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Suppose I have $X_t=\sinh (W_t+t)$. I am not sure how to show if this is a submartingale, and how to calculate its expectation.

I don't want to integrate this against the normal distribution to find the expectation. Does anyone have a short cut to it? Thanks!

1 Answers 1

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For $0\leq s\leq t$, simple calculations give \begin{eqnarray*}\mathbb{E}(\sinh(W_t+t)\ |\ {\cal F}_s) &=&{\mathbb{E}(\exp(W_t+t)\ |\ {\cal F}_s)-\mathbb{E}(\exp(-W_t-t)\ |\ {\cal F}_s)\over 2}\cr &=&{\exp(W_s+t)-\exp(-W_s-t)\over 2}\exp((t-s)^2/2)\cr &=&\sinh(W_s+t)\exp((t-s)^2/2). \end{eqnarray*}

  1. Setting $s=0$, we get the expectation $\mathbb{E}(\sinh(W_t+t))=\sinh(t)\exp(t^2/2)$.

  2. Also since $\sinh$ is an increasing function, we have $\mathbb{E}(\sinh(W_t+t)\ |\ {\cal F}_s)=\sinh(W_s+t)\ \exp((t-s)^2/2)\geq \sinh(W_s+s)$ which gives the submartingale property.