(28 December 2015)
ON THE DOUBLE LAPLACE TRANSFORM OF THE TRUNCATED VARIATION OF A BROWNIAN MOTION WITH DRIFT
RAFAL MARCIN LOCHOWSKI,∗ Warsaw School of Economics
Abstract
The aim of this paper is to find a formula for the double Laplace transform
of the truncated variation of a Brownian motion with drift. In order to find
the double Laplace transform we also prove some identities for the Brownian
motion with drift, which may be of independent interest.
Keywords: total variation, truncated variation, uniform approximation, Brownian motion with drift, Laplace transform
2010 Mathematics Subject Classification: Primary 60E10
Secondary 60G15
1. Introduction
Let X = (Xt )t∈[a;b] be a real valued stochastic process with càdlàg trajectories. In
general, the total path variation of X, defined as
T V (X, [a; b]) = sup
sup
n
X
Xt − Xt ,
i
i−1
n a≤t0 <t1 <...<tn ≤b
i=1
may be (and in many most important cases is) a.s. infinite. However, in the neighborhood of every càdlàg path we may easily find a function, total variation of which is
finite.
Let f : [a; b] → R be a càdlàg function and let c > 0. The natural question arises,
what is the smallest possible (or infimum of) the total variations of functions g : [a; b] →
R from the ball g : kf − gk∞ ≤ 12 c , where kf − gk∞ := sups∈[a;b] |f (s) − g (s)| . The
∗
Postal address: Department of Mathematics and Mathematical Economics, ul. Madalińskiego 6/8,
02-513 Warsaw Poland
∗ Email address: [email protected]
1
2
R. M. LOCHOWSKI
bound from below reads as
T V (g, [a; b]) ≥ T V c (f, [a; b]) ,
where
T V c (f, [a; b]) := sup
sup
n
X
max {|f (ti ) − f (ti−1 )| − c, 0}
(1)
n a≤t0 <t1 <...<tn ≤b
i=1
and follows immediately from the inequality
|g (ti ) − g (ti−1 )| ≥ max {|f (ti ) − f (ti−1 )| − c, 0}
holding for any t1 , t2 ∈ [a; b] and any function g : [a; b] → R from the ball g : kf − gk∞ ≤ 21 c .
In fact, in [3] it was proven that we have equality
inf T V (g, [a; b]) : kf − gk∞ ≤ 12 c = T V c (f, [a; b]) .
(2)
Remark 1.1. Since we deal with càdlàg functions, a more natural setting of our
problem would be the investigation of
inf T V (g, [a; b]) : g - càdlàg, dD (f, g) ≤ 12 c ,
where dD denotes the Skorohod metric. Since the total variation does not depend
on the (continuous and strictly increasing) change of the variable t and the function
f c minimizing T V (g, [a; b]) appears to be a càdlàg one, solutions of both problems
coincide.
The bound (1) is called truncated variation. Moreover, for any c ≤ sups,u∈[a;b] |f (s) − f (u)|
there exist unique càdlàg function f c : [a; b] → R such that kf − f c k∞ ≤ 21 c and for
any s ∈ (a; b]
T V (f c , [a; s]) = T V c (f, [a; s]) .
f c is a càdlàg function with jumps possible only at the points where the function f
has jumps and it may be represented in the following form
f c (s) = f c (a) + U T V c (f ; [a; s]) − DT V c (f ; [a; s]) ,
where
U T V c (f, [a; b]) := sup
sup
n
X
n a≤t0 <t1 <...<tn ≤b
i=1
max {f (ti ) − f (ti−1 ) − c, 0} ,
(3)
On the double Laplace transform
c
DT V (f, [a; b]) := sup
3
sup
n
X
max {f (ti−1 ) − f (ti ) − c, 0}
(4)
n a≤t0 <t1 <...<tn ≤b
i=1
are called upward and downward truncated variations of the function f respectively
(see the next section and for more general results concerning regulated functions see
[4, Theorem 4]). We also have
T V c (f, [a; b]) = U T V c (f, [a; b]) + DT V c (f, [a; b]) .
(5)
Properties of the truncated variation and two other quantities related - upward
and downward truncated variations of trajectories of stochastic processes are up to
some degree known (see [2], [5], [6]). In particular, in [7] an exact representation of
the truncated variation of a (shifted) standard Brownian motion B in terms of its
local times was given and in [2] there were calculated double Laplace transforms of
U T V c (W, [0; S]) and DT V c (W, [0; S]) for Wt = Bt + µt being a standard Brownian
motion with drift µ and [0; S] being a random interval, length of which is exponentially
distributed and independent from the underlying Brownian motion B.
Remark 1.2. In [2] the functionals U T V c (·, [a; b]) and DT V c (·, [a; b]) were defined
with slightly different formulas, but it is easy to see that both definitions coincide.
This also gives the full characterisation of the distributions of U T V c (W, [0; T ]) and
DT V c (W, [0; T ]) for deterministic time T . However, since the variables U T V c (W, [0; T ])
and DT V c (W, [0; T ]) are dependent, these results do not provide us with the full
characterisation of the distribution of T V c (W, [0; T ]) and the dependence structure
between U T V c (W, [0; T ]) and DT V c (W, [0; T ]) .
The aim of this paper is to find a formula for the Laplace transform of T V c (W, [0; S]) .
In order to find this double Laplace transform we will also prove some identities for
the Brownian motion with drift, which may be of independent interest.
The paper is organized as follows. In the next section we introduce some necessary
definitions, notation and results. In the last section we deal with the Laplace transform
of the truncated variation of a standard Brownian motion with drift, its moments and
the covariance between the upward and downward truncated variations of the Brownian
motion with drift.
4
R. M. LOCHOWSKI
2. Truncated variation, upward truncated variation and downward
truncated variation of a càdlàg function - their optimality and other
properties
2.1. Definitions and notation
Let −∞ < a < b < +∞ and let f : [a; b] → R be a càdlàg function. For c > 0,
applying the convention inf ∅ = ∞, we define two stopping times
(
)
c
TD
f = inf
s ∈ [a; b] : sup f (t) − f (s) ≥ c ,
t∈[a;s]
TUc f
= inf s ∈ [a; b] : f (s) − inf f (t) ≥ c .
t∈[a;s]
c
Assume that TD
f ≥ TUc f i.e. the first time f is at the distance greater or equal c
above its running minimum appears before the first time f is at the distance greater
or equal c below its running maximum, or both events do not occur (both times are
c
f < T c f we may simply consider the function −f.
infinite). Note that in the case TD
∞ U ∞
c
c
c
Now we define sequences TU,k
, TD,k
, in the following way: TD,−1
= a,
k=0
k=−1
c
TU,0
= TUc f and for k = 0, 1, 2, ...
c
TD,k
=
c
TU,k+1

