The weak rate of convergence for the Euler-Maruyama approximation of one-dimensional stochastic differential equations involving the local times of the unknown process

0.15 MB·

by  Mohsine Benabdallah

The weak rate of convergence for the
Euler-Maruyama approximation of
7
one-dimensional stochastic differential equations
1
0
involving the local times of the unknown process
2

n
a
J M.Benabdallah∗ and K.Hiderah†

6
1 Department of Mathematics, faculty of science, University of Ibn Tofail, Kenitra,

Morocco
]
A
N
.
h
t
a
m Abstract
[ In this paper, we consider the weak convergence of the Euler-Maruyama approximation

for one dimensional stochastic differential equations involving the local times of the unknown
2
v process. We use a transformation in order to remove the local time Lat from the stochastic
1 differential equations of type:
5
t
5 Xt =X0+ ϕ(Xs)dBs+ ν(da)Lat
0 Z0 ZR
0
. whereB isaone-dimensionalBrownian motion,ϕ:R→Risaboundedmeasurablefunction,
1 and ν is a bounded measure on R and we provide the approximation of Euler-maruyama for
0
the stochastic differential equations without local time. After that, we conclude the approxi-
:17 wmeaatkioncoonfveErugelenrc-emEarruroyram=aEX|Gtn(XofTt)h−e aGb(oXveTn)m|,efnotrioanneydfueqnuctaitoinonG, ainndawceerptarionvicdlaestsh.e rate of
v Keywords: Euler-Maruyama approximation, weak convergence, stochastic differential equa-
i tion, local time, bounded variation.
X
MSC 2010: 60H35, 41A25, 60H10, 60J55, 65C30
r
a
1 Introduction
Let X = {X : t ≥ 0} be a process stochastic involving the local time defined by the stochastic
t
differential equations:
t
X =X + ϕ(X )dB + ν(da)La (1)
t 0 s s t
Z0 ZR
where B is a one-dimensionalBrownianmotion, ϕ:R→R is a boundedmeasurable function, and
ν is a bounded measure on R,La denotes the local time at a for the time t of the semimartingale
t
X.
∗M.Benabdallah: e-mail: [email protected]
†K.Hiderah: e-mail: [email protected]
1
In[14],ifν isabsolutelycontinuouswithrespecttotheLebesguemeasureonR(i.e.ν(da)=g(a)da)
then (1) becomes the usual Itoˆ equation :
t t
X =X + ϕ(X )dB + (gϕ2)ds (2)
t 0 s s
Z0 Z0
In general, we consider the following stochastic differential equation(SDE) with coefficients b and
σ, driven by a Brownian motion B in R:
t t
X =X + σ(X )dB + b(X )ds (3)
t 0 s s s
Z0 Z0
where the drift coefficient b and the diffusion coefficient σ are Borel-measurable functions from R
into R and X is an R-valued random variable, which is independent of B and b.
0
The continuousEulerscheme {Xn,0≤t≤T}forthe SDE (3)onthe time interval[0,T]is defined
t
as follows: Xn =X , and
0 0
t t
Xn =Xn+ σ(Xn )dB + b(Xn )ds, (4)
t 0 ηk(s) s η(k)
Z0 Z0
For η ≤t ≤ η ,k =0,1,2,...,n−1, where 0= η ≤η ≤... ≤ η = T is a sequence of random
k k+1 0 1 n
partitions of [0,T]. The weak convergence of the stochastic differential equations has been stud-
ied by Avikainen [1], Bally and Talay[2], Mikuleviˇcius and Platen[16], D.Talay and L.Tubaro[21],
and the weak convergence of SDEs with discontinuous coefficients has been studied by Chan and
Stramer [8], Yan [23], Arturo Kohatsu-Higa [12].
We use the same criteria that mentioned in [11], that the definition of weak convergence with
order γ > 0 is that for all functions f in a certain class, there exists a positive constant C, which
does not depend on ∆t, such that:
|E[f(X )]−E[f(Xn)]|≤C(∆t)γ (5)
T T
The stochastic differential equations of the type (1) have been studied by Stroock and Yor [20],
Portenko [19], Le Gall [13,14], Blei and Engelbert [6], Bass and Chen[4] and the approximation
of Euler-Maruyama for SDE of type (1) has been studied by,e.g., Benabdallah, Elkettani and
Hiderah[5].
