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5 votes
Accepted

An example such that $(X_n)$ and $(Y_n)$ are martingales with respect to their natural filtration and $(X_n +Y_n)$ is no martingale

Let $(X_n)$ and $(Y_n)$ be two simple random walks defined on the same probability space, generated by a sequence of i.i.d. fair coin tosses $\omega_1,\omega_2,\dots $ with $\omega_i\sim U\{-1,1\}$ ...
Joseph Basford's user avatar
3 votes
Accepted

Counterexample that $(X_n)_n$ is not a martingale

A simple counterexample: $X_n=n$.
Will's user avatar
  • 7,027
2 votes
Accepted

Show $M^2$ is of class D if $M$ is a martingale bounded in $L^2$

After some searching, I figured out the solution to my problem. I post it below in case someone needs it in the future. Lemma 1. Suppose that $Y$ is a nonnegative random variable with $\mathrm{E}(Y)&...
Mingzhou Liu's user avatar
2 votes

Sum of hitting times and hitting times of sum of Brownian Motion

Let $\tau_a=\inf\{t:B_t=a\}$, $a>0$. We have $P(\tau_a\leq t)=P(\sup_{s\leq t}B_s\geq a)$. Let $\Phi,\phi$ be respectively the standard normal cdf and pdf. The reflection principle yields $P(\tau_a\...
Snoop's user avatar
  • 15.6k
1 vote
Accepted

Can we always find a martingale part in a càdlàg supermartingale?

Example: If $B_t$ is a 3-dimensional Brownian motion started at some point other than the origin, then $Z_t:=\|B_t\|^{-1}$ is a positive supermartingale and a (continuous) local martingale. The ...
John Dawkins's user avatar
  • 26.5k
1 vote

Quadratic covariation of martingale transforms to simple processes.

Consider stopping times $T>S$ and stopping times $T'>S'$. Moreover, $\xi_T$ shall be the coefficient for $(S,T]$ and $\varphi_{T'}$ the coefficient for $(S',T']$. Using Proposition 4.10, you ...
user408858's user avatar
  • 2,785
1 vote

How can I show that $\mathbb{E}(T)=\frac{a^2}{\sigma^2}$?

There are other justifications that have to be made if you want to do it rigorously. For example, what is allowing you to conclude that $E(T\wedge t)\xrightarrow{t\to\infty}=E(T)$? What is allowing ...
Mr.Gandalf Sauron's user avatar
1 vote
Accepted

Prove that for symmetric random walk $S_n$ it holds that $S_{min\{n,\tau\}}^2 - \min\{n,\tau\}$ is a uniform integrable martingale

Here's a way to do this by doing a bit of a roundabout argument. I'll denote $\min(a,b)$ by $a\wedge b$ for ease of writing. First see that $\tau\wedge n$ is an increasing sequence of random variables....
Mr.Gandalf Sauron's user avatar

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