Stochastic functional differential equations and sensitivity to their initial path

0.22 MB·

by  David R. Baños

STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS
AND SENSITIVITY TO THEIR INITIAL PATH
D.R.BAÑOS,G.DINUNNO,H.H.HAFERKORN,ANDF.PROSKE
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n ABSTRACT. Weconsidersystemswithmemoryrepresentedbystochasticfunctionaldifferentialequa-
a tions. Substantially,thesearestochasticdifferentialequationswithcoefficientsdependingonthepast
J historyoftheprocessitself. Suchcoefficientsarehencedefinedonafunctional space. Modelswith
2 memory appear in many applications ranging from biology to finance. Here we consider the results
2 ofsomeevaluations based onthesemodels (e.g. thepricesofsomefinancial products) andtherisks
connectedtothechoiceofthesemodels. Inparticularwefocusontheimpactoftheinitialcondition

] ontheevaluations. Thisproblemisknownastheanalysisofsensitivitytotheinitialconditionand,in
R
theterminologyoffinance,itisreferredtoastheDelta.Inthisworktheinitialconditionisrepresented
P bytherelevantpasthistoryofthestochasticfunctionaldifferentialequation.Thisnaturallyleadstothe
h. redesignofthedefinitionofDelta. Wesuggesttodefineitasafunctionaldirectionalderivative,thisis
t anaturalchoice. Forthiswestudyarepresentationformulawhichallowsforitscomputationwithout
a
requiringthattheevaluationfunctionalisdifferentiable. Thisfeatureisparticularlyrelevantforappli-
m
cations. OurformulaisachievedbystudyinganappropriaterelationshipbetweenMalliavinderivative
[ andfunctionaldirectionalderivative.Forthisweintroducethetechniqueofrandomisationoftheinitial

condition.
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1 1. INTRODUCTION
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0 Severalphenomenainnatureshowevidenceofbothastochasticbehaviourandadependenceonthe
.
1 pasthistory whenevaluating thepresent state. Examples ofmodels taking into account bothfeatures
0 comefrombiologyinthedifferentareasofpopulation dynamics, seee.g. [8,26],orgeneexpression,
7
seee.g. [27],orepidemiology, seee.g. [11]. Wefindseveralstochasticmodelsdealingwithdelayand
1
: memory also in the different areas of economics and finance. The delayed response in the prices of
v
bothcommodities andfinancialassetsisstudiedforexamplein[1,2,5,6,12,13,23,24,25,36,37].
i
X The very market inefficiency and also the fact that traders persistently use past prices as a guide to
r decisionmakinginducesmemoryeffectsthatmaybeheldresponsibleformarketbubblesandcrashes.
a
Seee.g. [3,22].
In this work we consider a general stochastic dynamic model incorporating delay or memory ef-
fects. Indeedweconsiderstochasticfunctional differentialequations(SFDE),whicharesubstantially
stochastic differential equations withcoefficients depending onthepasthistoryofthedynamicitself.
These SFDEs have already been studied in the pioneering works of [28, 29, 38] in the Brownian
framework. The theory has later been developed including models for jumps in [9]. From another
perspective modelswithmemoryhave been studied viatheso-called functional Itôcalculus asintro-
duced in [17] and then developed steadily in e.g. [14, 15]. For a comparison of the two approaches
werefer to e.g. [16, 18]. In the deterministic framework functional differential equations are widely
studied. See,e.g. [21].
Date:January22,2017.
1
2 BAÑOS,DINUNNO,HAFERKORN,ANDPROSKE
Bymodelriskwegenerically meanallrisksentailedinthechoiceofamodelinviewofprediction
or forecast. One aspect of model risk management is the study of the sensitivity of a model to the
estimates of its parameters. In this paper we are interested in the sensitivity to the initial condition.
In the terminology of mathematical finance this is referred to as the Delta. However, in the present
setting of SFDEs, the very concept of Delta has to be defined as new, being the initial condition an
initial path and not only a single initial point as in the standard stochastic differential equations. It
is the first time that the sensitivity to the initial path is tackled, though it appears naturally whenever
workinginpresence ofmemoryeffects.
