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### Using BDG inequality to show the solution to a BSDE belongs to $S^2_{\mathscr{F}}$

By an application of the BDG inequality or Doob's maximal inequality, one can show that $$\mathbb{E}\left[ \sup_{0 \leq t \leq T} |y_t|^2 \right] < \infty$$ This was shown in your other recent ...
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### On continuous modifications being indistinguishable for random fields

Right-continuity with regard to every variable $t_1,\ldots,t_d$ is sufficient to conclude. Indeed by your assumption, the event $A : = \bigcap_{t \in \mathbf{Q}^d}[X_t=Y_t]$ is almost sure. Assume ...
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### How to show that $(X,B)$ and $(Y,W)$ satisfy the same SDE if their joint law is equal?

Suppose $\Big(X,B,\Omega,\mathcal F, \big(\mathcal F_t\big)_{t\geq0}, \mathbb P\Big) \stackrel{\mathcal{Law}}{=} \Big(Y,W,\Theta,\mathcal G, \big(\mathcal G_t\big)_{t\geq0}, \mathbb Q\Big)$ Since $X$ ...
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### Malliavin derivative of stopped Brownian motion

Stopped Brownian motion is not Malliavin differentiable because if it was it would imply that $T$ is constant (see footnote pg.4 Locally Lipschitz BSDE driven by a continuous martingale path-...

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