In [14], when ν =βδ (δ(0) denotes the Dirac measure at 0) and ϕ=1, we get
(0)
X =X +B +βL0(X),|β|≤1 (6)
t 0 t t
The solution of equation (6) is the well-knownprocess called the skew Brownianmotion which has
been studied by Harrison and Shepp [10], Ouknine [17,18], Bouhadou and Ouknine[7], Lejay[15],
Barlow [3], E´tor´e and Martinez [9], Walsh[22].
Our goal of this paper is that under the assumption that the SDE (1) has a weak solution and
thatitisunique,westudytheconditionsunderwhichtheEulerscheme{Xn :0≤t≤T}converges
t
weakly to the exact solution {X :0≤t≤T} of the SDE (1).
t
We face two major problems: the presence of local time in equation (1) and the inability
to provide a simple discretization scheme. and the presence of a discontinuity for the diffusion
coefficient ϕ.
Ourpaperis dividedasfollows: Insection(2), wepresentthe importantpropositions,assumptions
andstochasticdifferentialequationsinvolvinglocaltime. Ourmainresultsontheweakconvergence
ofEulerschemeofone-dimensionalstochasticdifferentialequationsinvolvingthelocaltimearegiven
in section (3). Some numerical examples are given in section (4). All proofs of the theorems are
given in section (5).
2
2 Preliminaries and approximation
In this section, we provide the definitions, important propositions, assumptions and stochastic
differential equations involving local time.
Definition 2.1. Let
N
T (x):=sup |f(x )−f(x )|,
f j j−1
j=1
X
where the supremum is taken over N and all partitions −∞<x <x <...<x =x be the total
0 1 N
variationfunctionoff. Thenwesaythatf isafunctionofboundedvariation,ifV(f):= lim T (x)
f
x→∞
is finite, and call V(f) the (total) variation of f.
Definition 2.2. We say that a function f has at most polynomial growth in R is there exist an
integer k and a constant C >0 such that |f(x)|≤C(1+|x|k) for any x∈R
Remark 2.3. Let Ck(R) denote the space of all Ck-functions of polynomial growth (together with
p
their derivatives).
Note 2.4. BV(R) will denote the space of all functions ϕ:R→R of bounded variation on R such
that:
1. ϕ is right continuous.
2. There exists an ε>0 such that : ϕ(x)≥ε for all x.
If ϕ is in BV(R), ϕ(x−) will denote the left-limit of ϕ at point x and ϕ′(dx) will be the bounded
measure associated with ϕ.
Note 2.5. M(R) will denote the space of all bounded measures ν on R such that:
|ν({x})|<1,∀x∈R
We have the stochastic differential equation (1), if ϕ is in BV(R) and ν is in M(R), then the
stochastic differential equation (1) has a unique strong solution as soon as |ν(da)|<1.
Theorem 2.6. Let ϕ be in BV(R) and ν be in M(R). Then existence and pathwise uniqueness of
solution hold for (1).
Proposition 2.7. Let ν be in M(R). There exists a function f in BV(R), unique up to a multi-
plicative constant, such that:
f′(dx)+(f(x)+f(x−))ν(dx)=0 (7)
If we require that f(x)−−−−−→1, then f is unique and is given by:
x→−∞
1−ν({y})
f(x)=f (x)=exp(−2νc(]−∞;x])) (8)
ν
1+ν({y})
y≤x(cid:18) (cid:19)
Y
Where νc denotes the continuous part of ν.
Proposition 2.8. Let ϕ be in BV(R) and ν be in M(R) and f be define by(8) and set:
ν
x
F (x)= f (y)dy (9)
ν ν
Z0
Then X is a solution of equation (1), if and only if Y :=F (X) is a solution of:
ν
dY =(ϕf )◦F−1(Y )dB (10)
t ν t t
3
The Euler scheme Yn :0≤t≤T for the SDE (10) on the time interval [0,T] is defined as
t
follows: Yn =Y , and
0 0
Yn =Yn +(ϕf )◦F−1(Yn)(B −B ) (11)
t ηk ν ηk t ηk
for η < t ≤ η ,k = 0,1,2,...,n where 0 = η ≤ η ≤ ... ≤ η = T. This Euler scheme can be
k k+1 0 1 n
written as
t
Yn =Yn+ (ϕf )◦F−1(Yn)dB (12)
t 0 ν ηk s
Z0
In[4],theSDE(1)hasexistenceofstrongsolutionsandpathwiseuniquenesswhenϕisafunctionon
Rthatisboundedaboveandbelowbypositiveconstantsandsuchthatthereisastrictlyincreasing
function f on R such that
|ϕ(x)−ϕ(y)|2 ≤|f(x)−f(y)|,x,y ∈R
and ν is a finite measure with |ν({a})|≤1 for every a∈R.
