Numerical Approximations of Stochastic Differential Equations With Non-globally Lipschitz Continuous Coefficients

112 Pages·2015·0.8 MB·English

by  Martin Hutzenthaler

MEMOIRS
of the
American Mathematical Society
Volume 236 • Number 1112 (second of 6 numbers) • July 2015
Numerical Approximations of
Stochastic Differential Equations
with Non-Globally Lipschitz
Continuous Coefficients
Martin Hutzenthaler
Arnulf Jentzen
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 236 • Number 1112 (second of 6 numbers) • July 2015
Numerical Approximations of
Stochastic Differential Equations
with Non-Globally Lipschitz
Continuous Coefficients
Martin Hutzenthaler
Arnulf Jentzen
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Hutzenthaler,Martin,1978-
Numericalapproximationsofstochasticdifferentialequationswithnon-globallyLipschitzcon-
tinuouscoefficients/MartinHutzenthaler,ArnulfJentzen.
pages cm. – (Memoirs of the AmericanMathematicalSociety, ISSN 0065-9266; volume 236,
number1112)
Includesbibliographicalreferences.
ISBN978-1-4704-0984-5(alk. paper)
1.Stochasticdifferentialequations. 2.Differentialoperators. I.Jentzen,Arnulf. II.Title.
QA274.23.H88 2015
519.2–dc23 2015007761
DOI:http://dx.doi.org/10.1090/memo/1112
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Contents
Chapter 1. Introduction 1
1.1. Notation 5
Chapter 2. Integrability properties of approximation processes for SDEs 9
2.1. General discrete-time stochastic processes 9
2.2. Explicit approximation schemes 20
2.3. Implicit approximation schemes 34
Chapter 3. Convergence properties of approximation processes for SDEs 45
3.1. Setting and assumptions 45
3.2. Consistency 45
3.3. Convergence in probability 46
3.4. Strong convergence 54
3.5. Weak convergence 59
3.6. Numerical schemes for SDEs 65
Chapter 4. Examples of SDEs 79
4.1. Setting and assumptions 79
4.2. Stochastic van der Pol oscillator 80
4.3. Stochastic Duffing-van der Pol oscillator 81
4.4. Stochastic Lorenz equation 82
4.5. Stochastic Brusselator in the well-stirred case 82
4.6. Stochastic SIR model 84
4.7. Experimental psychology model 85
4.8. Scalar stochastic Ginzburg-Landau equation 86
4.9. Stochastic Lotka-Volterra equations 86
4.10. Volatility processes 89
4.11. Overdamped Langevin equation 92
Bibliography 95
iii

Abstract
Many stochastic differential equations (SDEs) in the literature have a super-
linearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately,
moments of the computationally efficient Euler-Maruyama approximation method
divergefortheseSDEsinfinitetime. Thisarticledevelopsageneraltheorybasedon
rareeventsforstudyingintegrabilitypropertiessuchasmomentboundsfordiscrete-
time stochastic processes. Using this approach, we establish moment bounds for
fully and partially drift-implicit Euler methods and for a class of new explicit ap-
proximation methods which require only a few more arithmetical operations than
theEuler-Maruyamamethod. Thesemomentboundsarethenusedtoprovestrong
convergence of the proposed schemes. Finally, we illustrate our results for several
SDEs from finance, physics, biology and chemistry.
ReceivedbytheeditorApril26,2013and,inrevisedform,April13,2013.
ArticleelectronicallypublishedonNovember14,2014.
DOI:http://dx.doi.org/10.1090/memo/1112
2010 MathematicsSubjectClassification. Primary60H35;Secondary65C05,65C30.
Keywordsandphrases. Stochasticdifferentialequation,rareevent,strongconvergence,nu-
mericalapproximation,localLipschitzcondition,Lyapunovcondition.
This work has been partially supported by the research project “Numerical solutions of
stochastic differential equations with non-globally Lipschitz continuous coefficients” and by the
researchproject “Numericalapproximationofstochasticdifferential equations withnon-globally
Lipschitzcontinuouscoefficients”bothfundedbytheGermanResearchFoundation.
