The Laws of Exponential Functionals of Brownian Motion, Taken at Various Random Times
With the help of several different methods, a closed formula is obtained for the laws of the exponential functionals of Brownian motion with drift, taken at certain random times, particularly exponential times, which are assumed to be independent of the B
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The Laws of Exponential Functionals of Brownian Motion, Taken at Various Random Times 1 C.R. Acad. Sci., Paris, Ser. I 314 (1992), 951-956
Abstract. With the help of several different methods, a closed formula is obtained for the laws of the exponential functionals of Brownian motion with drift, taken at certain random times, particularly exponential times, which are assumed to be independent of the Brownian motion.
Abridged Version2 Let (Bt, t ::::: 0) be a real-valued Brownian motion, starting from O. For v E JR, define:
A~lI) = lot dsexp2(B s + vs),
t::::: O.
Consider, furthermore, TA' an exponential time with parameter .x, independent from B. The main result of this Note is the following: the distribution of A~; is the same as that of Zl,a/2Zb, where
/-l+v 2 '
a=--
and Zo., {3 (resp.: Z-y) denotes a beta variable with parameters (n, {3) (resp.: a gamma variable with parameter "(), and these two random variables are assumed to be independent. This result may be deduced from the closed form of the generalized moments:
E[(A~;)'Y], which are finite for
.x > 2"(("( + v),
and may be expressed in terms of the gamma function, and>., ,,(, v (see [3]). Conversely, from the identification of the distribution of the law of A~; as that of the beta-gamma ratio presented above, one easily obtains the expression of the generalized moments of A~;. 1
2
This Note was presented by Paul-Andre Meyer. This is the Abridged English Version, which appeared in the original French paper.
M. Yor, Exponential Functionals of Brownian Motion and Related Processes © Springer-Verlag Berlin Heidelberg 2001
.
56
4. Laws of Exponential Functionals of Brownian Motion
Consider now the case v = 0, and write, for simplicity, At for A~O). The knowledge of the distribution of AT>. enables to recover Bougerol's result (see [1]): for fixed
t
~
0,
. h(B) t
SIn
= /3At'
dist.
where (/3u, u ~ 0) is a real-valued Brownian motion, which is independent of the variable At. In turn, Bougerol's result allows to obtain the laws of AT for a large class of random times T, assumed to be independent of B.
1.
Some Identities in Distribution
Let (Bt, t ~ 0), be a real-valued Brownian motion, starting at O. Let v E lR and denote
A~V) = lot dsexp2(Bs + vs).
(t
~ 0)
The explicit expression (1), below, for the moments of the functional (A~v») taken at an exponential time independent of B is the point of departure for the results of this paper. Theorem 1. (See [3]). Let 1 ~ 0, and A > 0; set J-l = V2A + v 2 . Suppose that A > 21(1 + v), which is equivalent to: J-l > v + 21. Then we have
E[(A(v)Pl = r(1 T>.
+ 1)r(((JL + v)/2) + l)r(((JL - v)/2) -1) 2,r((J-l- v)/2)r(1 + 1 + ((J-l + v)/2))
(1)
where T).. denotes an exponential time with parameter A, independent of B.
The identities in distribution (2) and (3), below, can be easily deduced from the identity (1) and, conversely, (1) follows immediately from the identity in distribution (2). Theorem 2. (Using the notation of Theorem 1). We have the following identity in distribution
(2) where
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