Suppose, that $\mathbb{H}^{2}$ is the space of all predictable stochastic processes X, such that: $||X||^{2}_{\mathbb{H}} = \mathbb{E} \left[ \int_{0}^{T} |X_{t}|^{2} dt\right] < \infty$.
We have the sequence of stochastic processes $(Y^{k})_{k \in \mathbb{N}}$, which satisfies following assumptions:
- $Y^{k}$ convergences to Y in $\mathbb{H}^{2}$.
- $\sum_{k} ||Y^{k+1} - Y_{k}||^{2}_{\mathbb{H}} < \infty$
Now the question is that if under these assumptions $Y^{k}$ converges to $Y$ almost sure?
In other words, how can I proof the almost sure convergence of the solution of backward stochastic differential equation (corollary 2.1 from http://horst.qfl-berlin.de/files/ElKaroui-Peng-Quenez-BSDEinFinance.pdf)