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• 16.1k
Accepted

Martingale / local martingale : some confusion

Before trying to understand the difference between martingales and local martingales on a technical level, it pays to have an intuitive understanding of the difference: that is what I will attempt to ...
• 30.3k
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Expected stopping time

The linear map $T(x):=(x_1,x_1+x_2,\ldots,x_1+\ldots+x_n)$ taking a vector in $[0,1]^n$ to its partial sums is volume preserving, as it corresponds to a triangular matrix of determinant 1 (or by ...
• 22.1k
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How to show the following process is a local martingale but not a martingale?

We have to show that $(X_t^{S_n})_{t \geq 0}$ is a martingale. Since $$X_t^{S_n} = W_{t/(1-t)}^{T \wedge n} 1_{\{t<1\}} + W_{T \wedge n} 1_{\{t \geq 1\}}$$ this follows if we can prove the ...
• 121k
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"Converse" of optional stopping theorem

Neat question! It is true, if I'm not mistaken. Let $n$ be arbitrary and set $A = \{E[X_{n+1} \mid F_n] > X_n\} \in F_n$. Set $\tau = (n+1) 1_A + n 1_{A^c}$; you may verify that $\tau$ is a ...
• 98.2k

Understanding the $\sigma$-algebra of a sum of random variables

$\sigma\{X+Y\}$ is contained in $\sigma\{X,Y\}$. This because measurability of $X$ and $Y$ implies measuribility of $g(X,Y)$ for any Borel-measurable $g:\mathbb R^2\to\mathbb R$. Involved here is the ...
• 151k
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• 2,727
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If $(X_n)$ is a martingale, prove that $(X_{n\wedge N})_n$ is a martingale, where $N$ is a stoping time

Note that  \begin{aligned} E(X_{N\wedge n+1}| \mathcal F_n) &= E(X_{N} 1_{N\leq n}+X_{n+1} 1_{N\geq n+1}| \mathcal F_n)\\ &= 1_{N\leq n} E(X_{N} | \mathcal F_n) + 1_{N\geq n+1}E(X_{n+1}| \...
• 35.8k
Accepted

Let's begin with a preliminary definition. Definition No free lunch with vanishing risk (NFLVR). We say that a process $S$ satisfies NFLVR if there does not exist a sequence of admissible integrands $... • 13.2k 8 votes Accepted How to apply Ito's Formula to show that this is a martingale? Note$f(x)=(1+\lambda |x|)e^{-\lambda |x|}$is twice continuously differentiable. Indeed:$f'(x)=-\lambda^2xe^{-\lambda |x|}$and$f''(x)=\lambda^2e^{-\lambda |x|}(\lambda |x|-1)$. Furthermore, if$...
• 15.6k
The claim in question is a corollary of a standard SLLN for martingale difference sequences (MDS). SLLN for MDS The statement of SLLN for MDS is as follows. Let $N_t$ be a martingale difference ...