hits b no later than when it hits a.(Note that if τ_{a} = ∞ = τ_{b}then you would have
[τ = τ]
a b
.)
Then the Burkholder Davis Gundy inequality is as follows. Generalizations will be
presented later.
Theorem 62.4.3Let
{M (t)}
be a continuous H valued martingale which is uniformlybounded, M
(0)
= 0, where H is a separable Hilbert space and t ∈
[0,T]
. Then if F is afunction of the sort described in the good lambda inequality above, there are constants, Cand c independent of such martingales M such that
∫ ( ) ∫ ∫ ( )
c F ([M ](T))1∕2 dP ≤ F (M ∗)dP ≤ C F ([M ](T))1∕2 dP
Ω Ω Ω
by the assumptions about F. Therefore, combining this result with the first part,
∫ ( ) ∫
(kC )−1 F ([M ](T))1∕2 dP ≤ F (M ∗) dP
2 Ω Ω∫
( 1∕2)
≤ C Ω F ([M ](T)) dP ■
Of course, everything holds for local martingales in place of martingales.
Theorem 62.4.4Let
{M (t)}
be a continuous H valued local martingale, M
(0)
= 0,where H is a separable Hilbert space and t ∈
[0,T ]
. Then if F is a function of the sortdescribed in the good lambda inequality, that is,
F (0) = 0,F continuous, F increasing,
F (αx) ≤ cαF (x),
there are constants, C and c independent of such local martingales M suchthat
∫ ( ) ∫ ∫ ( )
c F [M ](T )1∕2 dP ≤ F (M ∗)dP ≤ C F [M ](T)1∕2 dP
Ω Ω Ω
where
M ∗(ω) ≡ sup{||M (t)(ω)|| : t ∈ [0,T]}.
Proof:Let
{τn}
be an increasing localizing sequence for M such that M^{τn} is
uniformly bounded. Such a localizing sequence exists from Proposition 62.2.2. Then
from Theorem 62.4.3 there exist constants c,C independent of τ_{n} such that
∫ ∫
c F ([M τn](T)1∕2) dP ≤ F ((M τn)∗)dP
Ω Ω
∫ ( τn 1∕2)
≤ C Ω F [M ](T) dP
The Burkholder Davis Gundy inequality along with the properties of the covariation
implies the following amazing proposition.
Proposition 62.4.6The space M_{T}^{2}
(H )
is a Hilbert space. Here H is a separableHilbert space.
Proof:We already know from Proposition 61.12.2 that this space is a Banach space.
It is only necessary to exhibit an equivalent norm which makes it a Hilbert space.
However, you can let F
(λ)
= λ^{2} in the Burkholder Davis Gundy theorem and obtain for
M ∈ M_{T}^{2}