[F ρ(δm(h))] = 0 ,

hence F isalso orthogonalto E = {ρ(δh): h ∈ H},which istotalinL2₍_F

1(B),where

F1(B) is theσ-algebra generated bythegoverning Brownianmotion, therefore F = 0.

ConsequentlythespanofK isdenseinL2₍_F

1(X)).Theorem3andCorollary1allowusto

calculatetheconditionalexpectationsofthemultipleWienerintegralsw.r.t.F1(X) and

theresultwillbe multipleiteratedstochasticintegralsw.r.t.m in theformgivenbythe formula(2.5);herewehavetobecarefulastheoperatorvaluedprocess(Ps(X),s∈ [0,1])

is notdeterministic, thesymmetricinterpretationof theIto–Wienerintegralsw.r.t.the Brownian motionisnolongervalid inourcaseandtheyhaveto bewrittenas iterated Itointegrals. 2

Remark1.Inthistheoremwe needtomake thehypothesis Ft(X)⊂ Ft(B) forany t∈

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althoughwe haveTheorem 2 andwhen we iterateit we havea similar representation, but,consistingofafinite numberofterms.Itis notpossibleto pushthisprocedureup toinfinitysincewehavenocontrolatinfinity.

Remark2.Let us note that ifthere is no strongsolution to the equation deﬁning the process X,thechaotic representationproperty mayfail.For example,letU beaweak solutionof

dUt= αt(U )dt + dBt, (2.6)

withU0given.Assumethat(2.6) hasnostrongsolution,asitmayhappeninthefamous

exampleof Tsirelson ([9]), i.e., U isnot measurable w.r.t.the sigmaalgebragenerated byB,thenwehavenochaoticrepresentationpropertyfortheelementsofL2(F1(U )) in

termsoftheiterated stochasticintegralsofdeterministic functionsonCn,n∈ IN w.r.t. B;thecontrarywouldimplytheequalityofF1(U ) andofF1(B),whichwouldcontradict

thenon-existenceofstrongsolutions. 3. Innovationprocess

In this section we assume that the SDE 2.2 has unique strong solution. To ﬁx the ideas,we cansuppose thatσ andb satisfythe followingkind ofLipschitz conditionon thepathspace:forξ,η∈ C([0,1],IRn):

|γ(t, ξ) − γ(t, η)| ≤ K sup

s≤t|ξ(s) − η(s)|

forγ beingequaleithertoσ ortob withcorrespondingEuclideannormsatthelefthand side.

Assumethat ˙u∈ L2_(dt_{× dP,}_IRd_{) is} _a_process _adapted_to_the _ﬁltration_of_Brownian

motion,letU = (Ut,t∈ [0,1]) bedeﬁned asUt= Bt+

t

0 ˙usds.We denoteby X

U _the

strongsolutionoftheequation

dX_tU = σ(t, XU)dUt+ b(t, XU)dt

= σ(t, XU)(dBt+ ˙utdt) + b(t, XU)dt

SinceB isthecanonicalBrownianmotion,wehaveXU

t = Xt◦U a.s.Moreover,ifη∈ IRn

isanyvector,we have

(σ(t, XU) ˙ut, η) = ( ˙ut, σ(t, XU)η)

= (Pt(XU) ˙ut, σ(t, XU)η)

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where Pt(XU) denotestheorthogonal projectionfrom IRd onto σ(t,XU)(IRn). Hence,

in order to deﬁne a reasonably useful concept of innovation, we should estimate the perturbation ˙u simultaneously w.r.t.theboth projections,i.e.,with respectto the con-ditional expectation E[·|Fs(XU)] (which is a projection) and also w.r.t. Ps(XU): Let Z = (Zt,t∈ [0,1]) (to avoidtheambiguity we shall alsouse thenotation ZU if

neces-sary) bedeﬁnedas

Zt= Bt+ t

0

( ˙us− E[Ps(XU) ˙us|Fs(XU)])ds , (3.7)

where (Fs(XU),s∈ [0,1]) denotestheﬁltrationofXU.

Proposition 1.The process (₀tPs(XU)dZs,t ∈ [0,1]) is an (Ft(XU),t ∈ [0,1])-local martingale.

