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Approximating and Simulating Multivalued Stochastic Differential Equations
Dominique Lepingle, Thi Thao Nguyen
To cite this version:
Dominique Lepingle, Thi Thao Nguyen. Approximating and Simulating Multivalued Stochastic Dif- ferential Equations. Monte Carlo Methods and Applications, De Gruyter, 2004, 10, pp.129-152.
�hal-00003500�
Dierential Equations
Dominique LEPINGLE Thi Thao NGUYEN
MAPMO, Universitéd'Orléans,F-45067OrléansCedex2
email: lepinglelabomath.univ-orleans.fr
email: tnguyenlabomath.univ-orleans.fr
Abstrat
We propose a two-step simulation sheme for the solution of a singular stohasti
dierentialequationwithexplodingdrift. FirstweestimatethestrongorderoftheYosida
approximation. Thenweuseasemi-impliitEuler shemeto disretizetheapproximate
solution. Numerial experimentsare displayed for thepaths of Brownian partiles with
strong repulsive interation. We also present twosimple simulation shemes for Bessel
proesseswitharbitrarydimension.
Keywords: Multivaluedstohasti dierentialequations, Yosida approximation,interatingBrownian
partiles,semi-impliitsheme.
1 Introdution
Multivalued stohastidierential equations areusedto modelalarge lassofrandom evolu-
tionssuhasstohastidierentialequationswithreetionontheboundaryofadomain[22 ℄,
diusion equationswithhysteresis[16 ℄,or systemsofBrownianpartileswithrepulsive inter-
ation [10 ℄. Existene and approximationof suh equations, alsoalled stohastivariational
inequalities, have reeived muh attention in thereent years. Existene has been proved in
several papers ([3 ℄, [8℄, [9℄, [21 ℄). Numerial approximation has been takled in [1 ℄, [4℄ and
[19 ℄.
The last two authors have proposed the same projetion sheme whih is natural and
eient inthe aseofpurereetion([20 ℄, [18℄). In[19 ℄,theonvexfuntionisassumedtobe
ontinuous on the losureof the domain and in[4℄ its gradient satises a polynomial growth
ondition. In both papers, the diult ase of exploding drift is avoided. In the work [1℄,
these restritionsarereleased andtheauthorsuseasplittingupmethod. Theirshemeisnot
easy to implement beause at eahstep theyhave tosolvea multivaluedordinary dierential
equation. Our aimis to seta disrete simulation sheme whihalso worksfor sti equations.
AnexampleofsuhequationsisgivenindimensiononebytheBesselproesses. Letusmention
the reent work [13 ℄ where the weak error of an approximation sheme for these proesses is
estimated.
The paper is organized as follows. In Setion 2, we reall some material on multivalued
maximalmonotone operatorsassoiatedwithaproperlowersemi-ontinuousonvexfuntion
tointroduemultivaluedstohastidierential equationsandtheirapproximate equations. In
Setion 3, we estimate the strong order of approximation: note that in [21 ℄ the estimation
orderis" andin[1℄itis" . Improvingtheargumentsin[21℄,weobtain" . InSetion
4,weannot usean expliit Eulershemeto disretize theapproximate solutionbeause the
drift term is Lipshitz ontinuous with onstant 1=". So we resort to a more involved semi-
impliitshemewhihbehavesmuh better, aswasnotedfor stiequations in[14 ℄. Setion 5
isdevotedto the simulationof some systemsof Brownian partileswithrepulsive interation
whih aremultidimensional extensionsof Besselproesses.
2 Preliminaries
The purpose of this setion is to reall some results on the subdierentials of onvex lower
semi-ontinuous funtions being maximal monotone operators whih will be used all along
thiswork. Therelevant materialononvexanalysisandmaximalmonotone operatorsmaybe
found inV.Barbuand Th. Preupanu [2 ℄or inH.Brézis[6 ℄.
2.1 Subdierential of a onvex funtion
Let'bea onvexfuntion dened inR d
;d2N
. We denoteby
dom(')= n
x2R d
:'(x)<1 o
thedomain of '. Wesaythat ' is proper if Int(dom(')) 6= ;.