(cid:2)c2014 American Mathematical Society
v

CHAPTER 1
Introduction
This article investigates integrability and convergence properties of numerical
approximation processes for stochastic differential equations (SDEs). In order to
illustrate one of our main results, the following general setting is considered in this
introductory chapter. Let T ∈ (0,∞), d,m ∈ N := {1,2,...}, let (Ω,F,P) be
a probability space with a normal filtration (Ft)t∈[0,T], let W: [0,T]×Ω → Rm
be a standard (Ft)t∈[0,T]-Brownian motion, let D ⊂ Rd be an open set, let μ =
(μ1,...,μd): D → Rd and σ = (σi,j)i∈{1,2,...,d},j∈{1,2,...,m}: D → Rd×m be locally
Lipschitzcontinuous functionsandletX: [0,T]×Ω→D bean(Ft)t∈[0,T]-adapted
stochastic process with continuous sample paths satisfying the SDE
(cid:2) (cid:2)
t t
(1.1) X =X + μ(X )ds+ σ(X )dW
t 0 s s s
0 0
P-almost surely for all t ∈ [0,T]. Here μ is the infinitesimal mean and σ ·σ∗ is
the infinitesimal covariance matrix of the solution process X of the SDE (1.1). To
guarantee finiteness of some moments of the SDE (1.1), we assume existence of a
Lyapunov-typefunction. More precisely, letq ∈(0,∞),κ∈Rberealnumbers and
let V : D →[1,∞) be a twice continuously differentiable functionwith E[V(X )]<
0
∞ and with V(x)≥(cid:8)x(cid:8)q and
(cid:4) (cid:5) (cid:4) (cid:5)
(cid:3)d ∂V 1 (cid:3)d (cid:3)m ∂2V
(1.2) (x)·μ (x)+ (x)·σ (x)·σ (x)≤κ·V(x)
∂x i 2 ∂x ∂x i,k j,k
i i j
i=1 i,j=1k=1
for all x∈D. These assumptions ensure
(cid:6) (cid:7) (cid:6) (cid:7)
(1.3) E V(X ) ≤eκt·E V(X )
t 0
for all t ∈ [0,T] and, therefore, finiteness of the q-th absolute m(cid:6)oments(cid:7) of the
solution process X , t ∈ [0,T], of the SDE (1.1), i.e., sup E (cid:8)X (cid:8)q < ∞.
t t∈[0,T] t
Note that in this setting both the drift coefficient μ and the diffusion coefficient σ
of the SDE (1.1) may grow superlinearly and are, in particular, not assumed to be
globallyLipschitzcontinuous. Ourmaingoalinthisintroductionistoconstructand
to analyze numerical approximation processes that converge strongly to the exact
solution of the SDE (1.1). The standard literature in computational stochastics
(see, for instance, Kloeden & Platen [47] and Milstein [61]) concentrates on SDEs
with globally Lipschitz continuous coefficients and can therefore not be applied
here. Strong numerical approximations of the SDE (1.1) are of particular interest
forthecomputationofstatisticalquantitiesofthesolutionprocessoftheSDE(1.1)
through computationally efficient multilevel Monte Carlo methods (see Giles [20],
Heinrich [29] and, e.g., in Creutzig et al. [13], Hickernell et al. [32], Barth, Lang
& Schwab [5] and in the references therein for further recent results on multilevel
Monte Carlo methods).
1
2 MARTINHUTZENTHALERandARNULFJENTZEN
SeveralSDEsfromtheliteraturesatisfyth(cid:8)eaboves(cid:9)etting(seeSections4.2–4.11
below). For instance, the function V(x) = 1+(cid:8)x(cid:8)2 r, x ∈ D, for an arbitrary
r ∈(0,∞)servesasaLyapunov-typefunctionforthestochasticvanderPoloscilla-
tor (4.4), for the stochastic Lorenz equation (4.20), for the Cox-Ingersoll-Ross pro-
cess (4.74) and for the simplified Ait-Sahalia interest rate model (4.75) but not for
thestochastic Duffing-vander Pol oscillator (4.13), not for the stochastic Brussela-
tor(4.23),notforthestochasticSIRmodel(4.31),notfortheLotka-Volterrapreda-
tor prey model (4.60) and, in genera(cid:8)l, also not for the o(cid:9)verdamped Langevin equa-
tion (4.89). The function V(x) = 1+(x )4+2(x )2 r, x = (x ,x ) ∈ D = R2,
1 2 1 2
for an arbitrary r ∈(0,∞) is a Lyapunov-type function for the stochastic Duffing-
van der(cid:8)Pol oscillator (4.13). (cid:9)For the stochastic SIR model (4.31), the function
V(x)= 1+(x +x )2+(x )2 r, x=(x ,x ,x )∈D =(0,∞)3, for an arbitrary
1 2 3 1 2 3
r ∈(0,∞)servesasaLyapunov-type(cid:8)functionandforthest(cid:9)ochasticLotka-Volterra
system (4.48), the function V(x)= 1+(v x +...v x )2 r, x∈D =(0,∞)d, for
1 1 d d
an arbitrary r ∈ (0,∞) and an appropriate v = (v ,...,v ) ∈ Rd is a Lyapunov-
1 d
type function. More details on the examples can be found in Chapter 4.
