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

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Expectation of Martingale [on hold]

I am reading an article of American Put option here and I want to understand the proof of Theorem 1 (Main Decomposition of the American Put). It is stated that: $$\exp(-rT) \max[0,K-S_T]=P_0 - rK\...
3
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1answer
46 views

Law of large numbers for moving mean

Consider the following process: For $n = 1,\ldots$ $U_n \sim U[0, 1]$, that is, uniformly distributed on $[0, 1]$, $X_n = U_n 1_{U_n > q_n}$, where $q_n = \frac{1}{n-1} \sum_{i=1}^{n-1} X_i$, ...
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1answer
22 views

Martingale wrt natural filtration: $e^{ \sum_{i=1}^t X_i - t/2}$

Let all $X_i$ be standard normal and iid for $i \in [1,T]$, let $X_0 = 0$. Define for each $t \in [0,T]$ $S_t = e^{( \sum_{i=1}^t X_i) - t/2}$ Is this process a martingale wrt its natural filtration?...
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0answers
25 views

Time-changed Brownian Motion

Let $B_t$ be a standard Brownian motion and let $\tau_{-1}= \inf \{ t \geq 0: B_t(\omega) = -1\}$. By the Continuous Time Stopping Theorem we know that \begin{align} Z_t = B_{t \wedge \tau_{-1}} \...
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41 views

Concentration of a random variable in a random sequence

Given a sequence of random variables $X_0, X_1, \cdots$. Assume that $X_0=0$ and $X_{n+1}= X_n+U[0,1]$, where $U[0,1]$ denotes a uniform distribution over $[0,1]$. Define $N=\min\{n: X_n\geq 2\}$. Q:...
3
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1answer
32 views

Show that a cadlag adapted martingale is a local martingale (help using DCT to show uniform integrability)

EDIT 2: With the correct definition, I think I have a proof. Want to show $\lim_{M\to\infty} \sup_t E[|X_{t\wedge n}|; |X_{t\wedge n}|\ge M]=0$. Fix $n$. Note that $\sup_t E[|X_{t\wedge n}; |X_{t\...
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1answer
35 views

Doob Decomposition Theorem - submartingale iff increasing

Probability with Martingales To prove $b$ I tried: $$A_n \ge A_{n-1}$$ $$\iff E[X_{n} - X_{n-1} | \mathscr F_{n-1}] \ge 0$$ $$\iff E[X_{n} | \mathscr F_{n-1}] \ge X_{n-1}$$ That ...
-1
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1answer
23 views

Doob Decomposition Theorem in Williams is working backward? Unique modulo indistinguishability?

Probability with Martingales This is my understanding of what is going on in the proof above: We first assume $X$ has such Doob Decomposition in order to figure out what $A$ to use in ...
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0answers
41 views

Help with change of measure and martingales

Consider two three stochastic processes $X$, $Y$ and $Z$ in probability space $(\Omega, (\mathcal F_t)_{t \geq0},\mathbb P)$ such that $$ X_t = \exp\left(\int_0^t f_s ds\right), $$ $$ Y_t = \exp\...
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1answer
26 views

Prove $M_{S(k) \wedge n}$ is bounded in $\mathscr L^2$

Probability with Martingales: To prove $$\sup E[M_{S(k) \wedge n}^2] < \infty,$$ how can we use 12.12c? There aren't any stopping times there.
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18 views

Prove $\lim M_{S(k) \wedge n}$ exists a.s. if $S(k) = \infty$. Is $N_n \ge 0$?

Probability with Martingales: Why does $\lim M_{S(k) \wedge n}$ exist a.s.? Is it connected to $$\sup E[M_{S(k) \wedge n}^2] < \infty$$ ? What I tried: My approach is to use: If $\lim ...
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0answers
41 views

Showing that the following process is a martingale [closed]

Let $Nf(x) = \frac{1}{\sqrt{2\pi}}\int_\mathbb{R} e^{-\frac{|x-y|^2}{2}}f(y)dy, \;\; f \in b\mathcal{B}(\mathbb{R}), x \in \mathbb{R}.$ Let $X = (X_t, \mathcal{F}_t, \mathbb{P}^x)$ a pure jumps Markov ...
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0answers
32 views

How does $\langle M_{S(k) \wedge n}\rangle = A_{S(k) \wedge n}$ not follow by definition?