Thesubdierential of ', written as ', isanoperator in R d
dened byitsgraph
(x;y) 2 Gr(') , 8z 2 R d
;'(x) '(z) + hy;x zi (1)
For anyx2R d
,we note
'(x) = fy 2R d
:(x;y)2Gr(')g:
Thedomain of ' is
D(') = n
x 2 R d
: '(x) 6= ; o
: (2)
Thefollowing proposition isstated without proof.
Proposition 2.1 The subdierential ' isa multivalued maximalmonotoneoperatorin R d
.
Moreover, we have
Int(D(')) = Int(dom(')) D(') dom(') dom(') = D('); (3)
where we reall that Int(D) and D are respetively the interior and the losure (for the
Eulidean topology) of D in R d
.
Let D bea onvexlosed subset of R d
withnonemptyinterior. Itfollows that
I
D (x) =
(
0 if x 2 D;
+1 if x 2= D
is onvex, l.s.. and properwith dom(I
D
) = D. Itssubdierential is
I
D (x) =
n
y2R d
: hy;z xi 0; 8z2D o
;
i.e.
I
D (x) =
8
>
<
>
:
; if x 2= D;
f0g if x 2 Int(D);
x
if x 2 D;
where
x
is the normaloneat x.
Note: In the one-dimensional ase, every multivalued maximal monotone operator A with
Int(D(A)) 6= ; is the subdierential of a proper l.s.. onvex funtion. This property does
not holdinthe multidimensional ase.
2.2 Yosida approximation
Wewillonstrutinthispartasequeneofsinglevaluedapproximationsforthesubdierential
ofaproperl.s.. onvexfuntion. Letusreallthat,foreah ">0and x2R d
,theequation
x 2 (I +"A)(y) has one and only one solution y in the domain of A if A is a maximal
monotone operator of R d
. The Yosida approximation of the subdierential ' of a proper
l.s.. onvex funtion ' istheappliation
"
dened by
"
= 1
"
(I
"
);
where
"
x is the unique solutionof theequation y 2 (I + "') 1
(x); x2R d
.
Proposition 2.2 For eah ">0, we have :
i)
"
is a ontration from R d
to D(');
ii)
"
is a single valued maximal monotone operator dened on the whole R d
, Lipshitz
ontinuous withonstant 1
"
;
iii)for every x2R d
;
"
(x) 2 '(
"
x).
Proposition 2.3 For eah ">0, put
'
"
(x) = min
y2R d
1
2"
jx yj 2
+ '(y)
; x2R d
: (4)
Then '
"
is alled the Yosida approximation of the funtion ' and
i) '
"
: R d
! ( 1;+1) isonvex withdomain dom('
"
) = R d
;
ii) '
"
is of lass C 1
R d
;R with r'
"
=
"
;
iii) the inmumdening '
"
(x) is attained at
"
x and
'
"
(x) =
"
2 j
"
(x)j 2
+ '(
"
x); (5)
iv) letting "#0, we have '
"
(x) " '(x) for all x2R d
;
v) there exists >0 suhthat for any x2R d
,
(1 + jxj) '(
"
x) '
"
(x) '(x): (6)
If ' istheindiator ofa onvexnonemptysubset D,then
"
x = proj
D
(x) 8">0; 8x2R d
;
"
(x) = 1
"
(x proj
D
(x)) 8">0; 8x2R d
:
We onlude this subsetion by stating some properties whih will be needed in the sequel.
Theproof ofthe rstpropositionmaybefound in[8℄andtheproofof theseondone iseasy,
heneit willbeomitted.
Proposition 2.4 For any a2R d
,there existonstants r>0; >0 (depending onlyon ')
suh that, for eah "2R
;x2R d
,
h
"
(x);x ai rj
"
(x)j + "j
"
(x)j 2
(jxj + 1): (7)
Proposition 2.5 Let F be a onvex, ontinuous, dierentiable funtion, from R d
into R.
Thenthe impliit Euler sheme withstep size
y
n+1
= y
n
+ f(y
n+1
); 0nN 1; (8)
where f = rF, is well-dened (i.e the equation y f(y) = ; 2 R d
has a unique
solutionfor all >0). Moreover, the sequene fF(y
n )g
n2N
isdereasing.