Proof. Assume to begin that |u|2_H = ₀1| ˙us|2ds ∈ L∞(P ). Let us ﬁrst prove that the

processunderquestionisadaptedtotheﬁltration(Ft(XU),t∈ [0,1]):FromtheGirsanov

theorem,theprocess(₀tσ(s,XU_)dU

s,t∈ [0,1]) isadaptedtotheﬁltration(Ft(XU),t∈

[0,1]),usingthesamemethodasinTheorem2byreplacingB byU andtheprobability

dP by ρ(−δu)dP , we conclude that the process (₀tPs(XU)dUs,t ∈ [0,1]), and hence

theprocess(₀tPs(XU)dZs,t∈ [0,1]) isadaptedtotheﬁltration(Ft(XU),t∈ [0,1]).To

show the(local)martingalepropertyitsuﬃcestowrite that

t 0 Ps(XU)dZs= t 0 Ps(XU)dBs+ t 0 Ps(XU) ˙us− E[ ˙us|Fs(XU)] ds

from which the martingale property follows. The general casefollows from a stopping argument. 2

Remark.NotethatZ = ZU _is_not_a_Brownian_motion.

Theorem 5.Let ˙u∈ L2_(dt_{× dP,}_IRd_{) be}_such_that _E[ρ(_−δu)]_{= 1,}_then_we_have ζt= E[ρ(−δu)|Ft(XU)] = exp ⎛ ⎝− t 0 Ps(XU)E[ ˙us|Fs(XU)]· dZs− 1 2 t 0 |Ps(XU)E[ ˙us|Fs(XU)]|2ds ⎞ ⎠ . Proof. Assumeﬁrstthat|u|2

H =

1 0 | ˙us|

2_ds_{∈ L}∞_{(P ).}_Let_(XU

t ,t∈ [0,1]) bethe(strong)

solution of dXt = σ(Xt)dUt, where dUt = dBt+ ˙utdt and let f be a C2-function on

IRn. Using the Itô formula, we calculate the Doob–Meyer process associated to the semimartingales (ζt) and(f (XtU)):

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f ◦ XU_{, ζ} t=− t 0 Df (X_sU), σ(s, XU)Ps(XU)E[ ˙us|Fs(XU)] ds . (3.8)

Letf ∈ C2_,_using_again_the_Itô_formula_and_the_relation₍_3.8_{) we}_get

f (X_tU)ζt= t 0 f (X_sU)dζs+ t 0 ζs(Lf )(s, XU)ds + t 0 ζs(Df (XsU), σ(s, XU)dUs) − t 0 (Df (X_sU), σ(s, XU)Ps(XU)E[ ˙us|Fs(XU)])ds ,

whereLf isdeﬁnedbytherelation2.3.Thereforetheprocess ⎛ ⎝f(XU t )ζt− t 0 (Lf )(s, XU))ζsds, t∈ [0, 1] ⎞ ⎠

is aP -local martingale, therefore, by the uniquenessin law of thesolution, we should have

E[ζ1F (XU) = E[F (X) = E[F (XU)ρ(−δu)]

for any F ∈ Cb on the path space, and the last inequality follows from the Girsanov

theorem. It suﬃces then to remark that ζ1 is F1(XU)-measurable by Theorem 3 and

again by the Girsanov theorem which allows us to replace B by U . The general case follows from the usual stopping time argument: let Tn = inf(t :

t

0| ˙us|

2_ds _{≥ n) and}

deﬁne ˙un_s = 1[0,Tn](s) ˙us and letU

n _{= B +}·

0 ˙u

n

sds.Then XU

n

converges almost surely uniformlytoXU_,_hence_lim

nE[·|Ft(XU n

)]= E[·|Ft(XU)] dt-almostsurelyasbounded

operators on L1_{(P ) and} _(P

s(XU n

),n ≥ 1) converges to Ps(X)ds× dP -almost surely.

Moreover(ρ(−δun_),_n_{≥ 1) converges}_strongly_in_L1_{(P ) to}_ρ(_−δu),_therefore_the_general

casefollows. 2

The following result is the generalization of the celebrated innovation’s theorem to thedegeneratecase,cf. [8]:

Theorem6.Let(Mt,t∈ [0,1]) beasquareintegrable(P,(Ft(XU),t∈ [0,1]))-martingale, then it can be represented as a stochastic integral of an (Ft(XU),t ∈ [0,1])-adapted,

IRd-valuedprocess β(XU_{) in}_the_following_way:

Mt= M0+

t

0

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P -a.s., where

t

0

|Ps(XU)βs(XU)|2ds <∞ P -a.s., forany t∈ [0,1].

