numerical methods for approximating solutions to rough differential equations

155 Pages·2009·1.5 MB·English

NUMERICAL METHODS FOR APPROXIMATING
SOLUTIONS TO ROUGH DIFFERENTIAL EQUATIONS
LAJOS GERGELY GYURKO´
StHugh’sCollege
UniversityofOxford
(cid:0)
DPhilThesis
Michaelmas2008
ACKNOWLEDGMENTS
IwouldliketoexpressmygratitudetoProfTerryLyonsforbeingmysupervisor,mentor
and friend and also for the never-ending and illuminating discussions and debates on
SDEs,RDEs,C++andingeneralonmathsandlife.
IamgratefultoPe´terMedvegyevforhislong-terminspiration.
AmongmyfellowDPhilstudents,IowemanythankstoChristianLittererandMichael
Coulonforreadingandcommentingonthisthesis. Icannotmentionhereallthestudents
andotherresearchersIhavehadthechancetotalktoandlearnaboutmaths. Iamgrateful
toallofthem.
IowemanythankstoSamHowison,GlenysLuke,BenHambly,PierreTarre`s,Andrew
Dancer and Christoph Reisinger for the teaching opportunities at the Mathematical Insti-
tute,StCatherine’s,StHugh’sandJesusColleges. IamalsothankfultoMatthiasArnsdorf,
Re´gisGuichard,IanDowker,RobBastonandmanyotherformercolleaguesatUBSandJP
Morganfortheinternshipopportunitiesandsupport.
I thankfully acknowledge the financial support of the EPSRC and the Oxford-Man In-
stitute.
Finally, I am deeply grateful to my wife, parents, sisters and in-laws for their many
forms of devoted support and in particular to my wife Tu¨ndi for her love and (almost)
unlimitedpatience.
i
ii
DEDICATED TO MY SONS DOMONKOS AND FE´LIX
AND TO THE LOVING MEMORY OF MY LATE GRANDPARENTS
BOLDIZSA´R, BORBA´LA AND LAJOS
CONTENTS
Listoffigures vi
Listofnotations vii
Introduction 1
1 RoughDifferentialEquations 3
1.1 IntroductiontoRoughPaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Integrationalongroughpaths . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Differentialequationsdrivenby p-roughpaths . . . . . . . . . . . . . . . . . 8
2 DifferentialEquationsdrivenbyΠ-roughpaths 9
Π
2.1 -roughpaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Π
2.2 Almost -roughpaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Π
2.3 Integrationalong -roughpaths . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Integralofinhomogeneousdegreeofsmoothness . . . . . . . . . . . . . . . 25
Π
2.5 Differentialequationsdrivenby -roughpaths . . . . . . . . . . . . . . . . . 27
2.5.1 ProofoftheUniversalLimitTheorem . . . . . . . . . . . . . . . . . . 28
2.6 Resultsoncontinuityandglobalcontrol . . . . . . . . . . . . . . . . . . . . . 32
2.6.1 Controlofthesolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6.2 Continuityofthesolution . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6.3 Aremarkonsimplefunctionalsofroughpaths . . . . . . . . . . . . . 41
3 NumericalsolutionofRoughDifferentialEquations 42
3.1 Approximationsofroughpaths . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Globalconvergencetheorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 Approximatingtermscorrespondingtomulti-indicesinB . . . . . 48
1
3.2.2 Approximatingtermscorrespondingtomulti-indicesinB form > 1 50
