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# Questions tagged [local-martingales]

For questions about local martingales (in continuous time).

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### Showing that $M^T(N-N^T)$ is a continuous local martingale for all stopping times $T$ and continuous local martingales $M,N$

Given is that a cadlag adapted process $X=(X_t)_{t\geq 0}$ is a martingale if and only if $\mathbb{E}X_T=\mathbb{E}X_0$ and $X_T\in L^1$ for every bounded stopping time $T$. Now let $M,N$ be two ...
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2 votes
1 answer
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### Can we always find a martingale part in a càdlàg supermartingale?

Abstract: By Doob-Meyer decomposition theorem, for any càdlàg supermartingale $Z$, there exists a unique predictable increasing process $A$ starts from $A_0=0$ such that $Z+A$ is a local martingale ...
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2 votes
1 answer
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### A local martingale subtract half its quadratic variation tends to negative infinity

Let $M$ be a continuous local martingale with $M_0=0$ and $[M]_\infty=\infty$ almost surely. I am required to show that $M_t-\frac{1}{2}[M]_t\to-\infty$ almost surely as $t\to\infty$. Unfortunately I ...
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5 votes
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1 vote
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### Is there a problem of the proof about local martingale is Martingale?

In the definition of local martingale, an adapted process $M=(M_t)_{t\ge 0}$ with continuous sample paths and vanishing at $0$ is called a continuous local martingale if there exists a non-decreasing ...
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