Proof. Assume thatM isa(P,(Ft(XU),t∈ [0,1]))-martingale,thenfor anys< t and A∈ Fs(XU) wehave E Mt ζt 1Aρ(−δu) = E Mt ζt 1Aζt = E[Mt1A] = E[Ms1A] = E Ms ζs 1Aρ(−δu)

where ζ is the optional projection of ρ(−δu) w.r.t. the ﬁltration (Ft(XU),t ∈ [0,1])

as calculated in Theorem 5. Consequently (Mt/ζt,t ∈ [0,1]) is a (Q,(Ft(XU),t ∈

[0,1]))-martingale, where dQ = ρ(−δu)dP . As U is a Q-Brownian motion, from The-orem2,wecanrepresent(Mt/ζt,t∈ [0,1]) as

Mt ζt = c + t 0 (Ps(XU) ˙αs(XU), dUs) ,

then usingtheItôformula

Mt= Mt ζt ζt = c + t 0 ζs(Ps(XU) ˙αs(XU), dUs)− t 0 Ms ζs ζs(Ps(XU)E[ ˙us|Fs(XU)], dZs) − t 0 ζs(Ps(XU) ˙αs(XU), E[ ˙us|Fs(XU)])ds = c + t 0 ζs(Ps(XU) ˙αs(XU), dZs+ Ps(XU)E[ ˙us|Fs(XU)])ds − t 0 Ms(Ps(XU)E[ ˙us|Fs(XU)], dZs)

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− t 0 ζs(Ps(XU) ˙αs(XU), E[ ˙us|Fs(XU)])ds = c + t 0 Ps(XU) ζsα˙s(XU)− MsE[ ˙us|Fs(XU)] , dZs

andthiscompletes theproof. 2

4. Entropycalculation andMonge–Ampèreequation

Assumethatl(X) isaprobabilitydensitymeasurable w.r.t.F1(X),i.e., E[l(X)]= 1

andwithﬁnite entropy:E[l(X)log l(X)]<∞.WewanttoﬁndaprocessU = B + u=

B +₀· ˙usds whichisanadaptedperturbationoftheBrownianmotionB suchthat

l(X) = dX U_{(P )} dX(P ) ◦ X .

ThisproblemiscalledthecausalMonge–Ampèreproblem.Tosimplifythecalculations, we shall assumethatl◦ X isP -a.s. strictly positive.Assumethat sucha U (henceu)

existsandthatu satisﬁestheGirsanovtheorem,i.e.,E[ρ(−δBu)]= 1.ThentheGirsanov

theoremimpliesthat

l◦ XUE[ρ(−δu)|F1(XU)] = 1 (4.9)

P -a.s., which is the causal version of the Monge–Ampère equation. From Theorem 2,

l◦ X canberepresentedas l◦ X = exp ⎛ ⎝− 1 0 Ps(X) ˙vs(X)· dBs− 1 2 1 0 |Ps(X) ˙vs(X)|2ds ⎞ ⎠ ,

where (Ps(X) ˙vs(X),s ∈ [0,1]) is adapted to the ﬁltration of X and

1

0 |Ps(X) ˙vs(X)|

2_ds_<_{∞ P -a.s.} _Besides,_since_{U is}_a_Brownian_motion_under_the

prob-abilityρ(−δu)dP ,itfollowsfromTheorem 2thatl◦ XU canberepresentedas

l◦ XU = exp ⎛ ⎝− 1 0 Ps(XU) ˙vs(XU)· dUs− 1 2 1 0 |Ps(XU) ˙vs(XU)|2ds ⎞ ⎠ . (4.10) Insertingtherighthandsideof(4.10) andE[ρ(−δu)|F1(XU)] whichisalreadycalculated

inTheorem5intheMonge–Ampèreequation(4.9) andthentakingthelogarithmofthe ﬁnalexpression, weobtain

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1 0 Ps(XU) E[ ˙us|Fs(XU)] + ˙vs(XU) · dZs +1 2 1 0 |Ps(XU) ˙vs(XU) + Ps(XU)E[ ˙us|Fs(XU)]|2 ds = 0 .