m
3.3 Localbehavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.1 Algebraicsetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
iii
CONTENTS iv
3.3.3 Therough-Taylorexpansion . . . . . . . . . . . . . . . . . . . . . . . 59
3.3.4 Propertiesoftheextendedvectorfields . . . . . . . . . . . . . . . . . 65
3.4 LocalapproximationbasedonlocallyderivedODEs . . . . . . . . . . . . . . 67
3.4.1 Approximatingtermscorrespondingtomulti-indicesinB . . . . . 67
1
3.4.2 Approximatingtermscorrespondingtomulti-indicesinB form > 1 72
m
3.4.3 NotesontheLog-signaturetheorems . . . . . . . . . . . . . . . . . . 75
3.4.4 Notesonaspecialcase . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Localapproximationunderweakerconditions . . . . . . . . . . . . . . . . . 76
3.5.1 RoughpolynomialapproximationofRDEs . . . . . . . . . . . . . . . 77
3.5.2 ODE-basedapproximationofRDEsdeterminedbypolynomialvec-
torfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4 NumericalsolutionsofStochasticDifferentialEquations 82
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2 EstimatesofStratonovichintegrals . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 Approximatingtermscorrespondingtomulti-indicesinB . . . . . . . . . . 87
1
4.3.1 Globalconvergencetheorem . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.2 Thestochasticlog-signaturetheorem . . . . . . . . . . . . . . . . . . 91
4.4 Approximatingtermscorrespondingtomulti-indicesinB forh > 1 . . . . 94
h
4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.6 Implementinghighorderschemes . . . . . . . . . . . . . . . . . . . . . . . . 102
4.6.1 Practicalconsiderations . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.6.2 Anoteonthecomputationalexpense . . . . . . . . . . . . . . . . . . 106
4.7 Numericalexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.7.1 Estimatingtheorderofconvergence . . . . . . . . . . . . . . . . . . . 106
4.7.2 FirstorderapproximationoftheCIRprocess . . . . . . . . . . . . . . 108
4.7.3 Secondorderapproximationimplemented . . . . . . . . . . . . . . . 111
5 NumericalevaluationoftheKusuoka-Lyons-Victoirmethod 113
5.1 DescriptionoftheKLVfamily . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1.1 CubatureonWienerspace . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1.2 Kusuoka’sapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2 Practicalconsiderations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2.1 SolvingtheODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2.2 Unevenpartitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.2.3 Thecasewhenthesupportoftheglobalcubaturemeasureexplodes 122
5.3 Numericalresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3.1 LinearSDEinonedimension . . . . . . . . . . . . . . . . . . . . . . . 124
5.3.2 Non-linearSDEinonedimension . . . . . . . . . . . . . . . . . . . . 128
CONTENTS v
5.3.3 StockwithHestonvolatility . . . . . . . . . . . . . . . . . . . . . . . . 131
5.3.4 Treebasedbranchingalgorithmappliedin4dimensions . . . . . . . 133