Theorem2.9. SupposeϕisameasurablefunctiononRthatisboundedaboveandbelowbypositive
constants and suppose that there is a strictly increasing function f on R such that
|ϕ(x)−ϕ(y)|2 ≤|f(x)−f(y)|,x,y ∈R (13)
For any finite signed measure ν on R such that ν({x}) < 1 for each x ∈ R and everyx ∈ R, the
2 0
SDE
t
X =X + ϕ(X )dB + ν(da)La,t≥0 (14)
t 0 s s t
Z0 ZR
has a continuous strong solution and the continuous solution is pathwise unique.
Define
Let
−log(1−2x), if x∈(−∞,0)∪(0,1)
π(x):= 2x 2
(1, if x=0.
Let µ(dx):=π(ν({x})ν(dx) which is a finite signed measure. Define
x
S(x):= e−2µ(−∞,y]dy
Z0
′
Since µ is a finite measure, S is right continuous and strictly positive. Hence S is increasing and
one-to-one. Let S−1 denote the inverse of S, let S′ denote the left continuous version of S′, i.e.,
ℓ
the left hand derivative of s, and let σ′ denote the left hand derivative of S−1. Since µ is a finite
ℓ
′
signed measure, S is of bounded variation. Let {Y ,t≤0} solve
t
dY =(S′ϕ)◦S−1(Y )dB ,Y =S(X ) (15)
t ℓ t t 0 0
Let X =S−1(Y), and Y define by (15), in [4], they must show that X is a solution to (14).
t
3 The main results
In this section, we provide the main theorems. we study two cases: the continuous of function
(ϕ.f )◦F−1(.) and the discontinuous of function (ϕ.f )◦F−1(.).
ν ν
4
3.1 Main Theorems
Inthissection,wepresentthe followingresultsontheratesoftheEuler-Maruyamaapproximation.
Let X be defined as in equation (1), Xn be the Euler scheme for equation (1), let F is defined by
t t
equation (9), and let f be from the proposition(2.7). Here, we suppose that the assumptions of
ν
proposition(2.8) satisfied, and the constant C may change from line to line and from theorem to
theorem.
Theorem 3.1. For any function g : R → R, if for each G := g◦F−1 ∈ C2(γ+1)(R), and ψ(.) :=
P
(ϕ.f )◦F−1(.) is continuous, then, there exists a constant C > 0, which does not depend on n,
ν
such that
C
E|g(Xn)−g(X )|≤
T T nγ
holds.
Next we give a sufficientcondition under which the Euler scheme convergesweakly to the weak
solution of SDE (2) case: discontinuous of function (ϕ.f )◦F−1(.). Let ψ (z):=liminfψ2(y)>0
ν 1
y→z
Theorem 3.2. Suppose that ψ := (ϕ.f)◦F−1(.) has at most linear growth with D of Lebesgue
ψ
measure zero and E(Y )4 <∞. If ψ (z)>0 for z ∈D ,and G:=g◦F−1 ∈C2(γ+1)(R), then the
0 1 ψ P
Euler scheme of SDE(1) converges weakly to the unique weak solution of SDE(1), where D is the
ψ
set of discontinuous points of ψ(y).
If we suppose that the conditions of theorem (2.9) satisfied, we show the following note.
Note 3.3. The same results in the theorems (3.1,3.2) stay valid, if the conditions of theorem (2.9)
are hold.