n
h
i
 inf s ∈ T c ; b : sup
U,k
t∈[T c
U,k
;s] f (t) − f (s) ≥ c
o
c
if TU,k
< b,
 ∞
c
if TU,k
≥ b,

n
h
i
o
c
 inf s ∈ T c ; b : f (s) − inf
if TD,k
< b,
c
D,k
t∈[TD,k
;s] f (t) ≥ c
=
 ∞
c
if TD,k
≥ b.
c
c
= ∞.
= ∞ or TD,K
Remark 2.1. Note that there exists such K < ∞ that TU,K
∞
∞
Otherwise we would obtain two infinite sequences (sk )k=1 , (Sk )k=1 such that a ≤ s1 <
S1 < s2 < S2 < ... ≤ b and f (Sk ) − f (sk ) ≥ 21 c. But this is a contradiction, since f is
∞
∞
càdlàg and (f (sk ))k=1 , (f (Sk ))k=1 have a common limit.
h
c
c
Now let us define two sequences of non-decreasing functions mck : TD,k−1
; TU,k
∩
h
c
c
c
[a; b] → R and Mkc : TU,k
; TD,k
∩ [a; b] → R for such k that TD,k−1
< ∞ and
c
TU,k
< ∞ respectively, with the formulas
mck (s) =
inf
c
t∈[TD,k−1
;s]
f (t) , Mkc (s) =
sup
c ;s
t∈[TU,k
]
f (t) .
On the double Laplace transform
5
Next we define two finite sequences of real numbers (mck ) and (Mkc ) , for such k that
c
c
TD,k−1
< ∞ and TU,k
< ∞ respectively, with the formulas
mck
c
= mck TU,k
− =
Mkc
c
= Mkc TD,k
− =
inf
f (t) ,
c
c
t∈[TD,k−1
;TU,k
)∩[a;b]
f (t) .
sup
c ;T c
t∈[TU,k
D,k )∩[a;b]
Finally, let us define the function f c : [a; b] → R with the formulas

c

mc0 + c/2
if s ∈ a; TU,0
;