Asillustration, letusconsider theSFDE:
dx(t)= f(t,x(t),x )dt+g(t,x(t),x )dW(t), t ∈[0,T]
t t
((x(0),x0)=h
wherebyx(t)wemeantheevaluationattimet ofthesolutionprocessandbyx wemeanthesegment
t
ofpastthatisrelevantfortheevaluationatt. Letusalsoconsidertheevaluation p(h )att=0ofsome
value F (h x(T),h x ) at t =T of a functional F of the model. Such evaluation is represented as the
T
expectation:
(1.1) p(h )=E[F (h x(T),h x )].
T
Wehavemarkedexplicitly thedependence ontheinitialpathh byananticipated superindex.
Evaluations of this type are typical in the pricing of financial derivatives, which are financial con-
tracts with payoff Y written on an underlying asset with price dynamics S given by an SFDE of the
typeabove. Indeedinthiscasetheclassicalnonarbitragepricingruleprovidesafairpriceintheform
Y (h S(T),h S ) Y (h S(T),h S )
prisk−neutral(h )=Eh Q T =E h Z(T) T ,
N(T) N(T)
(cid:20) (cid:21) (cid:20) (cid:21)
whereh Z(T)= dh Q istheRadon-Nykodim derivativeoftherisk-neutral probability measureh Qand
dP
h
N(T)isachosen numéraire usedfordiscounting. Weobserve thatsuchpricing measure Qdepends
onh byconstruction.
Analogously, in the so-called benchmark approach to pricing (see e.g. [32]), a non-arbitrage fair
priceisgivenintheform
Y (h S(T),h S )
p (h )=E T ,
benchmark h G(T)
(cid:20) (cid:21)
h
where G(T)isthevalueofanappropriatebenchmarkprocess, usedindiscounting andguaranteeing
that the very P is an appropriate pricing measure. Here we note that the benchmark depends on the
initial path h of the underlying price dynamics. Both pricing approaches can be represented as (1.1)
and from now on we shall generically call payoff the functional F , borrowing the terminology from
finance.
Then, in the present notations, the study of the sensitivity to the initial condition consists in the
studyofsomederivativeof p(h ):
¶ ¶
p(h )= E[F (h x(T),h x )].
¶h ¶h T
anditspossiblerepresentations.
In this work we interpret the derivative above as a functional directional derivative and we study
formulae for its representations. Our approach takes inspiration from the seminal papers [19, 20].
HereMalliavin calculus is used toobtain anice formula, where the derivative isitself represented as
anexpectationoftheproductofthefunctional F andsomerandomvariable,calledMalliavinweight.
STOCHASTICFUNCTIONALDIFFERENTIALEQUATIONSANDSENSITIVITYTOTHEIRINITIALPATH 3
We remark immediately that the presence of memory has effects well beyond the expected and
theformulae weobtain willnotbe, unfortunately, soelegant. Therepresentation formulae wefinally
obtaindonotformallypresentorrequiretheFréchetdifferentiabilityofF . Thisisparticularlyrelevant
for applications e.g. to pricing. To obtain our formulae we shall study the relationship between
functionalFréchetderviativesandMalliavinderivatives. However,thisrelationshiphastobecarefully
constructed. Our technique is based on what we call the randomisation of the initial path condition,
whichisbasedontheuseofanindependent Browniannoiseto”shake”thepast.
Thepaper is organised as follows. In Section 2 weprovide a detailed background ofSFDEs. The
first part of Section 3 is dedicated to the study of the sensitivity to the initial path condition and the
technique of randomisation. We obtain a general representation formula for the sensitivity. Here we
seethatthereisabalancebetweenthegeneralityofthefunctionalF allowedandtheregularityonthe
coefficients ofthedynamics oftheunderlying. ThesecondpartofSection3presents further detailed
results in the case of a suitable randomisation choice. The Appendix contains some technical proof,
givenwiththeaimofaself-contained reading.
2. STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS
Inthis section wepresent ageneral setup for stochastic functional differential equations (SFDEs).
Ourframeworkisinspiredbyandgeneralises [5,6]and[25].
