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Conditional Expectation of 3 random variables

Let $X_1,X_0$ be iid Bernoulli fair variables: $P(X_i=0)=P(X_i=1)=1/2$ Let $X_2 = 2 X_1 X_0$. We get $E[X_2| X_1]= 2 X_1 E[X_0] = 2 X_1 \frac12 = X_1$ and $E[X_2]= E[E[X_2| X_1]]= 1/2$. Also $$E[X_1 | ...
leonbloy's user avatar
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Quadratic variation of integral of martingales

For the first question, yes, this is a martingale. This only requires that $\int \mathbb{E}[|M_t(x)|] |\phi(x)| dx < \infty$ for all $t$. Then, by Fubini's theorem, \begin{align*} \mathbb{E}\left[...
user6247850's user avatar
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$u(t,B_t)$ is a martingale if it satisfies a certain condition.

Your self-answer is unsatisfactory. Also: In OP, what do you mean by "a polynomial $u(t,x)$ in $t,x$ such that $$ \frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial^2 u}{\partial x^2}"\...
Kurt G.'s user avatar
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Is a stopped martingale always a martingale?

Yes, if $(M_t)_{t\in\mathbb R_+}$ is a right-continuous martingale and $T$ is a stopping time, then $(M_{t\land T})_{t\in\mathbb R_+}$ also is a martingale, see for instance Corollary 3.24 in Le Gall, ...
Maximilian Janisch's user avatar
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Why are there different definitions of admissibility in the literature, and why do we need admissibility?

Why do we need admissibility? The problem is that you need extra conditions for the first fundamental theorem of asset pricing to hold; the existence of an equivalent martingale measure is not enough ...
Jose Avilez's user avatar
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Dense subset of $L^2$ in the integral representation of $X \in L^2(\Omega)$

The set $\mathcal U$ is not dense in $L^2$ as it contains only non-negative random variables with unit expectation. However, this shows that $\text{span}(\mathcal U)$ is dense in $L^2$, or ...
user6247850's user avatar
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Martingale example: a martingale on the partitioned unit interval

I will work on $([0,1),\mathscr{B}[0,1),m)$ and on interval partitions of the form $([x_{k}^n,x_{k+1}^n))_{0\leq k < M_n}$ where $0=x_0^n<...<x_{M_n}^n=1$. You may want to prove that for all $...
Snoop's user avatar
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Conditional Expectation of 3 random variables

Take any $X_1\neq X_2$ such that $\mathbb E[X_2\vert X_1]=X_1$ and set $X_3=X_1+X_2-\mathbb E[X_1\vert X_2]$. Note that whatever example you take, you'll have $X_3\neq X_2$, see e.g. this question. ...
Will's user avatar
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1 vote

Let $Y_t=t\sin W_t$ where $W_t$ - Wiener process. Calculate $\left< Y \right>_t$.

The one dimensionale Ito's rule for the brownian motion states that for every $F\in C^{2}(\mathbb{R})$ it holds the following $$dF(W_t)=F'(W_t)dW_t+\frac{1}{2}F''(W_t)dt$$ Applying this formula with $...
Davide's user avatar
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Stopping time of a martingale $Y_n=e^{S_n}$

If I understand correctly, you assume that $q=\frac{1}{1+e}$ so $Y_n$ is a martingale? Then, note that $T = \inf\{n : Y_n > 100\} = \inf\{ n : Y_n = e^5\}$. Write $T_j = \min\{j,T\}$ and note that ...
Presage's user avatar
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1 vote

Help me find the formula for this problem please (probably easy!)

It doesn't matter what the odds are. You will always eventually win and if the pay out is 1 to 1 if you double the bet it will work. If the odds are against you that just means it will take longer. ...
fleablood's user avatar
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