h
c
c
Mkc (s) − c/2
if s ∈ TU,k
; TD,k
, k = 0, 1, 2, ...;
f c (s) =

h


 mc (s) + c/2 if s ∈ T c ; T c
k+1
D,k
U,k+1 , k = 0, 1, 2, ....
h
c
c
Remark 2.2. Note that due to Remark 2.1, b belongs to one of the intervals TU,k
; TD,k
h
c
c
or TD,k
; TU,k+1
for some k = 0, 1, 2, ... and the function f c is defined for every
s ∈ [a; b]. The substraction (resp. addition) of the term c/2 from Mkc (s) (resp. to
mck+1 (s)) joins pieces of the graph of the function f c so that it becomes continuous
(in the case when f is continuous).
Remark 2.3. One may think about the function f c as of the most ”lazy” function
possible, which changes its value only if it is necessary for the relation kf − f c k∞ ≤ c/2
to hold.
h
The function f c has finite total variation since it is non-decreasing on the intervals
h
c
c
c
c
TU,k
; TD,k
∩ [a; b], k = 0, 1, 2, ... and non-increasing on the intervals TD,k
; TU,k+1
∩
[a; b], k = 0, 1, 2, ... and since f c has finite total variation, we know that there exist
c
two non-decreasing functions fUc and fD
: [a; b] → [0; +∞) such that f c (t) = f c (a) +
c
c
c
(s) = 0 for s ∈ a; TU,0
,
(t) for s ∈ [a; b]. Setting fUc (s) = fD
fUc (t) − fD
 P
h
k−1
c
c
c
c
c
c

{M
−
m
−
c}
+
M
(s)
−
m
−
c
if
s
∈
T
;
T
;
i
i
k
k
U,k
D,k
i=0
h
fUc (s) =
P
k
c
c
c
c

if s ∈ TD,k
; TU,k+1
i=0 {Mi − mi − c}
and
 P
k−1 c
c

i=0 Mi − mi+1 − c
c
fD (s) =
 Pk−1 M c − mc − c + M c − mc (s) − c
i
i+1
k
k+1
i=0
h
c
c
if s ∈ TU,k
; TD,k
;
h
c
c
if s ∈ TD,k
; TU,k+1
we have
c
f c (s) = f c (a) + fUc (s) − fD
(s)
for s ∈ [a; b].
6
R. M. LOCHOWSKI
2.2. Important result
c
In the case TD
f < TUc f we may apply the definitions from the previous subsection
c
to the function −f and simply define f c = −(−f )c , fUc = −(−f )cU and fD
= −(−f )cD .
c
This way the mappings f 7→ f c , f 7→ fUc and f 7→ fD
are defined for any càdlàg
function f : [a; b] → R.
We have the following (cf. [3, Corollary 3.8, Theorem 4.1])
Theorem 2.1. The function f c is optimal i.e. if g : [a; b] → R is such that kf − gk∞ ≤
c/2 and has finite total variation, then for every s ∈ [a; b]
T V (g, [a; s]) ≥ T V (f c , [a; s]) .
It is unique in such a sense that if for every s ∈ [a; b] the opposite inequality holds
T V (g, [a; s]) ≤ T V (f c , [a; s])
and c ≤ sups,u∈[a;b] |f (s) − f (u)| then g = f c . Moreover, for any s ∈ (a; b] the following
equalities hold
U T V c (f, [a; s]) = fUc (s) ,
(6)
c
DT V c (f, [a; s]) = fD
(s) ,
(7)
c
T V c (f, [a; s]) = fUc (s) + fD
(s) .
(8)
3. The Laplace transform of truncated variation process of Brownian
motion with drift stopped at exponential time
Let Bt , t ≥ 0, be a standard Brownian motion, c > 0, µ ∈ R, and Wt = Bt + µt be
a standard Brownian motion with the drift µ. For typographical reasons instead of Wt
we will sometimes write W (t). In this section we will calculate the double Laplace
transform of the truncated variation process of W, i.e.
the Laplace transform of
the process T V c (W, s) := T V c (W, [0; s]) , s ≥ 0, stopped at (independent from W )
exponentially distributed time S.
On the double Laplace transform
7
3.1. The Laplace transform
We begin with some auxiliary observations. First let us notice that by Theorem
∞
c
c
2.1, on the set {TD
W ≥ TUc W } , applying the definition of sequences TU,k
and
k=0
∞
c
TD,k
(c.f. subsection 2.1) to the function f = W, for s ≥ 0 we obtain
k=−1


