I'm a beginner in stochastic processes and I want to solve the following stochastic equation:
$$X_t = 1 + \int_0^t X_s \sigma_s\, dB_s,$$
where $B_s$ is a standard Brownian motion. I want to apply Ito's formula and it gives me:
$$d X_t = X_t \sigma_s\, dB_t.$$
Is it a proper way to start solving it? How should I continue the calculation to solve this equation?
Best Answer
From here, I think the easiest way is to look at $\ln X_t$. Ito's formula gives \begin{align*}d\ln X_t &= \frac 1{X_t} dX_t - \frac 1{2X_t^2} dX_t dX_t \\ &= \sigma_t dB_t - \frac 12 \sigma_t^2 dt\end{align*} so we can just integrate to obtain $\ln X_t - \ln X_0 = \int_0^t \sigma_s dB_s - \frac 12 \int_0^t \sigma_s^2 ds$. Since $X_0 = 1$, by exponentiating we obtain $X_t = \exp(\int_0^t \sigma_s dB_s - \frac 12 \int_0^t \sigma_s^2 ds)$.