Let $X_t$ be a continuous adapted stochastic process and let $X_t^n$ be the mollified process defined by $$ X_t^n= n \int\limits_{(t-\frac{1}{n})^{+}}^{t} X_s \,ds$$ Prove that $X_t^n$ is a semimartingale.
2026-04-08 17:32:29.1775669549
Why is the mollified process a semimartingale?
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At fixed $n$,and for $t>\frac{1}{n}$ it is clear that this process is an adapted (continuous) finite variation process which entails that it is a (continuous) semi-martingale.
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