2.1. The model. On the complete probability space (W ,F,(F ) ,P) where the filtration satis-
t t∈[0,T]
fies the usual assumptions and is such that F =F , weconsiderW ={W(t,w ); w ∈W ,t ∈[0,T]}
T
anm-dimensional standard(F ) -Brownianmotion. HereT ∈[0,¥ ).
t t∈[0,T]
We are interested in stochastic processes x : [−r,T]×W → Rd, r > 0, with finite second order
moments and a.s. continuous sample paths. So, one can look at x as a random variable x : W →
C([−r,T],Rd)inL2(W ,C([−r,T],Rd)). Infact,wecanlookatxas
x:W →C([−r,T],Rd)֒→L2([−r,T],Rd)֒→Rd×L2([−r,T],Rd)
wherethenotation ֒→stands forcontinuously embedded in, whichholds since thedomains arecom-
pact.
Fromnowon,foranyu∈[0,T],wewriteM ([−r,u],Rd):=Rd×L2([−r,u],Rd)fortheso-called
2
Delfour-Mitter spaceendowedwiththenorm
(2.1) k(v,q )k = |v|2+kq k2 1/2, (v,q )∈M ([−r,u],Rd),
M2 2 2
where k·k stands for the L2-norm(cid:0) and |·| fo(cid:1)r the Euclidean norm in Rd. For short we denote
2
M :=M ([−r,0],Rd).
2 2
The interest of using such space comes from two facts. On the one hand, the space M endowed
2
with the norm (2.1)has a Hilbert structure which allows for a Fourier representation of its elements.
Onthe other hand, aswewillsee later on, the point 0plays an important role and therefore weneed
to distinguish between two processes in L2([−r,0],Rd) that have different images at the point 0. In
general the spaces M ([−r,u],Rd) are also natural to use since they coincide with the corresponding
2
spacesofcontinuous functions C([−r,u],Rd)completedwithrespecttothenorm(2.1),bytakingthe
naturalinjection i(j (·))=(j (u),j (·)1 )foraj ∈C([−r,u],Rd)andbyclosingit.
[−r,u)
4 BAÑOS,DINUNNO,HAFERKORN,ANDPROSKE
Furthermore, by the continuous embedding above, we can consider the random process x: W ×
[−r,u]−→Rd asarandomvariable
x:W −→M ([−r,u],Rd)
2
inL2(W ,M ([−r,u],Rd)),thatis
2
1/2
kxkL2(W ,M2([−r,u],Rd))= W kx(w )k2M2([−r,u],Rd)P(dw ) <¥ .
(cid:18)Z (cid:19)
Forlater use, wewriteL2(W ,M ([−r,u],Rd))forthe subspace ofL2(W ,M ([−r,u],Rd))ofelements
A 2 2
thatadmitan(F ) -adapted modification.
t t∈[0,u]
Todealwithmemoryanddelayweusetheconceptofsegmentofx. Givenaprocessx,somedelay
gapr>0,andaspecifiedtimet ∈[0,T],thesegment ofxinthepasttimeinterval [t−r,t]isdenoted
byx(w ,·):[−r,0]→Rd anditisdefinedas
t
x(w ,s):=x(w ,t+s), s∈[−r,0].
t
Sox (w ,·)isthesegmentofthew -trajectory oftheprocess x,andcontains alltheinformation ofthe
t
past down to time t−r. In particular, the segment of x relative to time t =0 is the initial path and
0
carriestheinformation abouttheprocess frombeforet =0.
Assume that, for each w ∈W , x(·,w )∈L2([−r,T],Rd). Then x (w ) can be seen as an element of
t
L2([−r,0],Rd)foreachw ∈W andt ∈[0,T]. Indeedthecouple (x(t),x )isaF -measurable random
t t
variablewithvaluesinM ,i.e. (x(t,w ),x (w ,·))∈M ,givenw ∈W .
2 t 2
Let us consider an F -measurable random variable h ∈L2(W ,M ). To shorten notation we write
0 2
M :=L2(W ,M ). Astochastic functional differential equation (SFDE),iswrittenas
2 2
dx(t)= f(t,x(t),x )dt+g(t,x(t),x )dW(t), t ∈[0,T]
t t
(2.2)
((x(0),x0)=h ∈M2
where
f :[0,T]×M →Rd and g:[0,T]×M →L(Rm,Rd).