Probability with Martingales: What is the relation between $\langle M_{S(k) \wedge n}\rangle \ = A_{S(k) \wedge n}$ and $\{N_n\}, \{ N_{ S(k) \wedge n } \}$ being martingales? It seems that $$\...
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0answers
41 views

Prove $A^{S(k)}$ is previsible

Probability with Martingales: I have a different attempt in mind, but I'm guessing it's wrong because if it were right, the book would've used it. It seems that we must show that $$A_{S_k \...
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1answer
28 views

Prove $S_k$ is a stopping based on $A$ being previsible

Probability with Martingales: It looks like we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ ...
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1answer
60 views

showing a process martingle from ito's lemma.. [closed]

**by ito formula, Ft: filtratoin $M(t) = (aB(t) - t) \exp( 2B(t) - 2t ) $ find constant a for $M(t)$ to be a martingale plz help!**
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1answer
26 views

Why is $M$ bounded in $\mathscr L^2$ iff $E[\lim A_n] < \infty$?

Probability with Martingales: About $(c)$ My understanding is that $(c)$ is equivalent to: $$\sup E[A_n] < \infty \iff E[\lim A_n] < \infty$$ Why is that so?
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18 views

Preservation of ui condition [closed]

I have a stopping time $\tau_n$ with $\mathbb{P}(\tau_n=\infty)\rightarrow 1$ for $n \to \infty $. With this stoppingtime $M^{\tau_n}$ is a uniformly integrable martingale. I deduced that $M$ is a ...
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22 views

Example of a semimartingale with special properties

I have to find an example of a semimartingale X such that $\lim_{t \rightarrow \infty} X_t$ exists a.s. and $X$ is not a semimartingale up to infinity. I think it could be a deterministic function ...
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40 views

martingale square integrable

Let $X_t=\int_0^te^{W_s}dW_s$ and $Y_t=\int_0^tW_sdX_s$. How to show that $X$ and $Y$ are martingale square integrable? ($W_t$ - Wiener) It it enough to show that $\mathbb{E}X_t^2<\infty$, $\...
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0answers
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Showing that Polya's Urn is a martingale - is this correct definition of Polya's urn?

Possibly related: Show rigorously that Pólya urn describes a martingale Suppose we had $b$ and $r$ blue/red balls in urn at time $0$, and at each $n\ge 1$ we draw a ball randomly and then put it ...
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0answers
17 views

Proving mean of random variables does not converge to zero almost surely. [duplicate]

We have a sequence of i.i.d random variables $(X_n)$ such that $$ P(X_n=n)=P(X_n=-n)=\frac{1}{2(n+1)\log(n+1)},$$ and $$P(X_n=0)=1-\frac{1}{(n+1)\log(n+1)}$$ I want to show that $\dfrac{S_n}{n}$ (...
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2answers
22 views

Natural filtration of martingales

I don't quite understand what the natural filtration really is. Imagine e.g. a sequence of independent and identically distributed random $N(0,1)$ variables. What is their natural filtration, and how ...
3
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1answer
43 views

Stochastic differential equation substitution reasoning?

I need help to explain reasoning behind why we choose certain substitutions to solve SDE. After choosing the correct substitution the solution of the SDE are often quite easy. However I have trouble ...
4
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1answer
147 views

Limit theorem for changed time

This post seems long, but its almost everything proofed in this post. Only one step seems to be left, for the desired proof. I would be very gratefull for any help. The setup Given a Levy-Process $U_{...
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1answer
27 views

From brownian bridge to brownian motion proof

Let $B_t$ be a brownian motion. and let $\{W_t=B_t-tB_1:0\le t\le 1\}$ be a brownian bridge. Now let $Y_t=(1+t)W_{t\over 1+t}$. Proof that $Y_t$ is a brownian motion in $[0, \infty)$ My attempt: 1) $...
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1answer
44 views

The Second Hearts Problem

According to the last part of these lecture notes, if we have a standard deck of playing cards and turn cards until the first heart appears, the probability that the next card is a heart is $\color{...
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0answers
38 views

Show that $W^2 _t - t$ is a $\mathbb{P}$-martingale.