0




Pk−1
Pk−1 

{Mic − mci − c} + i=0 Mic − mci+1 − c

i=0


T V c (W, s) =
+Mkc (s) − mck − c


Pk
Pk−1 c

c
c
c



i=0 {Mi − mi − c} +
i=0 Mi − mi+1 − c



 +M c − mc (s) − c
k
k+1
c
if s ∈ 0; TU,0
;
h
c
c
if s ∈ TU,k
; TD,k
;
h
c
c
if s ∈ TD,k
; TU,k+1
c
c
, TD,k
were defined for a function with a domain being the compact
(although TU,k
interval [a; b] , the extension of their definition to a function defined on a half-line is
c
straightforward). By the continuity of Brownian paths, on the set {TD
W ≥ TUc W } we
have
c
c
W TU,k
= mck + c, W TD,k
= Mkc − c,
hence


0




Pk−1 Pk−1 c

c
c

+ i=0 W TD,i
− mci+1

i=0 Mi − W TU,i


c
+Mkc (s) − W TU,k
T V c (W, s) =


Pk−1  Pk c
c
c


Mi − W TU,i
+ i=0 W TD,i
− mci+1

i=0



c
 +W T c
D,k − mk+1 (s)
Now let us define two sequences of stopping times
c
T̃U,k
∞
k=0
c
if s ∈ 0; TU,0
;
h
c
c
if s ∈ TU,k
; TD,k
;
h
c
c
if s ∈ TD,k
; TU,k+1
.
∞
c
, T̃D,k
in the
k=0
c
following way: T̃U,0
= TUc W and for k = 0, 1, 2, ...
c
T̃D,k
= inf