5.3.5 Somespecialissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
A Tree-basedbranchingalgorithm 141
LIST OF FIGURES
3.1 PropagationofthelocalerrorthroughtheRDEsandtheODEs . . . . . . . . 47
4.1 Weakapproximationresults(CIR) . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2 Estimatederror,comparisonofsolvers(CIR) . . . . . . . . . . . . . . . . . . 110
4.3 Implementinghighorderapproximations . . . . . . . . . . . . . . . . . . . . 112
5.1 ErrorofapproximatingE[S ]in1D . . . . . . . . . . . . . . . . . . . . . . . 126
T
5.2 ErrorofapproximatingE[A ]in1D . . . . . . . . . . . . . . . . . . . . . . . 126
T
5.3 ErrorofvaluingtheITMoptionin1D . . . . . . . . . . . . . . . . . . . . . . 127
5.4 ErrorofvaluingtheOTMoptionin1D . . . . . . . . . . . . . . . . . . . . . . 127
5.5 EstimatederrorofvaluingtheAsianoptionin1D . . . . . . . . . . . . . . . 128
5.6 ErrorofapproximatingE[r ]in1D . . . . . . . . . . . . . . . . . . . . . . . . 129
T
5.7 ErrorofapproximatingE[q ]in1D . . . . . . . . . . . . . . . . . . . . . . . 129
T
5.8 EstimatederrorofE[g (r )]in1D . . . . . . . . . . . . . . . . . . . . . . . . 130
3 T
5.9 EstimatederrorofE[g (r )]in1D . . . . . . . . . . . . . . . . . . . . . . . . 130
4 T
5.10 ErrorofapproximatingE[v ]in2D . . . . . . . . . . . . . . . . . . . . . . . 131
T
5.11 ErrorofapproximatingE[S ]in2D . . . . . . . . . . . . . . . . . . . . . . . 132
T
5.12 EstimatederroroftheAsianoption1in2D . . . . . . . . . . . . . . . . . . . 132
5.13 EstimatederroroftheAsianoption2in2D . . . . . . . . . . . . . . . . . . . 133
5.14 Estimatingthemaxin4D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.15 Estimatingthevalueofabasketspreadoptionin4D . . . . . . . . . . . . . . 135
5.16 Variance of TBBA and Monte-Carlo estimates corresponding to the basket
option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.17 Rombergextrapolationappliedtothe1Dlinearproblem(E[A ]) . . . . . . 137
T
5.18 Rombergextrapolationappliedtothe1Dlinearproblem(ITMoption) . . . 137
5.19 Rombergextrapolationappliedonthe1Dlinearproblem(Asianoption) . . 138
5.20 Testingunevenpartitionsonthe1Dlinearproblem . . . . . . . . . . . . . . 139
5.21 KLVwithdifferentODEsolversappliedtothe1Dlinearproblem(E[S ]) . 139
T
.
vi
LIST OF NOTATIONS
In the following list, R = (r ,...,r ) and Q = (q ,...,q ) are a k-multi-indices and Π is
1 l 1 m
apositivereal k-tuple(seeDefinition2.1.1). V andV1,...,Vk areBanachspacessuchthat
V = V1⊕···⊕Vk.
• OS(k ,...,k )seeDefinition1.2.3
1 n
• T((V))thespaceofformalseriesoftensorsofV (Definition1.1.1)
• AΠ = Ak setofallk-multi-indicesoffinitelength(Definition2.1.1)
• (cid:101)theemptyk-multi-index(Definition2.1.1)
• −R = −(r ,r ,...,r ,r ) = (r ,...,r ,r )(Definition2.1.1)
1 2 l−1 l 2 l−1 l
• R− = (r ,r ,...,r ,r )− = (r ,r ,...,r )(Definition2.1.1)
1 2 l−1 l 1 2 l−1
• R∗Q = (r ,...,r )∗(q ,...,q ) = (r ,...,r ,q ,...,q )(Definition2.1.1)
1 l 1 m 1 l 1 m
• V⊗R = Vr1 ⊗···⊗Vrl (Definition2.1.2)
• π : A⊕B → Ageneralformofthecanonicalprojection,specialcaseπ : T((V)) →