The standard method for approximating SDEs with globally Lipschitz con-
tinuous coefficients is the Euler-Maruyama method. Unfortunately, the Euler-
Maruyamamethodoftenfailstoconvergestronglytotheexactsolutionofnonlinear
SDEs of the form (1.1); see [41]. Indeed, if at least one of the coefficients of the
SDE grows superlinearly, then the Euler-Maruyama scheme diverges in the strong
sense. Moreprecisely,letZN: {0,1,...,N}×Ω→Rd,N ∈N,beEuler-Maruyama
approximations for the SDE (1.1) defined recursively through ZN :=X and
0 0
(cid:8) (cid:9)
(1.4) ZN :=ZN +μ¯(ZN)T +σ¯(ZN) W −W
n+1 n n N n (n+1)T nT
N N
for all n∈{0,1,...,N −1} and all N ∈N. Here μ¯: Rd →Rd and σ¯: Rd →Rd×m
are extensions of μ and σ given by μ¯(x) = 0, σ¯(x) = 0 for all x ∈ Dc and by
μ¯(x) = μ(x), σ¯(x) = σ(x) for all x ∈ D respectively (see also Sections 3.2 and 3.6
formoregeneralextensions). Theorem2.1of[40](whichgeneralizesTheorem2.1of
[41])thenimpliesinthecased=m=1thatifthereexistsarealnumberε∈(0,∞)
such that |(cid:6)μ¯(x)|+(cid:7)|σ¯(x)|≥ε|x|(1+ε) for all |x|≥1/ε and if P[σ(X0(cid:6))(cid:10)=0]>0, th(cid:7)en
limN→∞E |YNN|r =∞ for all r ∈(0,∞) and therefore limN→∞E |XT −YNN|r =
∞ for all r ∈ (0,q] (see also Sections 4 and 5 in [40] for divergence results for
thecorrespondingmultilevelMonteCarloEulermethod). Duetothesedeficiencies
of the Euler-Maruyama method, we look for numerical approximation methods
whosecomputationalcostisclosetothatoftheEuler-Maruyamamethodandwhich
converge strongly even in the case of SDEs with superlinearly growing coefficients.