c
s > T̃U,k

(
c
T̃U,k+1
= inf


: sup Wt − Ws ≥ c ,

c ;s
t∈[T̃U,k
]
)
c
s > T̃D,k
: Ws −
inf
c
t∈[T̃D,k
;s]
Wt ≥ c .
∞ ∞
∞ ∞
c
c
c
c
c
Notice that on the set {TD
W ≥ TUc W } the sequences TU,k
, TD,k
and T̃U,k
, T̃D,k
k=0
coincide.
k=0
k=0
k=0
8
R. M. LOCHOWSKI
Now for any 0 ≤ a ≤ b < +∞ we define two auxiliary functions
U [a; b] = sup Wt − Wa ,
a≤t≤b
D [a; b] = Wa − inf Wt
a≤t≤b
and for s ≥ 0 define two quantities
c
U (W, s)
=
∞
X
∞
h
i X
h
i
c
c
c
c
U T̃U,i ∧ s; T̃D,i ∧ s +
D T̃D,i
∧ s; T̃U,i+1
∧s ;
i=0
Dc (W, s)
=
i=0
∞
X
∞
i X
h
i
c
c
c
c
D T̃D,i
∧ s; T̃U,i+1
∧s +
U T̃U,i+1
∧ s; T̃D,i+2
∧s .
i=0
i=0
h
c
Notice that on the set {TD
W ≥ TUc W } we have
T V c (W, s) = U c (W, s) .
∞
c
c
W < TUc (W ) , then we change in the definitions of sequences T̃U,k
Similarly, if TD
k=0
∞
c
and T̃D,k
, W for −W and obtain
k=0
T V c (W, s) = U c (−W, s) .
What will be important to us is the fact that the conditional distribution of U c (W, s)
c
c
W ≥ TUc (W )) , is the same as the distribution
W ≥ TUc (W ) , L (U c (W, s) |TD
given TD
of U c (W, s) . This follows from the strong Markov property and the independence of
the increments of a Brownian motion.
Now let S be an exponential random variable, independent from W, with density
−νx
νe
. By MT V c (W,S) (λ) we denote the moment generating function of T V c (W, S) ,
i.e.
MT V c (W,S) (λ) := E [exp (λ · T V c (W, S))] .
We have the following equation
MT V c (W,S) (λ)
c
c
c
c
= E exp (λ · U c (W, S)) |S ≥ TU,0
, TD
W ≥ TUc W × P S ≥ TU,0
, TD
W ≥ TUc W
c
c
c
c
+E exp (λ · U c (−W, S)) |S ≥ TU,0
, TD
W < TUc W × P S ≥ TU,0
, TD
W < TUc W
c
+P (min {TUc W, TD
W } > S) .
(9)
On the double Laplace transform
9
By the lack of memory of the exponential distribution, the strong Markov property
and the independence of the increments of a Brownian motion we have
h
i
c
c
E exp (λ · U c (W, S)) |S ≥ T̃U,0
, TD
W ≥ TUc W
c
= E exp λ · U c W, S + T̃U,0
h
i
c
c
c
= E exp λ · U c W, S + T̃U,0
; S < T̃D,0
− T̃U,0
h
i
c
c
c
+E exp λ · U c W, S + T̃U,0
; S ≥ T̃D,0
− T̃U,0
h
h
i
i
c
c
c
c
= E exp λ · U T̃U,0
; S + T̃U,0
; S < T̃D,0
− T̃U,0
h
n
h
i
o
i
c
c
c
c
c
+E exp λ · U T̃U,0
; T̃D,0
+ λ · Dc W, S + T̃U,0
; S ≥ T̃D,0
− T̃U,0
h
h
i
i
c
c
c
c
= E exp λ · U T̃U,0
; S + T̃U,0
; S < T̃D,0
− T̃U,0
i
h
c
c
c
c
c
. (10)
E exp λ · Dc W, S + T̃D,0
− T̃U,0
; S ≥ T̃D,0
; TD,0
+E exp λ · U TU,0
Notice that in all the calculations above, except the first line, the starting value of
c
c
c
T̃U,0
≥ 0 is irrelevant, we need only to know the recursive definitions of T̃D,0
, T̃U,1
, ...;
c
thus we may set T̃U,0
= 0 and we have
h
i
c
c
c
c
c
E exp λ · U W, S + T̃U,0 ; S < T̃D,0 − T̃U,0 = E exp λ · sup Wt ; S < TD W ,
0≤t≤S
(11)
and
"
c
c
c
c
E exp λ · U TU,0 ; TD,0
; S ≥ TD,0
− TU,0
= E exp λ ·
!
sup
cW
0≤t≤TD
Wt
#
c
; S ≥ TD
W .
(12)
Similarly
c
E exp λ · Dc W, S + T̃D,0
h
i
c
c
c
= E exp λ · Dc W, S + T̃D,0
; S < T̃U,1
− T̃D,0
h
i
h
i
c
c
c
c
c
+E exp λ · D T̃D,0
; T̃U,1
; S ≥ T̃U,1
− T̃D,0
E exp λ · U c W, S + T̃U,1
(13)
from which we get
h
i
c
c
c
c
c
E exp λ · D W, S + T̃D,0 ; S < T̃U,1 − T̃D,0 = E exp −λ · inf Wt ; S < TU W
0≤t≤S
(14)
10
R. M. LOCHOWSKI
and
h
i
c
c
c
c
E exp λ · D TD,0
; TU,1
; S ≥ T̃U,1
− T̃D,0
