Sharp maximal inequalities for conditionally symmetric martingales and Brownian motion
Authors: Wang Gang 
Published: June 2010

Let

B
=

(

B
t

)

t
≥
0

B = {({B_t})_{t \geq 0}}

be a standard Brownian motion. For

c
>
0

c > 0

,

k
>
0

k > 0

, let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper T left-parenthesis c comma k right-parenthesis equals inf left-brace right-brace colon greater-than-or-equal-to greater-than-or-equal-to t 0 colon greater-than-or-equal-to greater-than-or-equal-to minus minus of maxmax less-than-or-equal-to less-than-or-equal-to stBs times times cBtk comma 2nd Row upper T Superscript asterisk Baseline left-parenthesis c comma k right-parenthesis equals inf left-brace right-brace colon greater-than-or-equal-to greater-than-or-equal-to t 0 colon greater-than-or-equal-to greater-than-or-equal-to minus minus of maxmax less-than-or-equal-to less-than-or-equal-to st vertical-bar vertical-bar Bs times times c vertical-bar vertical-bar Btk period EndLayout period">

T
(
c
,
k
)
=
inf
{
t
≥
0
:

max

s
≤
t

B
s

−
c

B
t

≥
k
}
,

T
∗

(
c
,
k
)
=
inf
{
t
≥
0
:

max

s
≤
t

|

B
s

|

−
c

|

B
t

|

≥
k
}
.

.

\begin {gathered} T(c,k) = \inf \{ t \geq 0:{\max _{s \leq t}}{B_s} - c{B_t} \geq k\} , \hfill \\ {T^*}(c,k) = \inf \{ t \geq 0:{\max _{s \leq t}}|{B_s}| - c|{B_t}| \geq k\} . \hfill \\ \end {gathered} .

\]
We show that for

c
>
0

c > 0

and

k
>
0

k > 0

, both

T
(
c
,
k
)

T(c,k)

and

T
∗

(
c
,
k
)

{T^*}(c,k)

are finite almost everywhere. Moreover,

T
(
c
,
k
)

T(c,k)

and

T
∗

(
c
,
k
)
∈

L

p

/

2

{T^*}(c,k) \in {L^{p/2}}

if and only if

c
>
p

/

(
p
−
1
)

c > p/(p - 1)

for

p
>
1

p > 1

, and for all

c
>
0

c > 0

when

p
≤
1

p \leq 1

. These results have analogues for simple random walks. As a consequence, if

T
T

is any stopping time of

B
t

{B_t}

such that

(

B

T
∧
t

)

t
≥
0

{({B_{T \wedge t}})_{t \geq 0}}

is uniformly integrable, then both of the inequalities
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row StartAbsoluteValue EndAbsoluteValue sup Underscript s less-than-or-equal-to upper T Endscripts upper B Subscript s Baseline StartAbsoluteValue EndAbsoluteValue Subscript p Baseline less-than-or-equal-to StartFraction p Over p minus 1 EndFraction StartAbsoluteValue EndAbsoluteValue upper B Subscript upper T Baseline StartAbsoluteValue EndAbsoluteValue Subscript p Baseline comma 2nd Row double-vertical-bar sup Underscript s less-than-or-equal-to upper T Endscripts StartAbsoluteValue upper B Subscript s Baseline EndAbsoluteValue double-vertical-bar Subscript upper P Baseline less-than-or-equal-to StartFraction p Over p minus 1 EndFraction double-vertical-bar upper B Subscript upper T Baseline double-vertical-bar Subscript p Baseline comma EndLayout">

|

|

sup

s
≤
T

B
s

|

|

p

≤

p

p
−
1

|

|

B
T

|

|

p

,

‖

sup

s
≤
T

|

B
s

|

‖

P

≤

p

p
−
1

‖

B
T

‖

p

,

\begin {array}{*{20}{c}} {||{{\sup }_{s \leq T}}{B_s}|{|_p} \leq \frac {p}{{p - 1}}||{B_T}|{|_p},} \\ {{{\left \| {{{\sup }_{s \leq T}}\left | {{B_s}} \right |} \right \|}_P} \leq \frac {p}{{p - 1}}{{\left \| {{B_T}} \right \|}_p},} \\ \end {array}

\]
are sharp. This implies that

q
=
p

/

(
p
−
1
)

q = p/(p - 1)

is not only the best constant for Doob’s maximal inequality for general martingales but also for conditionally symmetric martingales (in particular, for dyadic martingales), and for Brownian motion.