# Questions tagged [martingales]

For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.

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### Calculate the expectation of $\tau$=inf{$n\in\mathbb{N}_0|\exists{k}\lt{n}:S_{k}\le-1, S_n=1$}

Let $(Z_k)_{k\in\mathbb{N}} \text{ be a sequence of i.i.d. random variables on } (\Omega,\mathscr{F},\mathbb{P}) \text{ with }$ $$\mathbb{P}(Z_1=-1)=\frac{1}{2}=\mathbb{P}(Z_i=1)$$ and symmetric ...
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### Proving a claim that $Z_n = \max\{(X_n^{(1)})^2, X_n^{(2)}\}$ is a submartingale

Let $X_n^{(1)}, X_n^{(2)}, n \in \mathbb{N}$ be martingales. Let $\mathcal{F}_n$ be a filtration. I believe that $Z_n := \max\{X_n^{(1)}, X_n^{(2)}\}$ is a submartingale. I tried to show it by using ...
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### If $E[Y|X]=X$ and $Z$ independent of $(X,Y)$, what is the conditional distribution of $E[Y|X,X+Y+Z]$ given $(X,Y)$?

Let $X,Y$ be random variables such that $E[Y|X]=X$ and $Z$ independent of $(X,Y)$. Let $W=X+Y+Z$, and $V=E[Y|X,W]$. $V$ is a random variable that depends on $X$ and $W$. What is the conditional ...
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### Expected number of steps before 1-d biased random walk hits a number, given it's before another number [duplicate]

We have a biased 1-dimensional random walk on the number line. Each timestep, with probability $p$ such that $0<p<1$, it increments by $1$, otherwise it decrements by $1$. Conditioning on the ...
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### Probability that a biased 1-d random walk hits a number before another

We have a biased 1-dimensional random walk on the number line. Each timestep, with probability $p$ such that $0<p<1$, it increments by $1$, otherwise it decrements by $1$. Given positive ...
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### some details questions about doob's maximal inequality

Q1 I was so confused about this maximal sequence, I thought it was a constant random variable sequence at first, but then I realised it should be an increasing sequence (not really sure). Then I have ...
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### How to prove the Martingale's property if we have a special stopping time?

$(X_n)$ is a sequence of $(F_n)$-adapted integrable random variables, where $(F_n)$ is a Filtration and $X_0=0$. I have to prove that 1)$X_n$ is a martingale with a respect to $F_n$ iff 2)for any ...
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### Utilize the discrete-time martingale central limit theorem to prove the continuous-time case

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$. We know that if $(M_n)_{n\in\mathbb N_0}$ is a square-integrable ...
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### proving stochastic processes are martingale(Ito Integral)

I need to show that two random processes $\{X_t\}_{t\geq0}$ are martingale. (1) $X_t=\int^t_af(u)dB_u$ ($f$ is continuous) (2) $dX_t=b(t,X_t)dB_t$ ($b$ is continuous) My attempt: To prove the second ...
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### Can you factor out a known Brownian Motion during integration?

Say I have a problem which is of the form, where $W(t)$ is a Brownian Motion and $f(a,b)$ is just some function where we will plug in terms $u, W(u)$. For this example, let's say that $f(a,b) = b-a^2b$...
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### proof of the martingale central limit theorem

Let $(M_n)_{n\in\mathbb N_0}$ be a square-integrable martingale on a filtered probability spacce $(\Omega,\mathcal A,(\mathcal F_n)_{n\in\mathbb N_0},\operatorname P)$ with $M_0=0$ almost surely such ...
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1 vote
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### central limit theorem for continuous-time martingales

I'm searching for a proof of the following result: If $(M_t)_{t\ge0}$ is a square-integrable martingale with statinary increments such that $$\frac1t[M]_t\xrightarrow{t\to\infty}\sigma^2$$ in $L^1$ ...
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### What is the probability that a 2D symmetric random walk will reach a point $(x,y)$ before returning to the origin?

Let $p_n(\vec{x})$ be the probability that a symmetric random walk (SRW) on $\mathbb{Z}^n$ starting at the origin reaches the point $\vec{x}\in\mathbb{Z}^n-\{0\}$ before returning to the origin. What ...
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### Two questions about conditional expectations

When $Y$ is $\mathcal{F}$-measurable, what are the minimal conditions for $E[XY|\mathcal{F}]=YE[X|\mathcal{F}]$ to hold? Is it enough to know that both $X$ and $XY$ are $L^1$ or do we need some ...
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### Questions of the Proof of Kolmogorov's Strong Law of Large Numbers

From the Book of Jacod-Protter (Probability Essentials). Theorem: Let $(X_j)$ be an independent and identically distributed sequence with $E\{|X_j|\} < \infty$. Then \begin{equation*} \lim_{n \...
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### Jump process: Integrand Phi(s) is left-continuous and adapted

Below is the statement on a theorem left continuous process in a jump process (Poissoon): Source book: Stochastic Calculus Bk 2 Continuous Time model Theorem: Assume that the jump process X(s) (...
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