= E exp −λ ·
inf c
0≤t≤TU W
Wt ; S ≥ TUc W .
(15)
c
Now, substituting in (10) expression (13) for E exp λ · Dc W, S + TD,0
, and using
(11)-(12) and (14)-(15) we get
c
c
E exp (λ · U c (W, S)) |S ≥ TU,0
, TD
W ≥ TUc W
=
c
W]
E[exp(λ·sup0≤t≤S Wt );S<TD
h
i h
i
c W E exp −λ·inf
c
1−E exp λ·sup0≤t≤T c W Wt ;S≥TD
0≤t≤T c W Wt ;S≥TU W
U
D
h
i
c
c
E exp λ·sup0≤t≤T c W Wt ;S≥TD
W E[exp(−λ·inf 0≤t≤S Wt );S<TU
W]
D
h
i h
i.
+
c
cW
1−E exp λ·sup0≤t≤T c W Wt ;S≥TD W E exp −λ·inf 0≤t≤T c W Wt ;S≥TU
(16)
U
D
Using results of [8] we will be able to calculate all quantities appearing in (16).
c
To calculate E exp λ · sup0≤t≤S Wt ; S < TD
W we will use formulas appearing in
[8, page 236] . Denote τ (x) = inf t ≥ 0 : sup0≤s≤t Ws = x . We have equality (note
c
W is denoted
that in the notation of [8] S is denoted by ξ with parameter β = ν, TD
by T and c is denoted by a)
P
c
sup Wt > x, S < TD
W
0≤t≤S
c
c
= P (τ (x) < S < TD
W ) = P (τ (x) < S ≤ TD
W)
c
c
= P (τ (x) ≤ TD
W, τ (x) < S) − P (τ (x) ≤ TD
W < S)
c
exp (−θµ (ν) x) [1 − E exp (−νTD
W )]
= exp (−θµ (ν) x) 1 − e−µc Vµ (ν) /θµ (ν) ,
=
where we define
θµ (ν) =
p
p
µ2 + 2ν coth c µ2 + 2ν − µ
and
p
µ2 + 2ν
p
.
Vµ (ν) =
sinh c µ2 + 2ν
Thus, for λ such that < (λ) < θµ (ν) ,
θµ (ν) − e−µc Vµ (ν)
c
E exp λ · sup Wt ; S < TD W =
.
θµ (ν) − λ
0≤t≤S
c
Further, by the definition of TD
W we have sup0≤t≤TDc W Wt = WTDc W + c. By this and
On the double Laplace transform
11
c
by the independence of S from TD
W we calculate
"
!
#
"
E exp λ ·
sup
cW
0≤t≤TD
Wt
;S ≥
c
TD
W
!
= E exp λ ·
sup
cW
0≤t≤TD
Wt
#
c
exp (−νTD
W)
c
= E exp λ · WTDc W + c − νTD
W
c
= eλc E exp λ · WTDc W − νTD
W .
Now, utilizing the main result of [8] i.e. equation (1.1), we have
e−µc Vµ (ν)
c
.
eλc E exp λ · WTDc W − νTD
W =
θµ (ν) − λ
Similarly, using symmetry, for λ such that < (λ) < θ−µ (ν) ,
θ−µ (ν) − eµc Vµ (ν)
E exp −λ · inf Wt ; S < TUc W =
0≤t≤S
θ−µ (ν) − λ
and
E exp −λ ·
inf c
0≤t≤TU W
Wt ; S ≥
TUc W
=
eµc Vµ (ν)
.
θ−µ (ν) − λ
Substituting the above formulas into (16) and simplifying, for λ such that < (λ) <
min {θµ (ν) , θ−µ (ν)} we obtain
c
c
E exp (λ · U c (W, S)) |S ≥ TU,0
, TD
W ≥ TUc W
θµ (ν) − e−µc Vµ (ν) e−µc Vµ (ν) θ−µ (ν) − eµc Vµ (ν)
+
θµ (ν) − λ
θµ (ν) − λ
θ−µ (ν) − λ
=
−µc
θµ (ν) − e
Vµ (ν) θ−µ (ν) − eµc Vµ (ν)
1−
θµ (ν) − λ
θ−µ (ν) − λ
−µc
θ−µ (ν) + e
Vµ (ν) − λ
.
(17)
=1+λ 2
λ + 2ν + 2λµ − 2λθ−µ (ν)
c
c
To obtain formula for E exp (λ · U c (−W, S)) |S ≥ TU,0
, TD
W < TUc W we need only
c
c
to change µ into −µ in the formula for E exp (λ · U c (W, S)) |S ≥ TU,0
, TD
W ≥ TUc W .
To calculate probabilities appearing in expression (9) for MT V c (W,S) (λ) , i.e.
c
c
c
P S ≥ TU,0
, TD
W ≥ TUc W = P (S ≥ TUc W, TD
W > TUc W )
and
c
c
c
c
P S ≥ TU,0
, TD
W < TUc W = P (S ≥ TD
W, TD
W < TUc W ) ,
c
we will use results of [2]. Since S is independent from (TD
W, TUc W ) , we have
c
P (S ≥ TUc W, TD
W > TUc W )
P (S ≥
c
c
TD
W, TD
W
<
TUc W )
c
= Ee−νTU W I{T c W >T c W } ,
D
= Ee
c
−νTD
W
U
I{T c W >T c W } .
U
D
12
R. M. LOCHOWSKI
Using formula just below formula (19) in [2], with y = 0, we get
c
c
Ee−νTD W I{T c W >T c W } = 1 − L−W
(ν; c) Ee−νTD W ,
0
U
D
where (cf. [2, last but one formula on the page 389]) we have


