# Questions tagged [local-martingales]

For questions about local martingales (in continuous time).

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### Pointwise second moment of continuous local martingale

Let $(M_t)_{t \ge 0}$ be a continuous local martingale (as defined in LeGall). Let $\mathbb{E}$ denote expectation. Is $\mathbb{E}[M_t^2] < \infty$ for all $t \ge 0$? If yes, then how does one ...
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### Check that a process is a local martingale

Assume $\sigma:\mathbb{R}^d \times E$ is $C^2$ (on both parameters) and that $X_t=\int_0^t \sigma(X_s,I_s)dW_s$, where $(I_t)_{t \geq 0}$ is some discrete Markov Chain with finite state space $E$ that ...
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### Why is $\ln(\det(B^TB))$ a local martingale for B a Brownian matrix?

I am studying stochastic calculus on my own so this should be a basic question. I saw in https://link.springer.com/article/10.1007/BF01259552 the statement that for $B$ an brownian matrix (with ...
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### Differential form of local martingale and its quadratic variation

I have a question regarding the differential form of a local martingale and its quadratic variation (the source of the question is p. 136-137 in https://galton.uchicago.edu/~mykland/paperlinks/I.A.1-...
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### chapter 1 ex 4.22 from Brownian motion, martingales, and stochastic calculus by Jean-Francois Le Gall

This is ex 4.22 from chapter 1 of''Brownian motion, martingales, and stochastic calculus by Jean-Francois Le Gall''. exercise 4.22: Processes on defined on a probability space $(\Omega,\mathcal{F},P)$...
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### Show that $V_t = \|B_t\|^{-1}$ is a true local martingale

I have some difficulties with solving a problem of stochastic calculus related to local martingale. Here is the problem : Let be $B$ a 3-dimensional Brownian Motion starting at $B_0 = (0,0,1)$ and ...
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Consider the ito-process: $X(t)= \sigma(t) dW(t)$ for $t \in [0,T]$, where $\sigma$ is a predictable process and $\int_{0}^T \sigma^2 dt <\infty$. Consider $X^2(t)$: Appyling Ito's formula to $f(x)=... • 389 1 vote 0 answers 134 views ### Properties of Quadratic Covariation I want to show that for any two continuous local martingales and any stopping time$T$,$\langle M^T, N \rangle = \langle M, N \rangle ^T$. The method suggested in Revuz and Yor as an exercise is to ... • 1,388 1 vote 0 answers 60 views ### Understanding of definition of a local martingale In my lecture note, a local martingale is defined as the picture shows. I am a bit confused with the sequence of the stopping time. I understand this as every element in the sequence is a function ... 1 vote 0 answers 100 views ### Stochastic integral over an$L^2_\text{loc}$function is a continuous local martingale Let$B_t$be any standard Brownian motion and let$f \in L^2_\text{loc}$. Then$W = \int_0^{\cdot} f(\cdot,s)\, dB_s$is a continuous local martingale. This is stated in my lecture notes right after ... • 193 2 votes 1 answer 153 views ### Finding a stochastic process from Ito’s lemme on$Y_t$such that$\int_0^1Y_s\,\mathrm{d}B_s= B_1-2B_1\mathbb{I}_{|B_1|}$Problem: I would like to zoom in on a particular part of a question as a follow-up on a question I have previously asked, which I feel deserves its own space. I was looking at the random variable$$X=... • 2,138 1 vote 1 answer 101 views ### How to determine if the process is martingale? I have that$\{X_t; t=1,2... \}$is an i.i.d random variables such that$P(X_t=1)=p=1/4$and$P(X_t=-1)=1-p=3/4$. Let$S_0=0$and$S_t=\sum_{i=1}^tX_i$for$t=1,2...$Let finally$Y_t=a^{S_t}$for$t=...
I encoutered this problem while self-studying Karatzas&Shreve.I have worked out the equivalence relation between $\Lambda_t$ and $M_t$, which is setting $C_t = e^{-\alpha t}$ and \$C_t = e^{\alpha ...