2.3 Multivalued stohasti dierential equations
Letbegiven
i) d 2 N
and 0 < T < +1 ;
ii) ' : R d
! ( 1;+1℄ onvex, l.s.. and proper;
iii) b : R d
! R d
and : R d
! R d
R d
Lipshitz ontinuousmappings;
iv) (;F;P) a probability spaewith ltration F
t
satisfying theusualonditions;
v) B = fB
t
;F
t
;0t1g a d-dimensional standard Brownian motion dened on
(;F;F
t
;P) with B
0
= 0;
0
Withthepreviousdataandassumptions, we shallapproximate andsimulate theunique solu-
tion (X
t )
0tT
ofthe multivalued stohastidierential equation
(
dX
t
+ '(X
t
)dt 3 b(X
t
)dt + (X
t )dB
t
X
0
=
(9)
By denition, asolution to theabove equationis apair ofproesses (X;K) suh that:
i) X = fX
t
;0tTg is a ontinuous, adapted proess with values in dom(') and
X
0
=;
ii) K = fK
t
;0tTg isaontinuous, adaptedproesswithboundedvariationtaking
values in R d
with K
0
=0;
iii) dX
t
= b(X
t
)dt + (X
t )dB
t dK
t
;
iv) foreverypairof ontinuous,adaptedproess (;)takingvalues in R d
andsatisfying
(
u
;
u
) 2 Gr(') 8u2[0;T℄;
themeasure hX
u
u
;dK
u
u
dui is P-a.s. nonnegative on [0;T℄.
We nowgive the plan of thepaper. For eah ">0 xed, we rstly give an approximate
solution whihis the solutionof theordinarystohasti dierential equation
(
dX
"
t
= b(X
"
t
)dt
"
(X
"
t
)dt + (X
"
t )dB
t
X
"
0
= X
0
(10)
Next we estimate of the term E
"
sup
tT jX
t X
"
t j
p
#
; p 2, to obtain the order of the
approximation.
Weseondlyproposeasemi-impliitEulershemewithstepsize = T
N
;N 2N
onthe
timeinterval [0;T℄ to approximate the solution (X
"
t )
0tT
8
>
<
>
:
X
"
0
= X
0
P p.s.
X
"
(n+1)
= X
"
n
+ b(X
"
n
)
"
(X
"
(n+1)
)+(X
"
n )(B
(n+1) B
n )
X
"
t
= X
"
n
if t 2 [n;(n+1)):
(11)
and estimate the term E
"
sup
0tT
X
"
t X
"
t
p
#
; p 2. Putting together both approxima-
tions we obtain
E
"
sup
0tT
X
t X
"
t
p
#
(p)"
p
8
+
2 (p)
p
2
"
p :
Sine theestimationis valid for all p2,wemoreoverget thepathwiseonvergene.
Beforegoingto themainsetion,we notethatifthegradientofthefuntion' isrepulsive
enough, thenthereisno loaltimeatthe boundaryof thedomain. Ifmoreover ' belongs to
C (Int(dom('))), equation(9)turns into the ordinary stohastidierential equation
dX
t
= b(X
t
)dt r'(X
t
)dt + (X
t )dB
t
X
0
=
(12)
withexplodingdrift at theboundary ofthe domain.
3 Approximate solution
Severalauthorshaveprovedbythepenalizationmethodtheonvergenein L 2
ofthesequene
of the approximate solutions X
"
to the unique solution X of the multivalued stohasti
dierential equation (9)(see,e.g. [1℄,[8℄).
In[21℄, theestimation
E
"
sup
0tT jX
"
t X
t j
2
#
"
1
12
; 0 < T < +1:
wasproved. Wewill generalize theresult to the L p
-ase (p2) andnd alarger power.
3.1 Main estimates
We rstly reallthe main steps ofStorm's proofin[21 ℄.
