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We have $dB_t = rB_tdt$.

We are told that this corresponds to a "bank"..... how? When I insert money into a bank, how does this correspond to buying an asset for the price $B_t$?

It would make more sense to say it is a bond, yet my book insists this is a "bank".

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Well, I do not think there is a large difference: Given you deposit money at a Bank the value of this deposit changes according to $$\frac{dB_t}{B_t} = r dt$$ which simply means there is no uncertainty with respect to this evolution (instead of incorporating a risky component $dW_t$. If you really want to interpret the risk-less asset as a bond you are probably faced to some issues (the bond should not exhibit a maturity but instead should only pay interest). Therefore I would agree that it is best to interpret the risk-less term as a bank deposit.

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    $\begingroup$ Yes, for the Bond interpretation you would have to consider a strategy of constantly rolling over whichever bond(s)/bill(s) in a bond portfolio expire today, which gets messy to explain. $\endgroup$ – noob2 Feb 8 '17 at 20:33

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