2 2
2.2. Existence anduniquenessofsolutions. Undersuitable hypotheses onthefunctionals f and g,
one obtains existence and uniqueness of the strong solution (in the sense of L2) of the SFDE (2.2).
The solution is a process x∈L2(W ,M ([−r,T],Rd)) admitting an (F ) -adapted modification,
2 t t∈[0,T]
thatis,x∈L2(W ,M ([−r,T],Rd)).
A 2
Wesaythattwoprocesses x1,x2∈L2(W ,M ([−r,T],Rd))areL2-unique, oruniqueintheL2-sense
2
ifkx1−x2kL2(W ,M2([−r,T],Rd))=0.
Hypotheses(EU):
(EU1) (LocalLipschitzianity)Thedriftandthediffusionfunctionals f andgareLipschitzonbounded
setsinthesecondvariableuniformlyw.r.t. thefirst,i.e.,foreachintegern>0,thereisaLip-
schitzcontant L independent oft ∈[0,T]suchthat,
n
|f(t,j )− f(t,j )| +kg(t,j )−g(t,j )k 6L kj −j k
1 2 Rd 1 2 L(Rm,Rd) n 1 2 M2
forallt ∈[0,T]andfunctions j ,j ∈M suchthatkj k 6n,kj k 6n.
1 2 2 1 M2 2 M2
STOCHASTICFUNCTIONALDIFFERENTIALEQUATIONSANDSENSITIVITYTOTHEIRINITIALPATH 5
(EU2) (Lineargrowths)ThereexistsaconstantC>0suchthat,
|f(t,y )| +kg(t,y )k 6C(1+ky k )
Rd L(Rm,Rd) M2
forallt ∈[0,T]andy ∈M .
2
Thefollowingresultbelongsto[28,Theorem2.1]. Itsproofisbasedonanapproach similartothe
oneintheclassical deterministic casebasedonsuccessive Picardapproximations.
Theorem2.1(ExistenceandUniqueness). GivenHypotheses(EU)onthecoefficients f andgandthe
initialcondition h ∈M ,theSFDE(2.2)hasasolution h x∈L2(W ,M ([−r,T],Rd))whichisunique
2 A 2
inthesenseofL2. Thesolution(orbetteritsadaptedrepresentative)isaprocessh x:W ×[−r,T]→Rd
suchthat
(1) h x(t)=h (t),t ∈[−r,0].
(2) h x(w )∈M ([−r,T],Rd)w -a.s.
2
(3) Foreveryt ∈[0,T],h x(t):W →Rd isF-measurable.
t
Fromtheaboveweseethatitmakessensetowrite
h (0)+ t f(u, h x(u), h x )du+ tg(u, h x(u), h x )dW(u), t ∈[0,T]
h x(t)= 0 u 0 u
(h (t), t ∈R [−r,0]. R
Observethattheaboveintegrals arewelldefined. Infact,theprocess
(w ,t)7→(h x(t,w ), h x (w ))
t
belongs to M and is adapted since x is pathcontinuous and adapted and its composition with the
2
h
deterministiccoefficients f andgisthenadaptedaswell. Notethat xrepresentsthesolutionstarting
offattime0withinitialcondition h ∈M .
2
One could consider the same dynamics but starting off at a later time, let us say, s∈(0,T], with
initialcondition h ∈M . Namely,wecouldconsider:
2
dx(t)= f(t,x(t),x )dt+g(t,x(t),x )dW(t), t ∈[s,T]
t t
(2.3)
(x(t)=h (t−s), t ∈[s−r,s].
Again,under(EU)theSFDE(2.3)hasthesolution,
h (0)+ t f(u, h xs(u), h xs)du+ tg(u, h xs(u), h xs)dW(u), t ∈[s,T]
(2.4) h xs(t)= s u s u
(h (t−s)R, t ∈[s−r,s] R
Theright-hand sidesuperindex inh xs denotesthestartingtime. Wewillomitthesuperindex when
startingat0,h x0= h x. Theinterestofdefiningthesolutionto(2.3)startingatanytimescomesfrom
the semigroup property of the flow of the solution which we present in the next subsection. For this
reasonweintroduce thenotation
(2.5) Xs(h ,w ):=X(s,t,h ,w ):=(h xs(t,w ),h xs(w )), w ∈W , s6t.
t t
Inrelation to(2.3)wealsodefinethefollowingevaluationoperator:
r :M →Rd, r j :=v foranyj =(v,q )∈M .