There are a number of strong convergence results for temporal numerical ap-
proximations of SDEs of the form (1.1) with possibly superlinearly growing coeffi-
cientsintheliterature. Manyoftheseresultsassumebesideotherassumptionsthat
the drift coefficient μ of the SDE (1.1) is globally one-sided Lipschitz continuous
andalsoproveratesofconvergenceinthatcase. Inparticular,ifthedriftcoefficient
is globally one-sided Lipschitz continuous and if the diffusion coefficient is globally
Lipschitzcontinuousbesideotherassumptions, thenstrongconvergenceofthefully
drift-implicitEulermethodfollowsfromTheorem2.4inHu[37]andfromTheorem
5.3 in Higham, Mao & Stuart [35], strong convergence of the split-step backward
Euler method follows from Theorem 3.3 in Higham, Mao & Stuart [35], strong
convergenceofadrift-tamedEuler-MaruyamamethodfollowsfromTheorem1.1in
1. INTRODUCTION 3
[39]andstrongconvergenceofadrift-tamedMilsteinschemefollowsfromTheorem
3.2inGan&Wang[19]. Theorem2andTheorem3inHigham&Kloeden[34]gen-
eralize Theorem 3.3 and Theorem 5.3 in Higham, Mao & Stuart [35] to SDEs with
Poisson-driven jumps. In addition, Theorem 6.2 in Szpruch et al. [77] establishes
strong convergence of the fully drift-implicit Euler method of a one-dimensional
Ait-Sahalia-type interest rate model having a superlinearly growing diffusion coef-
ficient σ and a globally one-sided Lipschitz continuous drift coefficient μ which is
unbounded near 0. Moreover, Theorem 4.4 in Mao & Szpruch [56] generalizes this
result to a class of SDEs which have globally one-sided Lipschitz continuous drift
coefficients and in which the function V(x) = 1+(cid:8)x(cid:8)2, x ∈ D, is a Lyapunov-
type function (see also Mao & Szpruch [57] for related results but with rates of
convergence). A similar method is used in Proposition 3.3 in Dereich, Neuenkirch
& Szpruch [17] to obtain strong convergence of a drift-implicit Euler method for
a class of Bessel type processes. Moreover, Gy¨ongy & Millet establish in Theo-
rem 2.10 in [26] strong convergence of implicit numerical approximation processes
foraclassofpossiblyinfinitedimensionalSDEswhosedriftμanddiffusionσsatisfy
a suitable one-sided Lipschtz condition (see Assumption (C1) in [26] for details).
Strongconvergenceoftemporalnumerical approximationsfortwo-dimensional sto-
chastic Navier-Stokes equations is obtained in Theorem 7.1 in Brze´zniak, Carelli &
Prohl [10]. In all of the above mentioned results from the literature, the function
V(x) = 1+(cid:8)x(cid:8)2, x ∈ D, is a Lyapunov-type function of the considered SDE. A
result on more general Lyapunov-type functions is the framework in Schurz [76]
which assumes general abstract conditions on the numerical approximations. The
applicability of this framework is demonstrated in the case of SDEs which have
globally one-sided Lipschitz continuous drift coefficients and in which the func-
tion V(x) = 1+(cid:8)x(cid:8)2, x ∈ D, is a Lyapunov-type function; see [74–76]. To the
best of our knowledge, no strong numerical approximation results are known for
the stochastic van der Pol oscillator (4.4), for the stochastic Duffing-van der Pol
oscillator (4.13), for the stochastic Lorenz equation (4.20), for the stochastic Brus-
selator(4.23), forthestochastic SIRmodel(4.31), fortheexperimentalpsychology
model (4.40) and for the Lotka-Volterra predator-prey model (4.48).
In this article, the following increment-tamed Euler-Maruyama scheme is pro-
posedtoapproximatethesolutionprocessoftheSDE(1.1)inthestrongsense. Let
YN: {0,1...,N}×Ω→Rd, N ∈N, benumericalapproximationprocessesdefined
through YN :=X and
0 0
(cid:8) (cid:9) (cid:8) (cid:9)(cid:8) (cid:9)
μ¯ YN T +σ¯ YN W −W
(1.5) YN :=YN + (cid:8) n N n ((cid:8)n+N1)T nNT (cid:9) (cid:9)
n+1 n max 1, T(cid:8)μ¯(YN)T +σ¯(YN) W −W (cid:8)
N n N n (n+1)T nT
N N
for all n ∈ {0,1,...,N −1} and all N ∈ N. Note that the computation of (1.5)
requires only a few additional arithmetical operations when compared to the com-
putation of the Euler-Maruyama approximations (1.4). Moreover, we emphasize
that the scheme (1.5) is a special case of a more general class of suitable tamed
schemesproposedinSubsection3.6.3below. NextletY¯N: [0,T]×Ω→Rd,N ∈N,
(cid:8)be linearly(cid:9)interp(cid:8)olated c(cid:9)ontinuous-time versions of (1.5) defined through Y¯tN :=
n+1−tN YN+ tN−n YN forallt∈[nT/N,(n+1)T/N],n∈{0,1,...,N−1}
T n T n+1
and all N ∈ N. For proving strong convergence of the numerical approximation
processes Y¯N, N ∈ N, to(cid:6)the ex(cid:7)act solution process X of the SDE (1.1), we ad-
ditionally assume that E (cid:8)X (cid:8)r < ∞ for all r ∈ [0,∞) and that there exist
0