# Tagged Questions

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### On “for all” in if and only if statements in probability theory and stochastic calculus

1 In my friend's Probability Theory long test there was this question: Let $(\Omega, \mathfrak{F}, P)$ be a probability space on which is defined all sub-$\sigma$-algebras, events and random ...
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### Every Lipschitz function is the primitive of a measurable function

I was doing exercise 5 of this exercise sheet and I don't know how to conclude. I need to prove that if $f \colon [0,1]\to \mathbb{R}$ is Lipshitz, $X$ is a uniform$(0,1)$ random variable and ...
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### Expectations of martingales

Consider a martingale $(M_n)_{n \geq 0}$ adapted to a filtration $(\mathcal F)_{n \geq 0}$ on a probability space $(\Omega, \mathcal F, P)$. Prove that, for each $k \leq n$; $$E(M_n M_k) = E(M_k^2)$$ ...
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### Expectation of a stochastic integral conditioned on a particular σ-algebra

Suppose that $g$ is a simple process in the class $\mathcal{V}=\mathcal{V}[U,T]$. Using the notations $g_k=g(t_k)$, $\Delta B_k = B(t_{k+1})-B(t_k)$, and $\mathcal{F}_k=\mathcal{F}_{t_k}$, with the ...
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### Variance optimal martingale measure in discrete case

In a discrete market, the minimal variance (or variance optimal) martingale measure is defined by the martingale that minimizes the variance of the Radon-Nikodym derivative ...
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### Variance With Martingales Problem - Answered; Ignore the Bounty

Let $(X_{j})_{j \geq 1}$ be random variables such that $X_{j}$ is $\mathcal{F}$-measurable for each $j$, where $(F_{j})_{j\geq 1}$ is an increasing sequence of $\sigma$-algebras. Assume ...
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### Showing that $(X_n)$ is a submartingale if and only if $(-X_n)$ is a supermartingale

I was reading up on submartingales and supermartingales and saw this statement which I do not understand. A stochastic process $(X_n)_{n\geq 1}$ is a submartingale with respect to a filtration ...
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### Is the product of martingales a submartingale?

Suppose we have two square integrable continuous-time martingales X,Y: $X=\{X_t,\mathcal{F}_t; 0\leq t <\infty\}$ and $Y=\{Y_t,\mathcal{F}_t; 0\leq t <\infty\}$. If we consider the cases X=Y, ...
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### Azuma's inequality to McDiarmid's inequality?

I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from ...
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### Fail of optional sampling theorem

Could anyone help me see why the optional sampling theorem ($E(M_{\tau}\mid\mathcal{F}_{\sigma})=M_{\sigma}$ a.s.) fails for certain stopping times $\sigma\leq\tau$ for the not uniformly integrable ...
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### Doob's supermartingale inequality

I'm trying to prove that For a non-negative supermartingale $M$ it holds that for all $\lambda>0$ we have $$\lambda P\{\sup_{n}M_{n}\geq\lambda\}\leq E(M_{0})$$ My idea was to use Markov's ...
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### Martingale not uniformly integrable

I've come across a statement that implies that non-negative martingales for which $\{M_{\tau}\mid \tau \ \rm{stopping} \ \rm{time}\}$ is not uniformly integrable exist. I personally can't think of an ...
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### Integral Martingale

I've come across some statements in measure theory that I don't fully understand. Consider the unit interval $I=[0,1]$ equipped with the Borel-$\sigma$-algebra $\mathcal{B}([0,1])$ and the Lebesgue ...
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### Example of a martingale which is not jointly measurable

Suppose we have a measurable space $(\Omega,\mathcal{F})$ and an $\mathbb{R}$-valued continuous-time (but not necessarily continuous) stochastic process $X$. $X$ is jointly measurable if it is ...
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### Martingale preservation under independent enlargement of filtration

I think this is probably a very easy question but I haven't worked with $\sigma$-algebras in depth for a long time now so am finding myself a little rusty. Would be very grateful if someone could give ...
Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is ...
As I understand, martingale is a stochastic process (i.e., a sequence of random variables) such that the conditional expected value of an observation at some time $t$, given all the observations up to ...