Claim: $V_t = W^2 _t - t$ is a $\mathbb{P}$-martingale. I have shown via Ito's formula, that $dV_t = 2 W_t \, dW_t$. For reference, I will list this "Proposition": If $X$ is a stochastic process ...
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1answer
32 views

Find constants such that transformed simple symmetric random walk is martingale

Let $$S_0 :=0, \quad S_n = X_1 + ... + X_n \quad \forall n \in \mathbb{N}$$ be the simple symmetric random walk on $\mathbb{Z}$, i.e. the $X_i$ are i.i.d. with $$P[X_i = +1] = P[X_i = -1] = 1/2.$$ ...
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29 views

A stochastic process $X$ is a martingale $\iff$ $X$ is driftless.

A Collector's Guide to Martingales: If $X$ is a stochastic process with volatility $\sigma _t$ (that is, $dX_t = \sigma _t dW_t + \mu _t dt$) which satisfies $\mathbb{E}[(\int_{0}^{T} \sigma^2 _s \, ...
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1answer
33 views

Expected time until pattern (1,0,0,1)

Let $(X_n)_{n\geq 0}$ be i.i.d. with $\mathbb P(X_n = 0 ) = \mathbb P(X_n = 1) = \frac{1}{2}$. Let $\tau_a$ be the stopping times defined as $$\tau_a = \inf\{n: (X_{n-3}, ... , X_n) = (1,0,0,1)\}$$ I ...
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2answers
29 views

Divergence of asymmetric not-simple random walk

Consider a (not simple) random walk $S_n = \sum_{k=0}^n X_k$ where X_k are i.i.d and the mean $\overline{X}<0$. Is there is simple proof or a reference showing $P( \lim \limits_{k \to \infty} S_k = ...
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1answer
42 views

Functions of a random walk and martingales

Let $\xi_1,\xi_2,\ldots$ be a sequence of iid random variables, such that $$\mathbb{P}(\xi_i=1)=p\ne \frac{1}{2},\,\mathbb{P}(\xi_i=-1)=q=1-p.$$ Consider the corresponding random walk $X_n=\xi_1+\xi_2+...
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1answer
120 views

Hard Question in Stochastic processes - variance Martingales

I got some hard challenge to solve and I am looking for a small clue/help. My question goes like this: 10 Englishmen are trying to leave a pub in a rainy weather. They do it in the following ...
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33 views

$2D$ random walk stopping time

A $2D$ random walk starts at $(X_0, Y_0) = (k, k)$ where $k>0$ is an integer. At each step $(X_{n+1}, Y_{n+1}) = (X_{n}-1, Y_{n})$ or $(X_{n+1}, Y_{n+1}) = (X_{n}, Y_{n}-1)$ with the same ...
3
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0answers
30 views

Two Similar Measures on a Probability Space

Let $(\Omega,\mathscr{F}, P)$ be a probability space and let $Q$ be another probability measure on $\mathscr{F}$, and let $\mathscr{F}_n=\sigma(Y_1,\ldots,Y_n)$ be a non-decreasing sequence of $\...
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1answer
39 views

To test whether a process is a Martingale (Stochastic calculus)?

If $W_t$ is a standard Brownian motion, I was trying to prove $Y_t = \exp (\int_{0}^{t} s\cdot dW_s)$ is a martingale ! First I started finding $dY_t$ using Ito formula. But I am confused how to ...
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Prove that ($Y_n$,$F_n$) is martingale. What does ($Y_n$,$F_n$) mean in this problem?

