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

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14 views

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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18 views

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
16 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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25 views

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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28 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
38 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 $(X_t)_t$ $$A f (x) = \lim_{t \downarrow 0} \frac{\mathbf{E}^{x} [f(X_{t})] - f(x)}{t}$$ then we get that for any "nice" $f$ the process ...
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0answers
17 views

How many versions of martingale central limit theory? [closed]

Anyone knows exactly how many versions are there martingale central limit theory? I googled and nearly every link give me different version!!!!!
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1answer
28 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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11 views

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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2answers
24 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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24 views

Can all the theory of martingale with respect to itself applied to martingale with respect to other process?

Can all the theory of martingale with respect to itself applied to martingale with respect to other process? martingale with respect to itself: $E(X_n|\cal{F}_{n-1})=X_{n-1}$ where $\cal{F}_{n-1}$ ...
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29 views

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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0answers
17 views

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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2answers
35 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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2answers
25 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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0answers
9 views

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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47 views

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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1answer
36 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
17 views

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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2answers
86 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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1answer
42 views

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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1answer
35 views

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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1answer
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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0answers
25 views

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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1answer
55 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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1answer
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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1answer
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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0answers
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
60 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
116 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
34 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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0answers
15 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 $t\to\infty$?
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1answer
48 views

Show that $(W_t)_{t \geq 0}$ and $(W_t^2-t)_{t \geq 0}$ are not uniformly integrable

I'm considering the following martingale $M_t:=W_t^2-t,\ t\geq 1$, where the $W_t$ is a Brownian motion. I want to prove that this martingale and the Brownian motion are not uniformly integrable. I ...
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1answer
31 views

One Martingale problem

In the setting of Kolmogorov's maximal inequality, I need to prove the following $$P(\max_{1\leq m \leq n}|S_m| \leq x) \leq \frac{(x+K)^2}{var(S_n)}$$ Hint: Use the fact that $S_n^2 -s_n^2$ is a ...
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1answer
32 views

Martingale: Show $p\{T<+\infty \}=1$.

Let $(X_i)$ i.i.d. such that $p\{X_i=+1\}=p\{X_i=-1\}=\frac{1}{2}$ and let $(S_n)$ the martingale define by $S_0=0$ and $S_n= X_1+...+X_n$. Moreover, let $$T=\begin{cases}\inf\{n\geq 0\mid ...
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0answers
24 views

Discrete stochastic integral and optional sampling theorem

I want to prove the optional sampling theorem using the fact that discrete stochastic integrals for martingale integrators are still martingale. To prove: if $(M_t)_{t\in T}$ is a martingale and ...
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1answer
10 views

Show that for martingale and predictable process, it is not possible to gain almost surely in some step

Let $X_t, t = 0, 1,\ldots, T$ be a martingale and $V_t, t = 1,2,\ldots, T$ a predictable process, I want to show that for $t = 1,2,\ldots, T$ we have $$ V_t\cdot (X_t - X_{t-1}) \ge 0 \textrm{ ...
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0answers
16 views

probabilty of maximum of stochastic process

Given, $$ M_t=exp\left( \int_0^t f(s) dW_s - \frac{1}{2}\int_0^t f(s)^2ds \right) $$ where $W_t$ is a brownian motion. Let $Z_t=W_t-\int_0^tf(s)ds$. How do i show that the above may be used with ...
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1answer
34 views

BMO martingales

Let $(Y_t)_{t\leq 0}$ be a continuous uniformly integrable martingale. It can be shown that for any $p\geq 1$, the following two properties are equivalent : there is a constant $C$ such that for any ...
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27 views

Doob's optional sampling theorem

Say we have a right continues super-martingale $(X_t)_{t\geq 0}$ with filtration $F_t$ and a stopping time $\tau$ for which $P( \tau < \infty)=1$ why is it true to claim that $(X_{\tau\wedge ...
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3answers
64 views

Showing time changed brownian motion is martingale.

Let $W$ be a one dimensional Brownian motion and define, $$ X_t=W_{(\text{exp}(\beta t)-1)}\\ \hat{W}_t=\frac{1}{\sqrt{\beta}}\int_0^te^{-\frac{\beta s}{2}}dX_s $$ Show that $\hat{W}_t$ is a local ...
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2answers
50 views

“The first time a continuous local martingale grows in absolute value beyond $n$” is a localizing sequence

How can it be shown that, for a continuous local martingale $X$ defined w.r.t. the filtered probability space $(\Omega, \mathcal{A}, P; \mathcal{F})$, the stopping times $\tau_n := \inf \{t \geq 0 ...
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0answers
27 views

Probability of hitting zero

Suppose time is discrete. $X_{t+1} = X_t + x_t$. $x_t$ is of continuous value, iid with mean zero and finite variance. Let initial condition $X_0>0$, how can I prove that the probability of $X_t$ ...
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1answer
51 views

$e^{X_t - \frac{t^3}{6}}$ is a martingale - show it [closed]

I am trying to use Ito's integral properties to prove it is a martingale, but am getting stuck in the preliminaries. More so, I wanted to confirm, do I use this formula?
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0answers
17 views

Discrete-Time Stochastic Calculus and Stopping Times: Resources

In my measure-theoretic probability course we covered what the professor called "discrete-time stochastic calculus". Essentially, it was a three part method for computing certain quantities such as ...
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1answer
32 views

Proof that $\left<M\right>_{S_n}$ has finite expectation

Let $M$ be a continuous martingale with $M_0=0$ and $S_n:=\inf \{t:|M_t|>n\}$. Show using Ito's Isometry that $\langle M\rangle_{S_n}$ has finite expectation for each $n\in\mathbb{N}$. I know ...
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1answer
30 views

Martingale Strategy and Infinity [closed]

Given a game in which a gambler has the opportunity to bet that a fair coin, flipped randomly, will have the outcome of heads or tails... The gambler decides to always bet on heads and double their ...
0
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1answer
26 views

A uniformly bounded local martingale is a martingale

I was trying to prove that A uniformly bounded local martingale is a martingale. Clearly a bounded local martingale is integrable I know how to show that a lower bounded local martingale is a ...
2
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1answer
20 views

$MN-\langle M,N \rangle$ is uniformly integrable when M, N are $H^2$?

This proof below really seems to have little to do with uniform integrability, what does the DCT application give exactly? Any alternative proofs would be good too
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1answer
47 views

Deriving a closed form expression for stochastic integral (to show its a martingale)

I have $B_s = $ brownian motion at time $s$. $$ \int_0 ^t B_s \, dB_s$$ $$0 \leq t \leq T$$ And want to check if it is a martingale, first from its closed form expression, and then via conditions on ...