Consider a filtered probability space $(\Omega, \mathcal{F},(\mathcal{F}_t)_{t \geq 0},P)$ and $(M_t)_{t \geq 0}$ a continuous local martingale (CLM) w.r.t $(\mathcal{F}_t)_{t \geq 0}$.
Let $U$ denote a $\mathcal{F}_0$ measurable random variable.
I want to prove that $(UM_t)_{t \geq 0}$ is a CLM w.r.t $(\mathcal{F}_t)_{t \geq 0}$.
I have managed to prove continuity of sample paths as well as $UM_0 = 0$ without too much trouble.
As for finding a reducing sequence $(T_n)_{n \in \mathbb{N}}$ of stopping times, I have guessed that the (or one of the) reducing sequence for $(M_t)_{t \geq 0}$ will work. For a fixed $n \in \mathbb{N}$, I have also shown that $(UM^{T_n}_t)_{t \geq 0}$ is adapted to $(\mathcal{F}_t)_{t \geq 0}$ and that it has the martingale property.
What remains is showing that 1) $UM^{T_n}_t \in \mathcal{L}^1(P)$, as well as 2) the process is uniformly integrable, and I hope some of you can help me. I think maybe for the $\mathcal{L}^1(P)$ we can utilise that $M^{T_n}$ is uniformaly integrable but I can't quite make it work.
Based on the fact that you showed $UM_0 = 0$, I'm assuming that $M_0 = 0$ and $|U|<\infty$ a.s.
I would let $(\tau_n)$ be a localizing sequence for $M$ and define a new sequence of stopping times $T_n := \tau_n 1_{|U| \le n}$. Since $T_n \le \tau_n$, we have that $(T_n)$ is also a localizing sequence for $M$. Without loss of generality, we can assume that $\sup_{t \ge 0} |M_t^{\tau_n}| \le n$ for each $n$, so $|U M_t^{T_n}| \le n^2$. Hence $UM^{T_n}$ is bounded and therefore uniformly integrable.