0 2 0 2
Weobserveherethattherandomvariable h xs(t)isanevaluation at0oftheprocessXs(h ),t ∈[s,T].
t
6 BAÑOS,DINUNNO,HAFERKORN,ANDPROSKE
2.3. Differentiability of the solution. We recall that our goal is the study of the influence of the
initial path h on the functionals of the solution of (2.2). For this we need to ensure the existence
of an at-least-once differentiable stochastic flow for (2.2). Hereafter we discuss the differentiability
conditions onthecoefficients ofthedynamicstoensuresuchproperty ontheflow.
In general, suppose we have E and F Banach spaces, U ⊆ E an open set and k ∈ N. We write
Lk(E,F) for the space of continuous k-multilinear operators A:Ek →F endowed with the uniform
norm
kAk :=sup{kA(v ,...,v )k ,kvk 61,i=1,...,k}.
Lk(E,F) 1 k F i E
Then an operator f :U → F is said to be of class Ck,d if it is Ck and Dkf :U → Lk(E,F) is d -
Hölder continuous on bounded sets inU. Moreover, f :U →F is said to be of class Ck,d if it isCk,
b
Dkf :U →Lk(E,F)isd -Hölder continuous onU,and allitsderivatives Djf, 16 j6k areglobally
bounded onU. ThederivativeDistakenintheFréchetsense.
Firstofallweconsider SFDEsinthespecialcasewhen
g(t,(j (0),j (·)))=g(t,j (0)), j =(j (0),j (·))∈M
2
thatis,gisactuallyafunction [0,T]×Rd →Rd×m.
Forcompleteness wegivethedefinitionofstochastic flow.
Definition 2.2. Denote by S([0,T]):={s,t ∈[0,T]:06s<t <T}. Let E be a Banach space. A
stochastic Ck,d -semiflow on E is a measurable mapping X : S([0,T])×E×W → E satisfying the
followingproperties:
(i) Foreachw ∈W ,themapX(·,·,·,w ):S([0,T])×E →E iscontinuous.
(ii) Forfixed(s,t,w )∈S([0,T])×W themapX(s,t,·,w ):E →E isCk,d .
(iii) For06s6u6t,w ∈W andx∈E,thepropertyX(s,t,h ,w )=X(u,t,X(s,u,h ,w ),w )holds.
(iv) Forall(t,h ,w )∈[0,T]×E×W ,onehasX(t,t,h ,w )=h .
Inoursetup, weconsider thespaceE =M .
2
Hypotheses(FlowS):
(FlowS1) Thefunction f :[0,T]×M →Rd isjointlycontinuous;themapM ∋j 7→ f(t,j )isLipschitz
2 2
on bounded sets in M and C1,d uniformly in t (i.e. the d -Hölder constant is uniformly
2
boundedint ∈[0,T])forsomed ∈(0,1].
(FlowS2) Thefunction g:[0,T]×Rd →Rd×m is jointly continuous; the map Rd ∋v7→g(t,v) is C2,d
b
uniformlyint.
(FlowS3) Oneofthefollowingconditions issatisfied:
(a) ThereexistC>0andg ∈[0,1)suchthat
|f(t,j )|6C(1+kj kg )
M2
forallt ∈[0,T]andallj ∈M
2
(b) For all t ∈ [0,T] and j ∈ M , one has f(t,j ,w ) = f(t,j (0),w ). Moreover, it exists
2
r ∈(0,r)suchthat
0
f(t,j ,w )= f(t,j˜,w )
forallt ∈[0,T]andallj˜ suchthatj (·)1 (·)=j˜(·)1 (·).
[−r,−r0] [−r,−r0]
(c) Forallw ∈W ,
sup k(Dy (t,v,w ))−1k <¥ ,
M2
t∈[0,T]
STOCHASTICFUNCTIONALDIFFERENTIALEQUATIONSANDSENSITIVITYTOTHEIRINITIALPATH 7
wherey (t,v)isdefinedbythestochastic differential equation
dy (t,v)=g(t,y (t,v))dW(t),
(y (0,v)=v.
