Corrections for "Occupation and local times for skew Brownian motion with applications to dispersion across an interface"

0.07 MB·English

by  Thilanka Appuhamillage

TheAnnalsofAppliedProbability
2011,Vol.21,No.5,2050–2051
DOI:10.1214/11-AAP775
(cid:13)c InstituteofMathematicalStatistics,2011
CORRECTIONS
OCCUPATION AND LOCAL TIMES FOR SKEW BROWNIAN
2 MOTION WITH APPLICATIONS TO DISPERSION ACROSS
1 AN INTERFACE
0
2 Ann. Appl. Probab. 21 (2011) 183–214

n
a By Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann,
J Edward Waymire and Brian Wood

3
Oregon State University

]
R
The nonnegative parameter γ= (2α 1)v appearing in the formula for
P | − |
the transition probabilities for α-skew Brownian motion with drfitv in The-
.
h orem 1.3 should be replaced by the parameter γ=(2α 1)v ( v,v). The
t − ∈ −
a formula is then correct, as can be checked by (tedious) differentiations for
m the backward equation and interface condition in the backward variable x.
[ However, it is only in the cases when (2α 1)v 0 that the probabilistic
− ≥
interpretation of γ as a skew-elasticity parameter applies.
4
v In addition, the corrected display to Corollary 3.3 that follows from in-
0 tegration of the formula in Corollary 1.2 giving the trivariate density is as
1 follows:
4
5 Corollary 3.3. If x 0 we have
≥
9. Px(Bt(α) dy,ℓ(tα) dℓ)
0 ∈ ∈
0 2(1 α)(l y+x) (l y+x)2
− − exp − dydl,
1  √2πt3 (cid:26)− 2t (cid:27) if y 0,l 0,
v:  ≤ ≥
i 2α(l+y+x) (l+y+x)2
X  exp dydl
= √2πt3 (cid:26)− 2t (cid:27)
r 
a  1 (y x)2
+ exp −
√2πt(cid:20) (cid:26)− 2t (cid:27)


 (y+x)2

 −exp(cid:26)− 2t (cid:27)(cid:21)δ0(dl)dy, if y≥0,l≥0,




Received March 2011.
AMS 2000 subject classifications. Primary 60K35, 60K35; secondary 60K35.
Key words and phrases. Skew Brownian motion, advection-diffusion, local time, occu-
pation time, elastic skew Brownian motion, stochastic order, first passage time.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Applied Probability,
2011,Vol. 21, No. 5, 2050–2051. This reprint differs from the original in
pagination and typographic detail.
1
2 T. APPUHAMILLAGEET AL.
whereas if x 0, then
≤
(α) (α)
Px(Bt dy,ℓt dℓ)
∈ ∈
2α(l+y x) (l+y x)2
− exp − dydl,
 √2πt3 (cid:26)− 2t (cid:27)

 if y 0,l 0,

 ≥ ≥
2(α 1)(l y x) (l y x)2

 − − − exp − − dydl
= √2πt3 (cid:26)− 2t (cid:27)


1 (y x)2
+ exp −
 √2πt(cid:20) (cid:26)− 2t (cid:27)



 (y+x)2
 −exp(cid:26)− 2t (cid:27)(cid:21)δ0(dl)dy, if y≤0,l≥0.




Acknowledgments. The authors are grateful to Pierre Etor´e and Miguel
Martinez for sharing their preprint Etor´e and Martinez (2011), who pointed
out these mistakes. In particular, the formula with (2α 1)v in place of
−
(2α 1)v is given in Proposition 4.1 of their paper.
| − |
REFERENCES
Etor´e, P. and Martinez, M. (2011). Exact simulation of one-dimensional stochastic
differential equations involving local time at zero of the unknown process. Preprint.
Available at http://arxiv.org/abs/1102.2565.
T. Appuhamillage B. Wood
V. Bokil School ofChemical, Biological
E. Thomann andEnvironmentalEngineering
E. Waymire Oregon StateUniversity
Departmentof Mathematics Corvallis, Oregon 97331
Oregon StateUniversity USA
Corvallis, Oregon 97331 E-mail: [email protected]
USA
E-mail:[email protected]
[email protected]
[email protected]
[email protected]