4 Some examples
Example 4.1. In equation (1), let ν(dx)=αδ (dx),|α|<1 where (δ is the Dirac measure at 0 )
0 0
and
1+αexp −1−αx if x≥0
ϕ= 1−α 1+α
(exp(x) (cid:16) (cid:17) if x<0
Using proposition (2.7), we have
1−α if x≥0
f (x)= 1+α
ν
(1 if x<0
Using proposition (2.8), we have
1−αx if x≥0
F(x)= 1+α
(x if x<0
Then
exp(−y) if y ≥0
ψ(y):=(ϕ.f)◦F−1(y)=
(exp(y) if y <0
We note ψ is continuous, then for any function g : R → R, such that g◦F−1 ∈ C2(γ+1)(R), for
p
any γ >0, for example
1 1
g(x)= .1 + 1
2 x≥0 1+x2 x<0
1+ 1−αx
1+α
there exist a constant C >0, which does(cid:16)not de(cid:17)pend on n, such that
5
|E[g(Xn)]−E[g(X )]|≤ C
T T nγ
Example 4.2. In equation (1), let ν(x)=αδ (x),|α|<1, where δ is the Dirac measure at 0, and
0 0
E(X )4 =E(Y )4 <∞ we define ϕ=1, such that
0 0
Using proposition (2.7), we have
1−α if x≥0
f(x)=f (x)= 1+α
ν
(1 if x<0
And by using proposition (2.8), we have
1−αx if x≥0 1+αy if y ≥0
F (x)= 1+α F−1(y)= 1−α
ν (x if x<0 ν (y if y <0
Then, we obtain
1−α if y ≥0
ψ(y)=(ϕ.f )◦F−1(y)= 1+α (16)
ν
(1 if y <0
And
t
Y =Y + ψ(Y )dB (17)
t 0 s s
Z0
The Euler scheme {Yn :0≤t≤T} for the SDE (17), on the time interval [0,T] is defined as
t
follows: Yn =Y , and
0 0
Yn =Yn +ψ(Yn)(B −B ) (18)
t ηk ηk t ηk
For η < t ≤ η ,k = 0,1,2,...,n, where 0 = η ≤ η ≤ ... ≤ η = T, the coefficient of equation
k k+1 0 1 n
(17) is discontinuous at point 0, and, we have:
• D ={0}
ψ
• ψ (z):=liminfψ2(y)>0
1
z→y
• ψ(·) is at most linear growth with D
ψ
We have the Euler scheme of SDE(18) converges weakly to the unique weak solution of SDE((17)),
then the Euler scheme Xn converges weakly to the unique weak solutionX of SDE((1)).
t t
5 Proofs of theorems
Before proving the theorems below, we introduce some notations.
Note 5.1. The constant C may change from line to line and from theorem to theorem.
Let B be a one-dimensional brownian motion (B ) , and ϕ is in BV(R) and ν is in M(R)
t t∈[0,T]
and
t
X =X + ϕ(X )dB + ν(da)La. (19)
t 0 s s t
Z0 ZR
and Y is solution of the equation
t
dY =(ϕ.f )◦F−1(Y )dB . (20)
t ν t t
for which, as we get by the proposition(2.8), where F be defined by the equation (9), f get by
ν
the proposition(2.7), and let Yn is a solution by Euler scheme for equation(10) and defined by
t
equation(12).
6
5.1 Proof of theorem (3.1)
Definition 5.2. In [11], a time discrete approximation Xn converges weakly with order γ > 0 to
X at time T as n→∞, if for each G∈C2(γ+1)(R), there exists a constant C >0, which does not
P
depend on n, such that
C
|E(G(Xn))−E(G(X ))|≤ holds.
T T nγ
Using the above definition, we can prove theorem (3.1).
1. The proposition (2.8) is satisfied , then X =F−1(Y ) is solution uniqueness of the SDE (1).
t t
2. G:=g◦F−1 ∈C2(γ+1)
p
3. By using definition (5.2), we have
|E[g(X )]−E[g(Xn)]|= E[g◦F−1(Y )]−E[g◦F−1(Yn)]
T T T T
C
=(cid:12)|E[G(Y )]−E[G(Yn)]|≤ (cid:12)
(cid:12) T T nγ (cid:12)
5.2 Proof of theorem (3.2)
If we have SDE of type
t
Y =Y + σ(Y )dB (21)
t 0 s s
Z0
and the Euler scheme of equation (21) give by
t
Y =Y + σ(Y )dB (22)
t 0 ηn(s) s
Z0
Here, we present the following theorem(see [23]):
Theorem 5.3. Suppose that σ(y) has at most linear growth with D of Lebesgue measure zero and
σ
E(Y )4 <∞. If σ (z):=liminfσ2(y)>0 for z ∈D , then the Euler scheme of SDE(21) converges
0 1 ψ
z→y
weakly to the unique weak solution of SDE(21), where D is the set of discontinuous points of σ(y).
σ
Using the above theorem, we can prove theorem (3.2).
1. The proposition (2.8) is satisfied , then X =F−1(Y ) is solution uniqueness of the SDE (1).
t t
2. G:=g◦F−1 ∈C2(γ+1)
p
3. (ϕ.f )◦F−1(·) has at most linear growth with D of Lebesgue measure zero.