$X_1$, .... , $X_n$ are independent variables, $P(X_i=1)=p$, $P(X_i=-1)=q$, where $0<p<1$. $F_n=\sigma(X_1,.....,X_n)$ and $Y_n=(\frac{q}{p})^{X_1+....+X_n}$. The task is to prove that ($Y_n$,$...
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78 views

stochastic exponential uniformly integrable martingale

$N$ is a continuous local martingale and $T_c:=\inf\{t>0:[N]_t>c\}$, $c>0$ . I need to show that the stochastic exponential $\mathcal{E}(-N)$ is a uniformly integrable martingale if and only ...
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1answer
22 views

definitions of martingals

Let X - discrete time stochastic process A discrete-time martingale: is $E[|X_{n}|] < \infty$ $$ E[X_{n+1}|X_{1}, X_{2}, ..., X_{n}] = X_{n}. $$ The definition of martingale for filtration: $$ ...
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1answer
33 views

Buyer's price in terms of risk-neutral measures

Let us consider a finite arbitrage-free market model $(B,S)$, where $B$ is a bank account and $S$ is a share. Let $X$ be a claim. We define a buyer's price of $X$ as follows:$$\Pi^b_0(X)=\sup \lbrace ...
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37 views

$n$ times integrated Brownian motion martingale process

According to this post, we found that a $n$ times integrated Brownian motion could be expressed as, \begin{align} V_n(t) = \int_0^t V_{n-1}(s)\ ds = \frac{1}{n!} \int_0^t (t-s)^n\ dB_s, \end{align} ...
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1answer
113 views

Stochastic Integral with respect to Compensated Poisson Process

Proposition: Let $N_t$ be an $\mathcal{F}$-Poisson process and $M_t=N_t-\lambda t$ its compensated process. Then for any $\mathcal{F}$-predictable bounded process $H_t$, the stochastic integral $$(H\...
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58 views

Brownian Motion Third Power Martingale using Ito Integral

Let $(B_t)_{t \geq 0}$ be a standard Brownian motion and $M_t = B_t^2 - t$. According to this and this posts we know that \begin{align} [M] = [B^2] = 2 \int_0^t B_s^2\ ds. \end{align} Now, without ...
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0answers
13 views

Gaussian blur over (or random walk in) a surface mesh

Let $V$ be the set of mesh vertices, connected by edges $E$, forming a mesh that represents a surface embedded in $\mathbb{R}^3$. On this mesh a function $f:V\rightarrow\mathbb{R}$ is defined. For ...
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1answer
49 views

Quadratic Variation Brownian motion martingale (2)

Let $B_t$ be a standard Brownian motion and $M_t = B_t^2 -t$. From here we are aware of the identity \begin{align} [M]=[B^2]. \end{align} Now, I want to apply Itô's formula to $B_t^2$ and from that ...
0
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1answer
35 views

Radon-Nikodym derivative as a Martingale

Let $(\Omega,\mathscr{F}, P)$ be a probability space, let $\nu$ be a finite measure on $\mathscr{F}$, and let $\mathscr{F}_{1}$, $\mathscr{F}_{2}$,... be a non-decreasing sequence of $\sigma$-fields ...
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1answer
21 views

Quadratic Martingale Problem

Let $S_n=\sum_{i=1}^n X_i$, where the $X_i$ are identically and independently distributed with $P(X_i=1)=p$ and $P(X_i=-1)=1-p$. Show that there does not exist such $c(n)$ that makes $M_n=S_n^...
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2answers
57 views

If $M_n$ is a Martingale, How to Show that $\log M_n$ is a Supermartingale?

I think that the problem should use Jensen's inequality. This is because my textbook states that with Jensen's inequality, if $\phi$ is a convex function and $M_n$ a martingale, then $\phi(M_n)$ is a ...
2
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1answer
47 views

show that the process is a martingale [closed]

maybe you will have an idea how to show that : the process $(exp(X_t-\frac{1}{2}Y_t))$ is a martingale? Where $h \in L^2([0,T])$, $T< \infty$, $X_t=\int_0^th(s)dW_s$ and $Y_t=\int_0^th^2(s)ds$ for $...