Let $H$ be a previsible locally bounded process, and let $X$ be a continuous local martingale. If $T$ is a stopping time and $X^T=(X_{t+T}-X_{T},t\geq 0) $ then $$\int_T^{t+T}H_s.dX_s=\int_0^tH_{T+u}dX_u^{T} $$

I'll be grateful for any help in details (step by step ).

What i did :

$$\int_T^{t+T}H_s.dX_s=\int_O^{t}H_{s+T}.dX_{s+T} $$ or we know that $$\int_0^{t}H_s.dX_s=\lim_{n\to \infty}\sum_{i=0}^{P_n-1 }H_{t^n_i}(X_{t^n_{i+1}}-X_{t^n_i}).$$ Best regards, Educ


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