A V
V (Definition2.1.2)
• πR = πVr1⊗···⊗Vrl : T((V)) → V⊗R canonicalprojection(Definition2.1.2)
• π : T((V)) → T((Vi))canonicalprojection(Definition2.1.2)
T((Vi))
• aR = π afora ∈ T((V))(Definition2.1.2)
R
• v := (π v)⊗···⊗(π v) ∈ V⊗R forv ∈ V (Definition2.1.2)
R (r ) (r)
1 l
• n (R) := card{i|r = j,r ∈ R}(Definition2.1.3)
j i i
• deg (R) = ∑k nj,Π(R) (Definition2.1.3)
Π i=1 pj
• ΓΠ(R) = n1(R) !··· nk(R) !for R ∈ Ak (Definition2.1.3)
p p
1 k
(cid:16) (cid:17) (cid:16) (cid:17)
• AΠ := R = (r ,...,r ) l ≥ 1, deg (R) ≤ s (Definition2.1.3)
s 1 l Π
(cid:110) (cid:12) (cid:111)
(cid:12)
(cid:12) vii
LISTOFFIGURES viii
• BΠ = a ∈ T((V))|∀R ∈ AΠ, aR := 0 (Definition2.1.3)
s s
• T(Π,s)((cid:8)V) := T((V))/BΠ (Definition2.(cid:9)1.3)
s
• πΠ,s : T((V)) → R∈AΠVR canonicalprojection(Remark2.1.1)
s
• aΠ,s = πΠ,sa(Rem(cid:76)ark2.1.1)
• therestrictionofπ to T(Π,s)(V)isalsodenotedbyπ
V V
• SΠ = {0 = s < s < s < ···}(Remark2.1.2)
0 1 2
• V⊗(Π,sm) = πΠ,sm −πΠ,sm−1 a |a ∈ T((V)) (Remark2.1.2)
• dΠ(X,Y) :=(cid:8)(cid:0)maxR∈A1k supD∈P(cid:1)([0,T]) ∑D XtRl(cid:9)−1,tl −YtRl−1,tl 1/degΠ(R) degΠ(R) the Π-va-
riationalmetric(Definition2.1.6) (cid:18) (cid:13) (cid:13) (cid:19)
(cid:13) (cid:13)
(cid:13) (cid:13)
• dα (X,Y) := max sup ∑ XR −YR 1/degΠ(R) degΠ(R) (Definiti-
Π R∈Akα D∈P([0,T]) D tl−1,tl tl−1,tl
on2.1.6) (cid:18) (cid:13) (cid:13) (cid:19)
(cid:13) (cid:13)
(cid:13) (cid:13)
• a(cid:104)sm(cid:105) = ∑ aR fora ∈ T((V))(Definition2.3.1)
degΠ,1(R)=sm
• aV = π afora ∈ T((V))(Definition2.3.1)
V
• π(sm1,...,smn)(a) = ∑ R1,...,Rn∈Ak πR1∗···∗Rna. for sm1,...,smn ∈ S(Π,1) and a ∈
degΠ(Ri−)=smi,i=1,...,n
T((V))(Definition2.3.2)
• ∆D = {(s,t)|s,t ∈ D, s ≤ t}.forD = {0 = t < ··· < t = T}(Definition3.1.1)
T 0 n
• dα,D(Y,Y) := max ∑ YR −YR 1/degΠY(R) degΠY(R) (Definition3.1.4)
ΠY R∈Akα D tl−1,tl tl−1,tl
(cid:98) D⊆D (cid:18) (cid:13) (cid:13) (cid:19)
(cid:13) (cid:13)
(cid:98) (cid:98)
• (Bn)n>0 asequenceof(cid:98)tree-lik(cid:13)esets(Definition(cid:13)3.2.4)
y
• W ,i = 1,...,kvectorfields(Definition3.3.8)
i
• Φy analgebrahomomorphism(Definition3.3.8)
• Wy,B,i = 1,...,kvectorfieldsforatree-likesetB (Definition3.3.9)
i
• Φy,B analgebrahomomorphism(Definition3.3.9)
• XR g(YU) := g(YU )dX(r1)···dX(rl) ∈ R iterated integrals (see
s,t 0,· s<u1<···<ul<t 0,u1 0,u1 0,ul
after(cid:110)Lemma(cid:111)3.3.4)
(cid:82)
(cid:98) (cid:98) (cid:98)
• Exp(tF)(ξ)solutiontotheODEdrivenbythevectorfield F andstartedatξ (section
3.4.1)
LISTOFFIGURES ix
• BR := ◦dBr1◦···◦dBrk Stratonovichiteratedintegrals(Definition4.2.1)
s,t s<t1<···<tk<t t1 tk
• BsR,t(Y·)(cid:82):= s<t1<···<tk<tYt1 ◦dBtr11 ◦···◦dBtrkk Stratonovich iterated integrals (Defini-
tion4.2.1)
(cid:82)
• DR := dBr1···dBrk Itoˆ iteratedintegrals(Definition4.2.1)
s,t s<t1<···<tk<t t1 tk
• DsR,t(Y·)(cid:82):= s<t1<···<tk<tYt1dBtr11···dBtrkk Itoˆ iteratedintegrals(Definition4.2.1)
• H aclasso(cid:82)fItoˆ integrablefunctions(introducedinDefinition4.2.3)
R
• HR(ξ·) a class of Stratonovich integrable functions corresponding to the process ξ·
(introducedinLemma4.2.2)