p
p




µc
2
2
µ + 2ν
µ + 2ν
e θµ (ν)
−W
L0 (ν; c) =
−
p
2
p


2ν
 sinh c µ2 + 2ν
sinh c µ2 + 2ν 
Vµ (ν) µc
(e θµ (ν) − Vµ (ν)) .
2ν
=
Thus
c
c
P (S ≥ TD
W, TD
W < TUc W ) =
1−
Vµ (ν) µc
(e θµ (ν) − Vµ (ν))
2ν
e−µc Vµ (ν)
θµ (ν)
and similarly
P (S ≥
c
TUc W, TD
W
>
TUc W )
eµc Vµ (ν)
Vµ (ν) −µc
= 1−
e
θ−µ (ν) − Vµ (ν)
.
2ν
θ−µ (ν)
Now, by (9), (17) and calculations above, we have
MT V c (W,S) (λ)
θ−µ (ν) + e−µc Vµ (ν) − λ
c
=
1+λ 2
P (S ≥ TUc W, TD
W > TUc W )
λ + 2ν + 2λµ − 2λθ−µ (ν)
θµ (ν) + eµc Vµ (ν) − λ
c
c
+ 1+λ 2
P (S ≥ TD
W, TD
W < TUc W )
λ + 2ν − 2λµ − 2λθµ (ν)
c
c
c
+1 − P (S ≥ TUc W, TD
W > TUc W ) − P (S ≥ TD
W, TD
W < TUc W )
!
2
Vµ (ν)
Vµ (ν)
µc Vµ (ν)
e −
θ−µ (ν) + e
2ν
2ν
θ−µ (ν)
!
2
Vµ (ν)
Vµ (ν)
−µc
−µc Vµ (ν)
e
−
θµ (ν) + e
.
2ν
2ν
θµ (ν)
θ−µ (ν) + e−µc Vµ (ν) − λ
= 1+λ 2
λ + 2ν + 2λµ − 2λθ−µ (ν)
+λ
θµ (ν) + eµc Vµ (ν) − λ
λ2 + 2ν − 2λµ − 2λθµ (ν)
µc
Thus we have obtained
Theorem 3.1. Let W be a standard Wiener process with drift µ and S be an exponential random variable with density νe−νx I{x≥0} , independent from W. For any complex
λ such that < (λ) < min {θµ (ν) , θ−µ (ν)} one has
E [exp (λ · T V c (W, S))]
−µc
=
1+λ
+λ
(18)
θ−µ (ν) + e
Vµ (ν) − λ
2
λ + 2ν + 2λµ − 2λθ−µ (ν)
θµ (ν) + eµc Vµ (ν) − λ
+ 2ν − 2λµ − 2λθµ (ν)
λ2
2
!
Vµ (ν)
Vµ (ν)
Vµ (ν)
θ−µ (ν) + eµc
2ν
2ν
θ−µ (ν)
!
2
Vµ (ν)
Vµ (ν)
Vµ (ν)
e−µc −
θµ (ν) + e−µc
.
2ν
2ν
θµ (ν)
eµc −
On the double Laplace transform
13
3.2. Examples of applications of formula (18)
3.2.1. The first two moments of the truncated variation process of the Brownian motion
with drift stopped at exponential time Differentiating formula (18) we obtain
∂
ET V (W, S) =
MT V c (W,S) (λ)
∂λ
c
=
λ=0
Vµ (ν)
cosh (cµ)
ν
(19)
which agrees with the relation
T V µ (W, S) = U T V c (W, S) + DT V c (W, S)
(20)
and already obtained in [2] formulas
EU T V c (W, S) =
eµc Vµ (ν)
,
2ν
EDT V c (W, S) =
e−µc Vµ (ν)
.
2ν
(21)
Similarly, we calculate
2
ET V c (W, S)
∂2
c
M
(λ)
T V (W,S)
∂λ2
λ=0
Vµ (ν)
(Vµ (ν) + cosh (µc) θµ (ν) + eµc µ) .
ν2
=
=
(22)
Remark 3.1. Inverting formulas (19) and (22), similarly as it was done with formula (21) in [2, subsection 4.1] we may calulate the first and the second moment of
T V c (W, T ) , where T > 0 is deterministic. To invert (22) one may use formulas from
[1, page 642].
3.2.2. Covariance of the upward and downward truncated variation processes of the
Brownian motion with drift stopped at exponential time Using (20), (22) and (21)
as well as results of [2, subsection 4.3] we are able to calculate the covariance of
U T V c (W, S) and DT V c (W, S) . Indeed, we have
E (U T V c (W, S) · DT V c (W, S))
1
2
2
2
=
ET V c (W, S) − EU T V c (W, S) − EDT V c (W, S)
2
2
Vµ (ν)
=
,
2ν 2
(23)
14
R. M. LOCHOWSKI
where we have used (22), the folowing formula (cf. [2, subsection 4.3])
Z ∞
2
2
c
EU T V (W, S) =
EU T V c (W, t) P (S ∈ dt)
0
Z ∞
2
= ν
e−νt EU T V c (W, t) dt
0
p
µc
e Vµ (ν) µ2 + 2ν − ν 1 − cosh 2c µ2 + 2ν
p
=
2ν 2 θµ (ν) sinh2 c µ2 + 2ν
=
eµc Vµ (ν) θ−µ (ν)
2ν 2
(24)
and the symmetric formula
2
EDT V c (W, S) =
e−µc Vµ (ν) θµ (ν)
.
2ν 2
Now we have
Cov (U T V c (W, S) , DT V c (W, S))
= E (U T V c (W, S) · DT V c (W, S)) − EU T V c (W, S) · EDT V c (W, S)
2
=
=
Vµ (ν)
eµc Vµ (ν) e−µc Vµ (ν)
−
2
2ν
2ν
2ν
2
2
Vµ (ν)
µ + 2ν
=
2 > 0.
p
4ν 2
4ν 2 sinh c µ2 + 2ν
Thus we can observe that the correlation between U T V c (W, S) and DT V c (W, S) is
positive. This is due to the fact that the magnitude of U T V c (W, S) and DT V c (W, S)
is highly dependent on the value of S.
3.2.3. Covariance of UTV and DTV of the Brownian motion with drift Performing
similar calculations to those in [2, subsection 4.1] we may simply obtain formulas
for covariance between U T V c (W, T ) and DT V c (W, T ) , where T is deterministic.
Indeed, denoting by L−1
ν (g) the inverse of the Laplace transform of the function
R ∞ −νt
g (ν) = 0 e f (t) dt, i.e. the function f (t) , we get
L−1
ν −3 = t2 /2,
ν


