Given a process: $$ X_t = (1-t)\int_0^t \frac{1}{1-s} dB_s $$ I am asked to prove the existence of the $L^2$ limit as $t\rightarrow 1^- $ $\mathbb{P}$-a.s. and to calculate this limit. How would you go about doing this?
2026-04-30 01:08:26.1777511306
Finding mean-square convergence of a stochastic process
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$EX_t^{2}=(1-t)^{2}\int_0^{t} \frac 1 {(1-s)^{2}}ds=(1-t)^{2}(\frac 1 {1-t}-1)=1-t-(1-t)^{2}\to 0$. Hence $L^{2}$ limit is $0$.