Moreover, thereexistsaconstant Csuchthat
|f(t,j )|6C(1+kj k )
M2
forallt ∈[0,T]andj ∈M .
2
Then,[29,Theorem3.1]statesthefollowingtheorem.
Theorem2.3. UnderHypotheses(EU)and(FlowS),Xs(h ,w )definedin(2.5)isaC1,e -semiflowfor
t
everye ∈(0,d ).
Next,wecanconsider amoregeneral diffusion coefficient gfollowing theapproach introduced in
[29,Section5]. Letusassumethatthefunction gisoftype:
t
g(t,(x(t),x ))=g¯(t,x(t),a+ h(s,(x(s),x ))ds),
t s
0
Z
for some constant a and some functions g¯ and h satisfying some regularity conditions that will be
specified later. This case can be transformed into a system of the previous type where the diffusion
coefficientdoesnotexplicitlydependonthesegment. Infact,definingy(t):=(y(1)(t),y(2)(t))⊤where
y(1)(t):=x(t), t ∈[−r,T], y(2)(t):=a+ th(s,(x(s),x ))ds, t ∈[0,T]and y(2)(t):=0on [−r,0], we
0 s
havethefollowingdynamicsfory:
R
dy(t)=F(t,y(t),y )dt+G(t,y(t))dW(t),
t
(2.6)
(y(0)=(h (0),a)⊤,y0=(h ,0)⊤,
where
f(t,y(1)(t),y(1)) g¯(t,y(1)(t),y(2)(t))
(2.7) F(t,y(t),y )= t ,G(t,y(t))= .
t h(t,y(1)(t),yt(1))! (cid:18) 0 (cid:19)
The transformed system (2.6) is now an SFDEof type (2.2) where the diffusion coefficient does not
explicitely depend on the segment. That is the differentiability of the flow can be studied under the
corresponding Hypotheses (FlowS).Hereafter,wespecifytheconditions ong¯andhsothatHypothe-
ses(EU)and(FlowS)aresatisfiedbythetransformedsystem(2.6). Sincetheconditions(FlowS3)(a)
and(b)arebothtoorestrictive for(2.6),wewillmakesurethat(FlowS3)(c)issatisfied. Underthese
conditionswecanguaranteethedifferentiability ofthesolutionstotheSFDE(2.3)fortheaboveclass
ofdiffusion coefficientg.
Hypotheses(Flow):
(Flow1) f satisfies(FlowS1)andthereexistsaconstant Csuchthat
|f(t,j )|6C(1+kj k )
M2
forallt ∈[0,T]andj ∈M .
2
(Flow2) g(t,j )isofthefollowingform
g(t,j )=g¯(t,v,g˜(q )), t ∈[0,T], j =(v,q )∈M
2
whereg¯satisfiesthefollowingconditions:
8 BAÑOS,DINUNNO,HAFERKORN,ANDPROSKE
(a) Thefunctiong¯:[0,T]×Rd+k→Rd×m isjointlycontinuous; themapRd+k∋y7→g¯(t,y)
isC2,d uniformlyint.
b
(b) Foreachv∈Rd+k,let{Y (t,v)} solvethestochastic differential equation
t∈[0,T]
tg¯(s,Y (s,v))dW(s)
Y (t,v)=v+ 0 ,
0
(cid:18)R (cid:19)
where 0denotes the null-vector inRk. Then Y (t,v)isFréchet differentiable w.r.t. vand
theJacobi-matrix DY (t,v)isinvertible andfulfils,forallw ∈W ,
sup kDY −1(t,v,w )k<¥ ,wherek·kdenotesanymatrixnorm.
t∈[0,T]
v∈Rd+k
and,g˜:L2([−r,0],Rd)→Rk satisfiesthefollowingconditions:
(c) Itexistsajointlycontinuousfunctionh:[0,T]×M →Rks.t. foreachj˜ ∈L2([−r,T],Rd),
2
t
g˜(j˜ )=g˜(j˜ )+ h(s,(j˜(s),j˜ ))ds,
t 0 s
0
Z
wherej˜ ∈L2([−r,0],Rd)isthesegmentatt ofarepresentative ofj˜.
t
(d) M ∋j 7→h(t,j )isLipschitzonboundedsetsinM ,uniformlyw.r.t. t∈[0,T]andC1,d
2 2
uniformly int.
