#sde

A variance-exploding forward process $d\mathbf{x} = g(t)\,d\mathbf{w}$ uses the diffusion coefficient $g(t) = \lambda^t$ with $\lambda = 25$, for $t \in [0, 1]$. Given $\mathbf{x}_0$, the noisy state is $\mathbf{x}_t \sim \mathcal{N}(\mathbf{x}_0, \sigma_t^2 I)$ with $\sigma_t^2 = \int_0^t g(\tau)^2\,d\tau$. What is the noise standard deviation $\sigma_1$ at the end of the process?