Solve Ito integral for continuous contributions to stock portfolio

stochastic-processes stochastic-calculus stochastic-integrals

I believe the solution may be:

$$ P_T = \frac 1 \mu \left( \exp \left[ \left(\mu-\frac 1 2 \sigma^2\right)T + \sigma\sqrt{T} \mathcal N \right] -1 \right) $$

where $\mathcal N \sim \text{Normal}(0,1)$ so $P_T$ is a linear function of a LogNormal dsitribution.

This gives the correct result for the expected value and the case where $\sigma = 0$, but I'm not sure how to actually derive it.

$$ \mathbb E[P_T] = P_{T\ \sigma=0} = \frac 1 \mu \left( e^{\mu T} - 1 \right) $$

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