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

numerical methods for approximating solutions to rough differential equations

Similar numerical methods for approximating solutions to rough differential equations