# Questions tagged [martingales]

For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.

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### Two exponential processes are martingales w.r.t the same measure?

Consider a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F_t}\},P)$. Define a process $$M_t=\exp\left(\int_0^t \lambda_sW_s-\frac{1}{2}\int_0^t \lambda_s^2ds\right)$$ where $W_t$ is a $P-$...
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### Quadratic variation condition and square integrability

Assume $M$ is local continuous martingale started from $0$. How would one go about showing that if it is a martingale bounded in $L^2$, then $E[\langle M\rangle_{\infty}]\lt \infty$?
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### Stochastic process problem solve with R studio [closed]

I want to find a task within stochastic processes (martingales, SBM, stochastic integral, and so on), which I can solve on paper and in Rstudio. Can you suggest something with a solution? Thanks in ...
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### Give a complete list of the functions of Brownian motion alone, g(Wt), which are martingales [closed]

As it says in the title. Confused by the wording really, thanks.
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### construct a martingale from given conditions

If there exists a sequence of non-negative real numbers $a_n$ with $\sum_{n=1}^{\infty} a_n <\infty$ with $$\mathbb E[X_n\mid \mathcal F_{n-1}]\leq X_{n-1}+a_n,$$ prove $X_n$ converges almost ...
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### Show that the following are Martingale. [closed]

Having some trobule with the following problem. I think I need to expand W(t)= W(t)-W(s)+W(s) Show that the following are Martingale. 1.E[|X(t)|]< ∞ 2.E[X(t)|Fs]=X(s) W(t) is the standard Brownian ...
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### Showing a stochastic process is a (discrete) martingale

Given a sequence of iid random variables $(Y_i)_{i=1}^\infty$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ such that $\mathbb{E}|Y_i| < \infty$ and $\mathbb{E}Y_i = 0$, consider the ...
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### If $\tau_n,\tau$ are stopping times with $\tau=\inf_n\tau_n$, then $\text E[X\mid\mathcal F_{\tau_n}]\to\text E[X\mid\mathcal F_\tau]$
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space. Lemma 1: Let $Y\in\mathcal L^1(\operatorname P)$, $I\subseteq\mathbb R$ be countable and $(\mathcal G_t)_{t\in I}$ be a filtration on ...
Let $X$ be a right continuous Markov process with left limits and generator $L$. Why is $f(X_t)-f(X_0) - \int Lf(X_s) ds$ a martingale for every $f \in D(L)$? Let s<t. \$E^x[M_t^f|F_s]= E^x[f(X_t)-f(...