I am given a continuous martingale $M_t$ with $M_0 = 0$ and a trading strategy $H$ (a left-continuous stochastic process who is here bounded also). I have a sequence of partitions $t_i$ $\in \Pi^n$ $\in [0,T]$(who converges towards $0$ in the limit).
The statement I have to proof is:
For any sequence $ (\Pi^n)_{n \in N} $ it holds :
$\mathcal{P}\left[ \int_0^t H_s d M_s = lim_{n \to \infty} \sum_{t_i \in \Pi^n} H_{t_i} ( M_{t_{i+1} \land t} - M_{t_i \land t})\right] = 1 ,t \in [0,T]$
I can (and probably should) make use of the following statement:
If $lim_{n \to \infty}H^n \to 0 (pointwise)$ then it holds that $\mathcal{P}\left[ lim_{n \to \infty} \sup_{t \in [0,T]} |(H^n \bullet M )_t| = 0\right] = 1$
($X \bullet Y$) stands for the Itô-integral.
I am thinking about this for almost a week and do not really have any idea how to approach this proof. I do not know how the additional 'statement to use' helps (I guess I should somewhere arrive where I can either (or both) deduct that my statement to proof is smaller than a sum of supremas and/or that my H is also pontwise converging to 0 ). I assume it has to because I would directly conclude that it follows directly by the boundedness and continiuty of H and M, but there has to be some steps inbetween. A steering in the right direction would be really helpful (and I would be thankful for it).
Edit: With the hint of Brian, define a sequence of trading strategies given by $H_t^n=H_t−H_{max \left[ t_i∈Π^n∣t_i≤t\right] } $ and lets compute $H^n \bullet M = lim \sum H_{t_i}^n (M_{t_{i+1} \land t} - M_{t_i \land t})$ = $lim \sum (H_t−H_{max \left[ t_j∈Π^n∣t_j≤t_i\right] ) } ((M_{t_{i+1} \land t} - M_{t_i \land t})$
If we take this in probability we should arrive (somehow) at $\mathcal{P}\left[ \int_0^t H_s d M_s = lim_{n \to \infty} \sum_{t_i \in \Pi^n} H_{t_i} ( M_{t_{i+1} \land t} - M_{t_i \land t})\right] = 1$
I do not know how to arrive at this statement from the calculation above. Another hint would be appreciated.