# Tagged Questions

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### Uniformly integrable martingale

I have the following martingale. $M_n=\exp\left(aB_n-\frac{1}{2}a^2n\right)$ for $n\geq0$ and $a\neq0$, $B_n$ is a BM. I have to show that for $a>0$, $M_n\rightarrow0$ in probability. Is $M_n$ ...
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### Doob's decomposition of a brownian motion.

Let $B_n$ be a discrete Brownian motion. I need to find the Doob decomposition for ($B_n^2$). Can someone help me please. Thank you in advance.
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### A right-inverse of Brownian motion local time at zero has stationary independent increments

Let $L_0^t$ be the local time for a standard Brownian motion at $0$ and define $$X_t=\sup\{s\ge0:L_0^s\le t\}, t\ge0.$$ I would like to show that $(X_t)$ has stationary independent increments. That ...
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### Find the generator of Markov Process

Homework question: Consider the Markov process $X_t=B_t-t^2+t$ where $B_t$ is the Brownian motion. Find the generator $Q$ of this process. I am completely confused how to find the generator for ...
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### Convergence to integral: $\sum_{k=1}^{k_n}f\left(B_{t_{k-1}^{(n)}}\right)\left(B_{t_{k-1}^{(n)}}-B_{t_{k}^{(n)}}\right)^2 \to_p \int_0^Tf(B_t)dt$

The problem goes: Let $(B_t)$ be a standard Brownian motion, and $f:\mathbb{R}\to\mathbb{R}$ be continuous. Show that if $T>0$ and $(P_n)$ is a sequence of partitions of $[0,T]$: ...
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### Show that this is a stopping time

Show that $\sigma=\inf \{ t\ge 0 : |B_t|= \log t \}$ is a stopping time with respect to $(\mathcal F_t^B)_{t\ge0}$. I've been trying to put the set $\{\sigma\le t\}$ equal to a countable union and ...
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### Brownian motion minimisation problem

Let $B_t$ be a Brownian motion, let $\sigma > 0$ be fixed and let $X_t$ be a process with fixed beginning value $x_0$ that satisfies $dXt = u_tdt + \sigma X_tdB_t.$ Solve ...
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### Can anyone explain me this proof about a Brownian Motion?

Prove that the process $W_t=(1+t)U_{t/(1+t)}$ on $[0,\infty)$ is a Brownian motion. $\text{(b)}$ Clearly $Y_0=U_0=0$, and inherits continuity of sample paths from $U_t$ (and hence from $W_t$). ...
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### Distribution of Brownian Bridge

PROBLEM $U_t = B_t - tB_1$, $B_t$ is a Brownian motion on $[0,1]$. What is a Brownian Bridge and give the twodimensional distributions of the vector $(U_s, U_t)$. I think that a Brownian ...
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### Variance of stochastic integral of brownian motion

How do i compute this integral? $Var [\int_0^T W(t)dW(t)]$ I know the following $E [\int_0^T W(t)dW(t)]$ is 0 but i'm not sure how to apporch the above
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### The first two moments of $\int_0^1 B_s^2 \, ds$

I was trying to solve the following problem from Continuous Martingales and Brownian Motion by Daniel Revuz and Marc Yor, but got my solution back as the answer for variance was wrong. I have already ...
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### Expected value of brownian motion for all positive paths

I've got this question but I can't figure it out. Derive the expected value of $B(t_1)$ of all paths that are positive $t_1$ and calculate the expectation for $t_1=1$ and variance$=1$? Thanks
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### Checking for Martingales on Stochastic processes

I am confused about how to check whether a process is Martingale. I know, I have to check for clear drift but a bit confused about to approach this problems. I need to apply Ito's first i think. For ...
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### two r.v sharing the same law

I have a question: Let $X=B^{+}$ or $X=|B|$ where $B$ is the standard Brownian motion. Set $$J_p=\sup_{t\geq 0}(X_t-t^{\frac{p}{2}})$$ where $p>1$ and $q$ its conjugate ...
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### d-dimensional Brownian motion and martingales

I was solving questions from the Martingales chapter in "Stochastic Processes" by Richard Bass. There was a question regarding d- dimensional Brownian motions(BM): Let $(W_t^1,...,W_t^d)$ be a d ...