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

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Construction of a Martingale from a random walk

A martingale can be constructed from a random walk. Can someone give a numerical example of how this can be done together with some little proof to spice up the example? Thanks in anticipation
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Ito's representation for $L^1$ random variable

Given $(\Omega,\mathbb{F},P)$ where $\mathbb{F}$ is the $P$-complete filtration generated by Brownian motion $W$. Ito's representation says for $X\in L^2(\mathcal{F}_\infty,P)$ with zero mean,there ...
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The definition of Random time. [on hold]

I define a random time of a Martingale $ \lbrace Z_n: n \geq 1 \rbrace $ to be the random variable $ N $ for which there is a function $ f(Z_1, Z_2,..., Z_n) $ such that $ P \lbrace N=n | ...
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Martingale Convergence Theorem Random Walk [closed]

Why doesn't the martingale convergence theorem apply to a zero-mean random walk?
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Can someone, please, suggest some books for Stochastic Processes with exercises?

Can someone, please, recommend me some books about Stochastic Processes,Martingales and Brownian Motion with many exercises? (I would be very happy if some of them are for beginners :D) Thank you!
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1answer
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Expectation of quadratic variation

I got stuck in a step of a proof and need some help. The situation is the following: Let $M$ be a continuous local martingale (which satisfies $\mathbb{E}[\langle M\rangle(T)]<\infty$ - I don't ...
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1answer
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Adapted random variable

Let $\{Y_n\}_{n=1}^{\infty}$ be a sequence of random variables, and let $F_n = \sigma(Y_1,Y_2,\dots,Y_n)$ for each $n$. Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of random variables adapted with ...
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1answer
20 views

Doob Meyer decomposition for Super-martingales

Let $Z$ be a super-martingale with usual Doob-Meyer decomposition: $Z=M-A$. Is it true that : $A\leq M$ and therefore: $\mathbb{E}[A^2]\leq \mathbb{E}[M^2]$ ?
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1answer
48 views

Central limit theorem: where is the martingale in this proof?

Yet another question from the depths of Durrett. Again in the proof of Theorem 8.8.3, the author notes that "by the orthogonality of martingale increments," $$ E \left( \sum_{m=1}^{[nt]} t_{n,m} - ...
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1answer
28 views

Is it true that if $X(t)$ is a martingale then $X^{2}(t)-E[X^{2}(t)]$ is also a martingale?

I'm assuming that $X^{2}(t)-E[X^{2}(t)]$ is integrable.
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1answer
33 views

Martingale CLT: “without loss of generality”?

(Hopefully last in a long series of posts from the "I don't have Rick Durrett's brain" department... apologies.) In Durrett's proof of a a simple martingale CLT (Theorem 8.8.3, p. 341), he loses me ...
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Generalization of the Central Limit Theorem for Local Martingales

the Central Limit Theorem for Local Martingales states the following. Theorem Let $M_n = (M_n(s))_{s \geq 0}$ be a square integrable local martingale such that for all $T > 0$ $$ \lim_{n ...
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Stopped Brownian motion proof

I'm trying to work through a proof in Durrett's textbook of a martingale convergence theorem via an embedding of the martingale in Brownian motion, and am stuck verifying a detail as usual. I'm ...
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2answers
36 views

Show that $X_tY_t=X_0Y_0+\int_0^tX_sdYs+\int_0^tY_sdX_s+[X,Y](t)$, where $X_t,Y_t$ are Ito processes

So I have done this exercise and the proof holds, but I really don't believe it can be correct because the proof is worth twice as much as other exercises. I am also not 100% sure if $d_s ...
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Show that $\mathbb{E}\left(\int_0^1X_n(t)dW(t)-\int_0^1X(t)dW(t)\right)^2\to 0$

I tried to shove this question in another one of my posts as a follow-up question, but I deleted my comment and will post it instead. I would really appreciate if someone could help me spot out and ...
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31 views

Show that $\mathbb{E}[X_t]=X_0e^{-ct}$ if $X_t=X_0e^{-ct}+\sigma e^{-ct}\int_0^te^{cs}dW_s$, $X_0\in\mathbb{R}$

so I know the result is trivially correct, but I am being asked to prove it. I tried using a theorem, but it seems rather contradictory. Thanks in advance! Question: Show that ...
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1answer
20 views

Burkholder's inequality for elementary stochastic integral

An elementary Burkholder's inequality for simple stochastic integral says that given nonnegative martingale $M$ and simple bounded predictable process $H$, it holds that for all $c>0$, the tail ...
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1answer
11 views

Existence of a localizing sequence of stopping times for a continuous local martingale

I have a a question about continuous local martingales: the definition of continuous local martingale says that a continuous process $X_s$ is continuous local martingale if there is non decreasing ...
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1answer
23 views

Show that $\mathbb{E}\left[c_{\tau\wedge n}X_{\tau\wedge n}-\sum_{i=1}^{\tau\wedge n}c_i\mathbb{E}(X_i-X_{i-1}\mid\mathcal{F}_{i-1})\right]\le 0$

