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Recently started Steele's Stochastic Calculus text and as I was going through the first chapter.

excerpt

While making a case for why $\tau$ is finite he introduces a super-script of $d$ on the variables without any prior mentions of the same.

Anybody who has studied the text before, could you perhaps afford some clarity on what this denotes and means in this context.

Would be extremely grateful for any responses.

Jose Avilez
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Vivaan Gupta
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    In the first sentence he writes "we can even decide from equation (1.3) that $\tau$ has moments of all orders". Based on this it seems to me that $\tau^d$ is simply $\tau$ to the power $d$ with $d \in \mathbb{N}$. He wants to show that $E[\tau^d] <\infty$, for all $d\in \mathbb{N}$, i.e. $\tau$ has finite moments of all orders. – Oscar Mar 24 '24 at 14:22
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    Also, if you are initiating your study in Stochastic Calculus, consider having a look at the book: "Introduction to Stochastic Integration", by Hui-Hsiung Kuo. It's a great introductory book. – Oscar Mar 24 '24 at 14:25
  • Thanks @Oscar, same question that I asked Jose, what would proving the same achieve? Isn't proving that stronger than what he has set out to prove, i.e $E(\tau)$ is finite. – Vivaan Gupta Mar 24 '24 at 16:06
  • Showing that all moments exist would show in particular that the first moment, i.e. $E[\tau]$, exists. – Oscar Mar 25 '24 at 08:21

1 Answers1

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The superscript $d$ on $\tau$ means the standard thing; i.e. $\tau^d$ is $\tau$ to the power of $d$.

Jose Avilez
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