2
2ν
+
µ

p

L−1
ν
sinh2 c 2ν + µ2


2
2
ν
+
µ
/2

p

= L−1
ν
sinh2 c 2 (ν + µ2 /2)
!
2ν
−µ2 t/2 −1
√ = e
Lν
sinh2 c 2ν
On the double Laplace transform
15
and, by the second formula on page 642 in [1]
L−1
ν
2ν
√ 2
sinh c 2ν
=
2
∞
X
Γ (2 + k) e−(2c+2kc) /(4t)
2c + 2kc
√
√
4
D3
t
2πt2 Γ (2) k!
k=0
=
∞
2 2
8c X
2 4 (k + 1) c − 3t −2(k+1)2 c2 /t
√
e
,
(k + 1)
t7/2
2π k=0
!
where D3 denotes the parabolic cylinder function of order 3.
Now, by (23) and the Borel convolution theorem we obtain
E (U T V c (W, T ) · DT V c (W, T ))




Z T
2
2
2
2ν
+
µ
1
(T
−
t)
2ν
+
µ


p
 =
p
 dt
= L−1
L−1
ν
ν
2
2
2
3
2
0
2ν sinh c 2ν + µ
sinh2 c 2ν + µ2
Z T
∞
2 2
2c X
2
2 4 (k + 1) c − 3t −µ2 t/2−2(k+1)2 c2 /t
= √
(k + 1)
(T − t)
e
dt.
(25)
t7/2
2π k=0
0
Finally, by (25) and [2, formula (28)] (notice that in [2, formula (28)] and in [2, formula
(27)] the term µ2 t shall be changed into µ2 t/2)
Cov (U T V c (W, T ) , DT V c (W, T ))
Z T
∞
2 2
2c X
2
2 4 (k + 1) c − 3t −µ2 t/2−2(k+1)2 c2 /t
= √
(k + 1)
(T − t)
e
dt
t7/2
2π k=0
0
!2
∞ Z
2
1 X T
(2k + 1) c2 − t −µ2 t/2−(2k+1)2 c2 /(2t)
−
(T − t)
e
dt .
(26)
2π
t5/2
k=0 0
Numerical experiments show that formula (26) gives negative numbers, but the strict
proof of this fact is not known to the author.
Remark 3.2. Inverting formula (24) - using the second formula on page 642 in [1] and
just calculated covariance - it is possible to obtain formula for VarU T V c (W, T ) and
hence Cor (U T V c (W, T ) , DT V c (W, T )). However, numerical experiments show that
obtained formulas are rather unstable for small cs. On the other hand, results of [5]
give the exact value of Cor (U T V c (W, T ) , DT V c (W, T )) as c → 0+, namely
1
lim Cor (U T V c (W, T ) , DT V c (W, T )) = − .
2
c→0+
16
R. M. LOCHOWSKI
References
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