I am trying to go through a past exam paper but I don't know how to deal with stopping times since we only did 2 exercises in class... I got stuck, so I would really appreciate if someone could help ...
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1answer
29 views

Calculate $\mathbb{E}[M_{\alpha}^{p}(t)]$ for all $p>0$ and $t>0$, where $M_{\alpha}(t):=e^{\alpha W_t-\frac{\alpha^2}{2}t}$, $t\ge 0$

I am going through this solved problem but I don't understand some steps. My professor is notorious for making errors very often so don't hold back if you think he's wrong... Or if I am wrong. I am ...
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Using Ito theory to decide whether $M^f$ is martingale or a local martingale

I came across the following while reading Ikeda & Watanabe book Stochastic differential equations and Diffusion processes, in page 163-164 At first the sentence $$f(X_t)- f(X_0) - ...
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Martingale and quadratic variation inequality

I have the following inequality $$\mathbb{E}(\mid[M^{\Pi^m},M^{\Pi^m}]_T^{1/2}-[M^{\Pi^n},M^{\Pi^n}]_T^{1/2}\mid^p)\leq \mathbb{E}([M^{\Pi^m}-M^{\Pi^n},M^{\Pi^m}-M^{\Pi^n}]_T^{p/2}),$$ where $M$ is a ...
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1answer
18 views

Martingale and independent increment

I know that in $L^2$ martingale a have independent increments. In particular that $\mathbb{E}[(X_m-X_n)^2]=\mathbb{E}[X^2_m-X^2_n]$ if X is a martingale. Does this extend also for general $p\geq 1$ in ...
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central limit theorem for complex-valued martingale

Does martingale strong law of large numbers and martingale central limit theorem extends to complex-valued martingale?
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33 views

Burkholder-Davis-Gundy inequalities

I want to prove these inequalities, i.e.: For $p\geq 1\ \exists 0<c_p\leq C_p$ such that for any martingale $M$ we have the following inequality: $$c_p\mathbb{E}[[M,M]^{p/2}_\infty]\leq ...
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1answer
58 views

Why is $f(X_t)-\int_0^t Af(X_s) \, ds$ a martingale for a Markov process $(X_t)_{t \geq 0}$?

I think if $A$ is the usual generator for the Markov process $(X_t)_t$ $$A f (x) = \lim_{t \downarrow 0} \frac{\mathbb{E}^{x} [f(X_{t})] - f(x)}{t}$$ then we get that for any "nice" $f$ the process ...
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1answer
30 views

What is a martingale array - its definition and importance?

What is a martingale array? What is the importance of defining such an array, instead of using a martingale itself? A common example of this definition is a martingale difference array.
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Why are functions of semimartingales again semimartingales?

I am trying to prove the Itō's lemma, and need to show that if $X$ is a semimartingale and $f$ is a $\mathcal{C}^2$-function, then $f(X_t)$ is again a semimartingale. How do I do that? I cannot see ...
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27 views

Prove that a succession of random variables is a martingale

I've been working on the following problem: Let $\{{Y_n:n\in \mathbb{N}}\}$ be independent identically distributed random variables with mean $\mu$ and variance $\sigma^2>0$. Define ...
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Stochastic Integral of Simple Predictable Process is a Martingale

Take $H\in S$ to be a simple process defined as: $$H_t:=\sum_{i=1}^{n-1} H_i1_{(T_i,T_{i+1}]}(t),\ \ H_i\in \mathcal{F}_{T_i}, \ (T_1\leq...\leq T_n \ stopping\ times),$$ and $X$ a Martingale. I ...
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Non-martingale with respected to the natural filtration, and satisfies $E[M_{n+1} | M_n]=M_n$ [duplicate]

I am thinking about the exercise: Exercise 5. Give an example of a random sequence ($M_n$) such that $E[ M_{n+1} | M_n ] = M_n$ for all $n\ge0$, but which is not a martingale w.r.t. the filtration ...
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37 views

Show that $\{X_n\}_{n\ge 1}$ is a submartingale with respect to $\{F_n\}_{n\ge 1}$, where $X_n=\left(Z_1+Z_2+…+Z_n\right)^2$

I am trying to do the following exercise from a past exam paper and I am really stuck in it. I know the theory and can prove other cases, but I am not too sure about this one. Any help would be ...
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32 views

Uniform integrability of stopped martingale

Let $(M_t,\mathcal{F}_t)_{t\geq 0}$ be a martingale with continuous paths and $(\tau_k)_{k\geq 0}$ stopping times. Hence we know that $M_{t\wedge\tau_k}=\mathbb{E}[M_t|\ \mathcal{F}_{t\wedge\tau_k}]$. ...
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Martingales and Stochastic Analysis James Yeh Th 8.13

Can someone check if the proof of theorem 8.13 of the book Martingales and Stochastic Analysis by James Yeh is correct (link here: https://goo.gl/ivxJnv, the Google Book version). Note line 11, page ...
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Brownian motion and associated martingales

Under the Wiener measure $\Bbb{W}$ the process $x(t)$ is a brownian motion. This means that $\Bbb{E}[{x(t)-x(s)\mid \mathcal{F}_s}]=0$. Let $P$ be a measure in $C([0,\infty),\Bbb{R}^d)$ such that ...
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40 views

Stochastic process $\exp(W_t - t/2)$ approaches zero for large $t$, but it is a martingale?

