# Tagged Questions

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### Is there an example that shows that the optional stopping theorem fails for finite (unbounded) stopping times?

Is there a martingale $M=(M_t)_{t\geq 0}$ and finite stopping times $S,T$ with $S \leq T$ a.s. such that $\mathrm{E}(|M_T|)<\infty$, but $M_S \neq \mathrm{E}(M_T|\mathcal{F}_S)$ a.s.? I found a ...
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### martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
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### Showing that a certain stochastic process does not have normal distributed increments

Edit: Question Resolved. See below. As a part of my bachelor thesis, I have to work through a paper about fake Brownian motion by Oleszkiewicz. In this paper he defines a stochastic process. Let ...
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### Finding a pre-visible process

Question: Let $W_t$ be a standard brownian motion under P with filtration $\mathscr F_t$. Let: $$M_t=\mathbb E[W_T^2|\mathscr F_t]$$ Show that $M$ is a P martingale. This is simple enough using ...
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### Lookback option with floating strike: boundary condition

I am trying to make sense of one of the boundary conditions of a look-back option with floating strike. Some notation first: let $v(t,x,y)$ denote the price at time $t$ of the option under the ...
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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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### Expectation of $e^{-4B_\tau}$, where $\tau$ is an extended stopping time

This is an specific example so with a bit of luck I can get some general methodology from your answers. I have this stopping time: $$\tau = \inf\{t \geq 0; B_t < t-2 \}$$ This is a clear ...
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### Submartingale example: proof

I am trying to prove if the process $M_t = e^{W_t^2-t}$ is a submartingale ($W_t$ is the Wiener Process). The proof becomes a bit difficult, to the point where I am unsure how to move forward. Let ...
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### Computing expectation of

I am reading a paper and got stuck on this simple equation: $$\mathbb{E}_t[e^{-cS_T}]$$ where $dS_t=\sigma W_t$ with $W_t$ standard 1 dimensional Brownian motion, $S_t=s$ and c some constant. I ...
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### Find the distribution of the maximum of a Wiener Process with negative drift

So.. what I have now is Let $M=max\{W_t; 0\leq t <\infty\}$ since $W_0=0$, $M\geq 0$ with probability 1. So, $P(M>x)=P(T_x<\infty)$ where $T_x$ is the stopping time, so we now use the ...
### $4^{Brownian(t)}$ martingale proof
Let $B(t)$ a Brownian motion. I like to prove that $4^{B(t)}$ = martingale I rewrote the expression into an exponential form (like $\exp(\ln(4) B)$), but then I don't know how to proceed.