ν σ
4. E(Y )4 <∞
0
5. By using definition (5.2) and theorem (5.3) , we have
|E[g(X )]−E[g(Xn)]|= E[g◦F−1(Y )]−E[g◦F−1(Yn)]
T T T T
C
=(cid:12)|E[G(Y )]−E[G(Yn)]|≤ (cid:12)
(cid:12) T T nγ (cid:12)
7
References
[1] R.Avikainen,OnirregularfunctionalsofSDEsandtheEulerscheme, FinanceStoch.13(2009)
381-401.
[2] V.Bally and D.Talay, The law of the Euler scheme for stochastic differential equations (I):
convergence rate of the distribution function,Probab, Theory Related Fields 104 (1996) 43-60.
[3] M.T.Barlow, Skew Brownian motion and a one-dimensional stochastic differential equation.
Stochastics 25(1988), no. 1, 162.
[4] RichardF.BassandZhen-QingChen One-DimensionalStochasticDifferentialEquations with
Singular and Degenerate Coefficients,Sankhya¯: The Indian Journal of Statistics, Volume 67,
Part 1, pp 19-45(2005).
[5] M. Benabdallah, Y. Elkettani and K.Hiderah, Approximation of Euler-Maruyama for one-
dimensional stochastic differential equations involving the local times of the unknown process,
Monte Carlo Methods and Applications, Volume 22, Issue 4, Pages 307-322,(2016).
[6] S.Bleiand H.J.Engelbert, One-dimensional Stochastic Differential Equations with Generalized
and Singular Drift, September 28, 2012,http://arxiv.org/abs/1209.6159v1
[7] S.Bouhadou, Y.Ouknine, On the time inhomogeneous skew Brownian motion, Bulletin des
Sciences Math´ematiques, Pages 835-850, (2013).
[8] K.S.Chan and O.Stramer, Weak Consistency of the Euler Method for Numerically Solving
StochasticDifferentialEquationswithDiscontinuousCoefficients,Stoch.Proc.Appl.76(1998),
33-44.
[9] P.E´tor´e and M.Martinez, on the existence of a time inhomogeneous skew Brownian motion
and some related laws, Electron. J.Probab. 17, no. 19, 1-27, (2012).
[10] J.M.Harrison and L.A.Shepp, On skew brownian motion. Annals of probability 9 (1981) p.
309-313.
[11] P.E.Kloeden and E.Platen, Numerical Solution of Stochastic Differential Equations, Applica-
tions of Mathematics (New York) 23. Springer, Berlin.(1992)
[12] A.Kohatsu-Higa and all, On Weak Approximation of Stochastic Differential Equations with
Discontinuous Drift Coeffcient,hal-00670123, version 1-15 Feb 2012
[13] J.F.Le Gall, Applications du temps local aux ´equations diff´erentielles stochastiques uni- di-
mensionnelles. In Seminar on probability, XVII, volume 986 of Lecture Notes in Math., pages
15-31.Springer, Berlin, 1983.
[14] J.F.Le Gall, One-dimensional stochastic differential equations involving the local times of the
unknown process. In Stochastic analysis and applications (Swansea, 1983), volume 1095 of
Lecture Notes in Math., pages 51-82. Springer, Berlin, 1984.
[15] A.Lejay, Onthe constructionsof theskew Brownian motion,Probability Surveys ,Vol. 3(2006)
413-466.
[16] R.Mikuleviˇcius and F.Platen, Rate of convergence of the Euler approximation for diffusion
processes, Math. Nachr. 151 (1991), 233-239, DOI 10.1002/mana.19911510114.
8
[17] Y.Ouknine, ”Skew-Brownian motion” and derived processes, Theory Probab. Appl. 35, 163-
169, (1990).
[18] Y.Ouknine and all , On countably skewed Brownian motion with accumulation point.
https://hal.inria.fr/hal-00850095 (2013)
[19] N.I.Portenko, Generalized Diffusion Processes, Naukova Dumka, Kiev, 1982. English transl.:
American Mathematical Society, Providence, R.I., 1990.
[20] D.W.StroockandM.Yor, Some remarkable martingales. InSeminaronProbability,XV(Univ.
Strasbourg, Strasbourg, 1979/1980) (French), volume 850 of Lecture Notes in Math.,pages
590-603.Springer, Berlin, 1981.
[21] D.TalayandL.Tubaro, Expansion of the global error for numerical schemes solving stochastic
differential equations, Stochastic Anal. Appl. 8, No. 4, 483-509, 1990.
[22] J.W.Walsh, A diffusion with discontinuous local time, Ast´erisque 52-53 (1978) p. 37-45.
[23] B.L.Yan, The Euler scheme with irregular coefficients, Ann. Probab. 30 (2002) 1172-1194.
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