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As long as your market is complete and $\tau$ is measurable w.r.t. the filtration generated by the market the continuous cash flow paid until $\tau$ is a hedgeable contingent claim and you have to work under the risk neutral measure.


Pricing always takes place under the risk neutral probability measure. In fact, this would make the price more conservative (i.e. lower) with respect to risk; if you priced it under the true measure you would be putting a smaller hazard rate for this random time. Completeness make the risk neutral probability measure unique. In your case you might have ...

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