# Questions tagged [local-martingales]

For questions about local martingales (in continuous time).

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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 ...
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### Apply Ito’s formula to $F(X_t,Y_t,Z_t)$, where $Z_t = X_t Y_t$.

Let $(X)_{t \geq 0}$ and $(Y)_{t \geq 0}$ be continuous semimartingales with values in $\mathbb{R}$. Apply Ito’s formula to $F(X_t,Y_t,Z_t)$, where $Z_t = X_t Y_t$ and $F$ is a $C^2$ function. ...
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### 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 ...
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### 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 ...
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### Expectation of the Inverse Bessel-3 Process

I am studying the inverse Bessel-3 process. I understand that there have been several posts already about this process, but I have a conceptual question about why this process is (strictly) a local ...
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### d-dimensional Brownian Motion

Let $B_{t} = (B_{1}(t), \ldots B_{d}(t))$ be a d-dimensional Brownian motion with $d>3$. Fix $0 < r < R < \infty$. Define $$A_{r}(R) = \{x \in \mathbb{R}^{d} : r < |x| < R\},$$ where,...
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### When is a this stochastic integral a Martingale, stopping time in the integral.

There are lots of questions on this site asking when Stochastic Integrals are Martingales. I have a similar question : Fix $0<T<\infty$. Given a Brownian Motion $W_t \in \mathbb{R}^d$ on a ...
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### Could an integral w.r.t a local martingale be a proper martingale?

The following is taken from "Stochastic differential equations and diffusion processes" by Ikeda & Watanabe. Let $Y,X\in \mathcal M_2^{c,loc}$ (the space of continuous, square integrable, ...
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### Difference Between Local Martingale and Martingale

I would like to try and distinguish between the two concepts of martingale and local martingale. I have read this answer in Martingale / local martingale : some confusion which was a good start and ...
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### Prove that $(Y_{t})_{t\in[0,\epsilon]}$ is a martingale.

I am struggling the following exercise: Processes $X$ and $Y$ are defined as: $$X=\Bigg( \int_{0}^{t}e^{W_{s}^{2}}\mathrm{d}W_{s}\Bigg)_{t\ge 0},$$ Y=\Bigg(\int_{0}^{t}W_{s}^{5}\mathrm{d}X_{s}...
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### How to show this is not a martingale.

Be advised that I cross-posted this question on MathOverflow. You can find it in this link: https://mathoverflow.net/questions/352152/show-that-this-process-is-not-a-martingale Assume we have the ...
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### Local martingale property for stochastic integrals.

First we start with a Brownian Motion $B(t)$, $a\leq t\leq b$ and an admissible filtration for the BM $\{\mathcal F_t\}$. Let $f(t,\omega)$ be a stochastic process satisfying: $f(t)$ is adapted to ...
Knowing that M and N are two local martingales. Can I do the following simple reasoning for claiming that $N^\tau(M^\tau - M)$ is a local martingale? : Since a stopped local martingale is still a ...
In the book "Semimartingale theory and Stochastic Calculus" from He, Wang and Yan I found the following integration by parts formula for semimartingales (Theorem 9.33): If $X$ and $Y$ are ...