# Questions tagged [local-martingales]

For questions about local martingales (in continuous time).

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### About continuous local martingales, question on Le-Gall's book

Background Hello, I'm working on question 4.24 on Le-Gall's Brownian motion(...) and I would ask you to check if my ideas are correct. The question is as follows: $(M_t)$ is a cont. local ...
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### Applyin Itô's formula in function of quadratic variation

I am learning some basic stochastic calculus and came across the following exercise: Consider a local martingale $M$ with continuous trajectories. Let $Z_t = \exp(M_t −0.5[M]_t)$. Show that Z ...
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### $X=M+A$ bounded semimartingale, then $M$ and $A$ are bounded? Counterexample?

Let be $X=M+A$ a semimartingale with $M$ being a local martingale and $A$ an adapted process a finite variation. If $M$ and $A$ are bounded, then of course $X$ is bounded as well. Is the converse true?...
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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 ...
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### expectation involving quadratic variation inequality

Let M be a local martingale and let $\tau_n$ be a localising sequence. $1)$ $\liminf_nE[ \langle M \rangle_{\tau_n \land t}] \le E[\langle M_t \rangle]$, is this actually an inequality or an equality?...
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### Stopping Time with local martingale

I am working on Karatzas&Shreve Brownian Motion and Stochastic Calculus P.36, Problem 5.19. • (i) A local martingale of class DL is a martingale. • (ii) A nonnegative local martingale is a ...
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### When is the limit of a parameterized family of martingales also a martingale?

Suppose I have a family of martingales continuous martingales $(X^\alpha)_{\alpha \in (0,\infty)}$, that is $X^\alpha = (X^\alpha_t)_{t \geq 0}$ is a continuous martingale, for each $\alpha$. Are ...
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### Bound on supremum of local martingale

Let $M$ be a continuous local martingale starting at $0$. How can I prove $$P(\sup_{s\leq t}M_s>a,\langle M\rangle_t\le b)\leq 4\frac b{a^2}$$ for all $a,b>0$ and $t>0$?
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### Martingality of a Doleans-Dade exponential local martingale.

A paper I read recently seems to make the following statement: if $\gamma_t$ is a progressively measurable process, and that $\exp\left(\int_0^T\gamma_s dW_s\right) \in L^p$ for some $p>1$, then ...
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### If $\mathbb{E}[\underset{0 \leq s \leq t}{\sup} \rvert X_s \rvert] < \infty$ for $X_t$ a local martingale, is $X_t$ a proper martingale?

Let $X_t$ be a continuous local martingale. Suppose $\mathbb{E}[\underset{0 \leq s \leq t}{\sup} \rvert X_s \rvert] < \infty$ for any $t > 0$. Is it true that $X_t$ is a proper martingale? We ...
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### Prove that a process (given through rsdes) is a martingale.

i have a rather complicated problem for a process which is connected to a kalman-bucy filter with a riccati equation. More precisely, let $dX_t =A(X_t)dt + R_1^{1/2} dWt, dY_=BX_tdt + R_2^{1/2}dV_t$ ...
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### How can I show that the stochastic integral of a jump process w.r.t. Brownian motion is a local martingale by using this special localizing sequence?

Suppose that $Y$ is a pure jump process with $N_t$ jumps in $(0,t]$ and $E[N_t]<\infty$. Denote the jump times by $T_i$. Let $W$ be a Brownian motion. If $T_0=0$, then \begin{equation} M_t=\int_0^...
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