Referring to the Doob's Maximal inequality we have that letting M t be a continous martingale, λ &gt; 0, p &gt; 1 and T &gt; 0 we have P ( sup 0 ≤ t ≤ T | M t | &gt; λ ) ≤ 1 λ p E [ | M T | p ]. I did not understand if the sup is the sup of the events ( | M t | &gt; λ : t ∈ [ 0 , T ]) or if it is the sup of the random variable | M t | itself.

Question

Referring to the Doob&#039;s Maximal inequality we have that letting       M    t   be a continous martingale,   λ  &amp;gt;  0,   p  &amp;gt;  1 and   T  &amp;gt;  0 we have   P  (      sup          0      ≤      t      ≤      T            |        M    t        |    &amp;gt;  λ  )  ≤      1          λ      p            E    [      |        M    T              |        p    ]. I did not understand if the sup is the sup of the events (      |        M    t        |    &amp;gt;  λ :   t  ∈  [  0  ,  T  ]) or if it is the sup of the random variable       |        M    t        |   itself.

Lilliana Burton · Accepted Answer

Let       M    T    ∗    =      sup          0      ≤      t      ≤      T            |        M    t        |  . Then the claim is that   P  (      M    T    ∗    &amp;gt;  λ  )  ≤      λ          −      p            E    [      |        M    T              |        p    ]. In other words, it is the sup of the random variables       |        M    t        |   for   0  ≤  t  ≤  T.

sedeln5w · Accepted Answer

Thank you

Referring to the Doob's Maximal inequality we have that letting M t </msub> be a

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