Corollary 2.4. UnderHypotheses (Flow),thesolution Xs(h )=X(s,t,h ,w ),w ∈W ,t >sto(2.3)is
t
aC1,e -semiflow foreverye ∈(0,d ). Inparticular, j 7→X(s,t,j ,w )isC1 intheFréchetsense.
3. SENSITIVITY ANALYSIS TO THE INITIAL PATH CONDITION
Fromnowon, weconsider astochastic process xwhichsatisfies dynamics (2.2),wherethecoeffi-
cients f andgaresuchthatconditions (EU)and(Flow)aresatisfied.
Ourfinalgoalistostudythesensitivity ofevaluations oftype
(3.1) p(h )=E F (X0(h )) =E[F (h x(T), h x )], h ∈M
T T 2
totheinitialpathinthemodelh x. (cid:2)Here,F :M(cid:3)→RissuchthatF (X0(h ))∈L2(W ,R). Thesensitivity
2 T
willbeinterpreted asthedirectional derivative
d p(h +e h)−p(h )
(3.2) ¶ p(h ):= p(h +e h) = lim , h∈M .
h de (cid:12)e=0 e→0 e 2
(cid:12)
Henceweshallstudypertubationsdirection(cid:12)h∈M . Thefinalaimistogivearepresentationof¶ p(h )
(cid:12) 2 h
inwhichthefunction F isnotdirectlydifferentiated. Thisisinthelinewiththerepresentation ofthe
sensitivityparameterDeltabymeansofweights. See,e.g. theMalliavinweightintroducedin[19,20]
fortheclassicalcaseofnomemory. Forthisweimposesomestrongerregularity conditions on f and
g:
Hypotheses(H):
STOCHASTICFUNCTIONALDIFFERENTIALEQUATIONSANDSENSITIVITYTOTHEIRINITIALPATH 9
(H1) (GlobalLipschitzianity) j 7→ f(t,j ),j 7→g(t,j )globallyLipschitzuniformlyint withLip-
schitzconstants L andL ,i.e.
f g
|f(t,j )− f(t,j )| 6L kj −j k
1 2 Rd f 1 2 M2
kg(t,j )−g(t,j )k 6L kj −j k
1 2 L(Rm,Rd) g 1 2 M2
forallt ∈[0,T]andj ,j ∈M .
1 2 2
(H2) (LipschitzianityoftheFréchetderivatives)j 7→Df(t,j ),j 7→Dg(t,j )aregloballyLipschitz
uniformlyint withLipschitzconstants L andL ,i.e.
Df Dg
kDf(t,j )−Df(t,j )k6L kj −j k
1 2 Df 1 2 M2
kDg(t,j )−Dg(t,j )k6L kj −j k
1 2 Dg 1 2 M2
forallt ∈[0,T]andj ,j ∈M .
1 2 2
Thecorresponding stochastic C1,1-semiflowisagaindenotedbyX.
Before proceeding, we give a simple example of SFDE satisfying all assumptions (EU), (Flow)
and(H).
Example3.1. ConsidertheSFDE(2.2)wherethefunctions f andgaregivenby
0
f(t,j )=M(t)j (0)+ M¯(s)j (s)ds,
−r
Z
0
g(t,j )=S (t)j (0)+ S¯(s)j (s)ds,
−r
Z
whereM:[0,T]→Rd×d,M¯ :[−r,0]→Rd×d,S :[0,T]→L(Rd,Rd×m),andS¯ :[−r,0]→L(Rd,Rd×m)
areboundeddifferentiable functions, S¯(−r)=0ands7→S¯′(s)= d S¯(s)arebounded aswell.
ds
Obviously, f and gsatisfy (EU)and (H)and therefore also (Flow1). In order to check conditions
(Flow2),wenotethat
g(t,j )=g¯(t,j (0),g˜(j (·))),
where
0
g¯(t,y)=S (t)y(1)+y(2),y=(y(1),y(2))⊤, and g˜(j (·))= S¯(s)j (s)ds.