This relationimpliesthat

Ps(XsU)

˙vs(XU) + E[ ˙us|Fs(XU)]

= 0 (4.11)

ds× dP -almostsurely,whichisaquiteelaborate nonlinearequation.From theMonge– Ampèreequation(4.9) wecancalculatetherelativeentropybetweenXU_{(P ) and}_{X(P ),}

denotedbyH(XU_{(P )}_{|X(P )):}

Theorem7.Supposethatl isanX(P )-almostsurelystrictlypositivedensity.Thereexists some ˙u∈ L2_(ds_{× dP ) with}_E[ρ(_−δu)]_{= 1 with}

dXU_{(P )} dX(P ) = l if andonly if Ps(XsU) ˙vs(XU) + E[ ˙us|Fs(XU)] = 0.

In this casewealso have

H(XU(P )|X(P )) = 1 2E 1 0 |Ps(XU)E[ ˙us|Fs(XU)]|2ds = 1 2E 1 0 |Ps(XU) ˙vs(XU)|2ds . Proof. H(XU(P )|X(P )) = logdX U_{(P )} dX(P ) dX U_{(P )} = logdX U_{(P )} dX(P ) ◦ X U_dP = log l◦ XUdP = 1 2E 1 0 |Ps(XU)E[ ˙us|Fs(XU)]|2ds ,

provided that ˙u ∈ L2_(ds_{× dP ) and} _the _ﬁrst _equality _follows, _the _second _one _is _a

consequence of the relation (4.11), it can be also proven directly from the Girsanov theorem. 2

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Proposition2.Assumethat l andu are givenasabove.Suppose furthermorethat H(XU(P )|X(P )) = 1 2E 1 0 |Ps(XU) ˙us|2ds . (4.12) Thenthefollowingequation holdstrue:

Ps(XU)dUs+ Ps(XU) ˙vs◦ XUds = Ps(XU)dBs (4.13) almostsurely.In particular,XU satisﬁes thefollowingstochasticdiﬀerential equation:

dX_tU = σ(t, XU)(dBt− ˙vt◦ XUdt) + b(t, XU)dt, (4.14) withthesame initialconditionas X.

Proof. Thehypothesis(4.12) impliesthattheprocess(Pt(XU) ˙ut,t∈ [0,1]) isds-almost

surelyadaptedtotheﬁltration(Ft(XU),t∈ [0,1]),hencewegetfromtheequality(4.11)

therelation

Pt(XU)( ˙vt◦ XU+ ˙ut) = 0 ,

whichimpliesatoncetherelation(4.13).Toseethenextone,notethat

dX_tU = σ(t, XU)(dBt+ ˙utdt) + b(t, XU)dt

= σ(t, XU)(dBt+ Pt(XU) ˙utdt) + b(t, XU)dt

= σ(t, XU)(dBt− Pt(XU) ˙vt◦ XUdt) + b(t, XU)dt

= σ(t, XU)(dBt− ˙vt◦ XUdt) + b(t, XU)dt

where we haveused the factthat σ(t,XU_{)η = σ(t,}_XU_)P

t(XU)η forany vector inIRd

sincePt(XU) istheorthogonalprojectionofIRd ontoσ(XtU)(IRn). 2

Theorem7canbe extendedas follows Theorem8.Assumethatu∈ L2

a(dt×dP,H) anddenotebyU theprocess(Bt+

t

0 ˙usds,t∈

[0,1]). assume also, asbefore, the Lipschitz hypothesisabout thedrift and diﬀusion co-eﬃcients,thenthefollowinginequalityholdstrue:

H(XU(P )|X(P )) ≤ 1 2E 1 0 |Ps(XU)E[ ˙us|Fs(XU)]|2ds . (4.15)

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Proof. Ifu∈ L∞a (dt×dP,H),thentheclaimwithequality(insteadofinequality)follows

fromTheorem7.Forthecaseu∈ L_a2(dt×dP,H),deﬁneTn= inf(t> 0:

t 0| ˙us| 2_ds_{> n),} then un _deﬁned_by un(t) = t 0 1[0,Tn](s) ˙usds is inL∞_a (dt× dP,H),hencewehave H(XUn(P )|X(P )) = 1 2E 1 0 |Ps(XU n )E[1[0,Tn](s) ˙us|Fs(X Un_)]_|2_{ds .} _(4.16)

As n → ∞, (XUn_{(P ),}_n _{≥ 1) converges} _weakly _to _XU_{(P ) and} _the _weak_lower

semi-continuityoftheentropyimpliesthat

where thelimit oftheright handsideof theequation(4.16) followsfrom theLipschitz hypothesis. 2

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