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# Tutorial on Additive Lévy Processes - Lecture #1

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##### Formal sciences

Informations DK.Saltsan(nevihoshCI)hatU,ytiCekaL1/06etlatSeA,SA
International Conference on Stochastic Analysis and Its Applications August 711, 2006
20
Tutorial on Additive Le´ vy Processes Lecture #1
Department of Mathematics University of Utah http://www.math.utah.edu/˜davar
Davar Khoshnevisan An “(N,d)random eldXhasNparameters and takes values in Rd(Adler, 1981); i.e.,X(t)Rdfor allt:= (t1, . . . ,tN)RN.
Some Terminology
02/2at)hytU,ekiCtlaLe06attlA,SeICSArtdocuitnoInN+inplaculdhaveRoCBrntdeenepndNi,X...,1X.cte,NRfoerowniveBdditd.AisRnitnonaomwoin Some Terminology
An “(N,d)random eldXhasNparameters and takes values in Rd(Adler, 1981); i.e.,X(t)Rdfor allt:= (t1, . . . ,tN)RN. Could haveR+Nin place ofRN, etc.
tioncudortnINind..,XX1,.):=X1(t1ion:X(tnwaimntotiviBeord.nRddAtimosionworBnainnepetned for allt= (t1, . . . ,tN)RN+.
0
An “(N,d)random eldXhasNparameters and takes values in Rd(Adler, 1981); i.e.,X(t)Rdfor allt:= (t1, . . . ,tN)RN. Could haveRN+in place ofRN, etc. X1, . . . ,XNindependent Brownian motions inRd. “Additive Brownian motion:
X(t) :=X1(t1) +∙ ∙ ∙+XN(tN)
IortnonolygomeTermiductionS D.)IahUty,iteCaktLlaS(nasivenhsohKAACSea,Slett62002/
X(t) :=X1(t1) +∙ ∙ ∙+XN(tN)for allt= (t1, . . . ,tN)R+N.
An “(N,d)random eldXhasNparameters and takes values in Rd(Adler, 1981); i.e.,X(t)Rdfor allt:= (t1, . . . ,tN)RN. Could haveRN+in place ofRN, etc. X1, . . . ,XNindependent Brownian motions inRd. “Additive Brownian motion:
Some Terminology
tcoiorudnntI ABM and the Local Dynamics
of
Brownian
Sheet
˙ W
:=white noise onR+.
tLaleCaky,itahUtSCI)S,AAttae0el ABM and the Local Dynamics of Brownian Sheet
˙ W:=white noise onR+. I.e., Gaussian, and ˙ ˙ IfAB=thenW(A)andW(B)are indept. ˙ W:=white noise onR+. I.e., Gaussian, and ˙ ˙ IfAB=thenW(A)andW(B)are indept. ˙ ˙ E W(A) =0 and VarW(A) =meas(A).
ABM and the Local Dynamics of Brownian Sheet lDcaamynsoicheftoLlntotaag6m0saee0(2s/n3A<
˙ W:=white noise onR+. I.e., Gaussian, and ˙ ˙ IfAB=thenW(A)andW(B)are indept. ˙ ˙ E W(A) =0 and VarW(A) =meas(A).
)
ABM and the Local Dynamics of Brownian Sheet
foralln.D.Khoshnei.vsa.sa(nWa˙S,t=lna[Le!kniAC1tåyn,=Ut˙aWh=)1InC)S(AAAs,lS,eaa.er,2..naodonrndisjmandthenointemL,AA1IfmaNteeShanniowBr NDK.oh
as long as meas(An)<for all n.
ABM and the Local Dynamics of Brownian Sheet
Lemma If A1,A2, . . .are nonrandom and disjoint then a.s., W˙n=[1An!=n=å1 ˙ W(An),
˙ W:=white noise onR+. I.e., Gaussian, and ˙ ˙ IfAB=thenW(A)andW(B)are indept. ˙ ˙ E W(A) =0 and VarW(A) =meas(A).
3/20e06attlA,SeASCI)hatU,ytiCekLaltSan(savineshmanyoscioLDlacSheenianBrowfthet 60elttaeS,AASCI
˙ (N,1)“Brownian sheetB:=the dF ofW; i.e.,
for allt= (t1, . . . ,tN)R+N.
˙ B(t) :=W([0,t1]× ∙ ∙ ∙ ×[0,tN])
ABM and the Local Dynamics of Brownian Sheet