Proposition 3.1 Let p 2 and assume X
0 2 L
p
. Then for every " > 0 there exists a
onstant independent of " suh that
E
"
sup
tT jX
"
t j
p
+
Z
T
0 j
"
(X
"
s )jds
p
2
+
Z
T
0
"j
"
(X
"
s )j
2
ds
p
2
#
: (13)
Proposition 3.2 Let p2 andassume X
0 2L
2p
. Thenforevery 0<"<1 and Æ>0, we
have
E
"
sup
tT
X
"
t X
Æ
t
p
#
E
"
sup
tT jX
"
t
"
X
"
t j
p
# 1
2
+ E
"
sup
tT
X Æ
t
Æ X
Æ
t
p
# 1
2
: (14)
3.2 Convergene rate
Proposition 3.3 Let p2andassume X
0 2L
5p=2
and E[j'(X
0 )j
p
℄ < 1. Thenforevery
0<"<1 we have
E
"
sup
tT jX
"
t
"
X
"
t j
2p
#
"
p
2
: (15)
Proof: Sine '
"
is only C 1
, we shall need the easy following lemma whih an be proved
by regularizingthe funtionf and usingtheIt formula.
Lemma 3.4 Let f be a onvex funtion of lass C (R ;R) whose gradient is Lipshitz
ontinuous withonstant . Thenfor all p1, we have almost surely
f(X
"
t^
k )
2p
jf(X
"
0 )j
2p
+ 2p Z
t^
k
0 D
j f(X
"
s )j
2p 1
rf(X
"
s
);(X
"
s )dB
s E
+ Z
t^
k
0
jf(X
"
s )j
2p 2
jrf(X
"
s )j
2
+ jf(X
"
s )j
2p 1
(jX
"
s j + 1)
2
ds
+ Z
t^
k
0
jf(X
"
s )j
2p 1
jrf(X
"
s )j(jX
"
s
j + 1)ds
2p Z
t^
k
0 D
j f(X
"
s )j
2p 1
rf(X
"
s );
"
(X
"
s )
E
ds:
For every k >0, we dene
k
= infft : jX
"
t
j kg . Applying the lemma to the onvex
funtion '
"
whose gradient is Lipshitz ontinuous with onstant 1
"
, taking the supremum
over tand takingexpetations, weget
E
sup
st^
k '
"
(X
"
s )
2p
E
'
"
(X
"
0 )
2p
+
"
E
Z
t^
k
0 j'
"
(X
"
s )j
2p 1
(jX
"
s j + 1)
2
ds
+E
Z
t^
k
0 j'
"
(X
"
s )j
2p 2
j
"
(X
"
s )j
2
(jX
"
s j + 1)
2
ds
+2pE
sup
st^
k Z
s
0
j'
"
(X
"
u )j
2p 1
"
(X
"
u );(X
"
u )dB
u
+E
Z
t^
k
0 j'
"
(X
"
s )j
2p 1
j
"
(X
"
s )j(jX
"
s
j + 1)ds
:
(16)
Step 1: Estimate ofthe seond term intheright hand side of(16).
Theonvexfuntion ' isboundedbelowbyananefuntion. Moreover
"
isaontration
and
"
(x) 2 '(
"
x). So we have,for every x2R d
,
(1 + jxj) '(
"
x) j
"
(x)jjx aj + :
Hene fromthe denitionof '
"
and Proposition2.4, we have
j'
"
(x)j =
"
2 j
"
(x)j 2
+ '(
"
x)
"
2 j
"
(x)j 2
+ + jxj + (1 + jxj)j
"
(x)j:
Applying thisinequalityto the seondterm on theright hand of(16), we obtain
"
E
Z
t^
k
0 j'
"
(X
"
s )j
2p 1
(jX
"
s j + 1)
2
ds
"
2p+1
E
"
sup
st^
k
"
2p 2
j'
"
(X
"
s )j
2p 2
1 + sup
st^
k jX
"
s j
2 Z
t^
k
0
"j
"
(X
"
s )j
2
jds
#
+"
2p+1
tE
"
sup
st^
k
"
2p 2
j'
"
(X
"
s )j
2p 2
1 + sup
st^
k jX
"
s j
3
#
+"
2p+1
E
"
sup
st^
k
"
2p 2
j'
"
(X
"
s )j
2p 2
1 + sup
st^
k jX
"
s j
3 Z
t^
k
0 j
"
(X
"
s )jds
#