The stochastic process $$ S_t = \exp\left( W_t - \frac{1}{2} t \right) $$ is a martingale (for example this could be seen by noting that it solves the SDE $dS_t = S_t dB_t$, which has no drift). But ...
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1answer
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Can I show that a process which a supermartingale above a certain value and a submartingale below it converges?

In my work, I have many times encountered dynamic stochastic systems which are a submartingale (increasing in expectation) below a certain value of the variable, $X^*$ and a supermartingale ...
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91 views

Show rigorously that Pólya urn describes a martingale

We work with the famous Pólya urn problem. At the beginning one has $r$ red balls and $b$ blue ball in the urn. After each draw we add $t$ balls of the same color in the urn. $(X_n)_{n \in \mathbb ...
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Is this martingale identically zero?

Let $X_t$ be such that $X_t$ is bounded continuous martingale adapted to the filtration $\mathcal{F}_t$ such that $$\Bbb{E} \bigg[\int_0^t e^{X_s} \, d\langle X\rangle_s\bigg] = 0$$ and $X_0=0$. ...
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Proving that $Z_t=g(X_t)$ is a martingale if and only if $\mathbb{E}(Z_t|Z_s)=Z_s \ \forall t>s \geq 0$. ($X_t$ Markov)

I want to prove the next property: Let $X_t$ be a Markov process (so $\mathbb{E}(X_t|\mathcal{F}_s)=\mathbb{E}(X_t|X_s)$ where $\mathcal{F_t}=\sigma(X_s, s\leq t)$). Suppose that $Z_t=g(X_t)$ where ...
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34 views

Prove that $\tilde{X}_{\tilde{\theta}}(t)$ is a martingale

Let me introduce the objects: 0) $(\Omega, \mathcal{F},\Bbb{P})$ is a probability space 1)$S_N $ is the set of symmetric, non-negative definite $N\times N$ matrices 2)$a:[0, \infty) \times \Omega ...
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Is the Martingale property still true for $\xi$ not necessarily $C^1$?

Denote $$M(t) = f(t, \alpha(t))\exp \bigg\{-\int_0^t g(u, \alpha (u)) \, du - \int_0^t h(u, \alpha(u)) \, d\xi(u)\bigg\}$$ Here $\xi: [0,\infty) \times \Omega \to \Bbb{R}$. If for each $\omega$ the ...
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56 views

Martingales and Integrals question

I'm stuck with an martingales exercise here: $$\lim_{n\to\infty}\int_0^1\int_0^1\cdots\int_0^1\sin\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)dx_1dx_2\cdots dx_n$$ I tried to do it without martingales ...
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47 views

Doob decomposition of $|S_n|$ where $S_n$ is simple random walk.

Let $X_n$, $n\geqslant 1$ be iid Rademacher random variables, i.e. $X_1$ takes values $\pm 1$ each with probability $\frac12$. Define $S_0=0$ and $S_n=\sum_{i=0}^n X_i$, and $\mathcal F_n = ...
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33 views

Is a martingale with bounded variance therefore bounded in $L^2$?

If a martingale $W_n$ has bounded variance, does this mean that $W_n$ is automatically bounded in $L^2$? I feel like this ought to be obvious but I don't see how to prove it and I haven't been able to ...
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24 views

Local martingale and integral condition

Suppose $M^i_t = X^i_t - X^i_0 - \int_0^t b_i(s,X)\, ds$ where $b_i:[0,\infty)\times \Omega \to \mathbb{R}$ is a progressively measurable functional and $X^i_t: C[0,\infty)^d \to \mathbb{R}$ ( ...
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1answer
66 views

Conditional probability branching process

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
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1answer
126 views

What are some easier books for studying martingale?

What are some easier books for studying martingale? They are defined to be comprehensive but easier than Roger and William's martingale book. For example, to study Q and F martingales? It should ...
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1answer
35 views

Why use stopping times rather than a deterministic sequence to localise a martingale?

I am a beginner on stochastic processes I am wondering why , to localise a martingale, require the existence of one non-decreasing sequence of stopping times [$ \tau_1 , \tau_2$,...] such that the ...
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16 views

Strong law of large numbers for continuous time martingales?

Is there a theorem/reference that states if $M(t)$ is a martingale, then under certain mild conditions, $M(t)/t\to 0$ a.s. as $t\to\infty$?