−r
Z
The function g¯ satisfies condition (Flow2)(a) as S is bounded and continuous. Let us check con-
dition (Flow2)(b) in the case d =m =1. Then g¯(t,y) = s (t)y(1)+y(2), where s is a real valued,
differentiable functionandY fulfilsthetwo-dimensional stochastic differential equation
Y (1)(t,v)=v(1)+ ts¯(s)Y (1)(s,v)+v(2)dW(s),
0
(Y (2)(t,v)=v(2), R
whichhasthesolution
t t
Y (1)(t,v)=Y˜(t) v(1)− s (s)v(2)Y˜−1(s)ds+ v(2)Y˜−1(s)dW(s) , Y (2)(t,v)=v(2),
0 0
(cid:18) Z Z (cid:19)
with
t t
Y˜(t)=exp − s 2(s)ds+ s (s)dW(s) .
0 0
(cid:26) Z Z (cid:27)
10 BAÑOS,DINUNNO,HAFERKORN,ANDPROSKE
Therefore,wegetthat
1+Y˜(t) Y˜(t) − ts (s)Y˜−1(s)ds+ tY˜−1(s)dW(s)
DY (t,v)= 0 0
0 1
(cid:18) (cid:0) R R (cid:1)(cid:19)
and
1 − Y˜(t) − ts (s)Y˜−1(s)ds+ tY˜−1(s)dW(s)
DY −1(t,v)= 1+Y˜(t) 1+Y˜(t) 0 0
0 (cid:0) R 1 R (cid:1)!
UsinginfactthatY˜(t)>0andapplying theFrobeniusnormk·k ,weobtain w -a.e.
F
kDY −1(t,v)k =tr (DY −1(t,v))⊤DY −1(t,v)
F
(cid:16) t (cid:17) t 2
62+Y˜2(t) − s (s)Y˜−1(s)ds+ Y˜−1(s)dW(s) <¥ ,
0 0
(cid:18) Z Z (cid:19)
fort ∈[0,T],v∈R2. BythisHypothesis(Flow2)(b)isfulfilled.
Moreover, a simple application of partial integration and Fubini’s theorem together with the fact
thatS¯(−r)=0showsthat
0 0 t 0
g˜(j˜ )= S¯(s)j˜ (s)ds= S¯(s)j˜ (s)ds+ S (0)j˜(u)− S¯′(s)j˜ (s)ds du
t t 0 u
−r −r 0 −r
Z Z Z (cid:26) Z (cid:27)
t
=g˜(j˜ )+ h(t,j˜(u),j˜ )du.
0 u
0
Z
It can be easlily checked that h(t,j )=S (0)j (0)− 0 S¯′(s)j (s)ds satisfies the conditions given in
−r
(Flow2)(c)and (d).
R
Wearenowreadytointroduce twotechnical lemmasneededtoproveourmainresults.
Lemma3.2. Assumethatthesolutionto(2.3)existsandhasaC1,1-semiflowXs(h ,w ),s6t,w ∈W .
t
Then,thefollowingequality holdsforallw ∈W andalldirections h∈M :
2
DXs(h ,w )[h]=(Dh xs(t,w )[h],Dh xs(t+·,w )[h])∈M .
t 2
Proof. NotethatDXts(h ,w )[h]∈M2. Let{ei}i∈N beanorthonormal basisofM2. Then,
¥ ¥
DXs(h ,w )[h]= (cid:229) hDXs(h ,w )[h],ei e = (cid:229) DhXs(h ,w ),ei [h]e
t t i M2 i t i M2 i
i=0 i=0
¥ 0
= (cid:229) D xs(t,w )e(0)+ xs(t+u,w )e(u)du [h]e
i i i
i=0 (cid:18) Z−r (cid:19)
¥ 0
= (cid:229) Dxs(t,w )[h]e(0)+ Dxs(t+u,w )[h]e(u)du e
i i i
i=0(cid:18) Z−r (cid:19)
¥
= (cid:229) h(Dh xs(t,w )[h],Dh xs(t+·,w )[h]),ei e
i M2 i
i=0
=(Dh xs(t,w )[h],Dh xs(t+·,w )[h]).
Thisfinishestheproof. (cid:3)