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

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Distribution of local time Brownian motion

I am working on this the following problem related to Tanaka's formula. I understand the theory behind it but I have no clue where to begin. Let $B_t$ is Brownian motion, $B_0=0$ and $L_t$ is the ...
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1answer
13 views

local martingale bounded below by a DL process

Let a continuous adapted process $Z= (Z_t)_{t \geq 0}$ be of class DL if \begin{equation} \{ Z_{\tau \wedge t} : \, \tau \text{ is a stopping time } \} \end{equation} is uniformly integrable, for each ...
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8 views

would the following game be beatable with martingale

I want to mix 2 games with weight on game one 51.5% and 48.5% on game two player will be presented with 3 coins and he will be asked to click on 1 of the 3 coins and then all 3 coins are turned ...
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28 views

Lévy's upward theorem and $\mathcal{L}^p$ convergence.

Lévy's upward theorem: Let $Y \in \mathcal{L}^1(\Omega, \mathcal{F}, P)$, $(\mathcal{F}_n)_{n=1}^{\infty}$ a filtration of $\mathcal{F}$ and $\mathcal{F}_{\infty} = \sigma( \bigcup_{n=1}^{\infty} ...
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1answer
13 views

show that the solution is a local martingale iff it has zero drift

Most financial maths textbook state the following: Given an $n$-dimensional Ito-process defined by \begin{equation} X_t = X_0 + \int_0^{t} \alpha_s \,d W_s + \int_0^{t} \beta_s \,d s, \end{equation} ...
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1answer
13 views

Asymmetric simple random walk?

It comes from the book Probability: Theory and Example. I don't understand the part marked with red line. Why it cannot converge to an interior point of $(a,b)$? Can anyone help? Thanks so much!
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1answer
28 views

Property of submartingale and supermartingle?

Is it true that for a submartingale, $$E(X_n) \le E(X_m)$$ for $n \le m$. And for a supermartingale, $$E(X_n) \ge E(X_m)$$ for $n \le m$. If it is true, then why? I feel confused because the ...
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Where is the assumption of right continuity used in the following proof?

Lemma:If $X$ be a right-continuous positive local martingale then , $X$ is a generalized super martingale Proof: $\forall s<t$ $$E[X_t\mid F_s]=E[\lim_{n\to\infty} X_{t \wedge\tau_n}\mid F_s] \leq ...
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2answers
35 views

Problem similar to Kolmogorov's inequality using martingale.

Suppose that $X_k$ is a sequence of independent random variables with mean zero and variance $1$. Let $S_k=X_1+\cdots+X_k$ and let $$ h(\lambda)=\limsup_{n \rightarrow \infty}P\left(\max_{1\leq k\leq ...
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fractional Brownian motion is not a semimartingale. How to apply Ergodic theorem in the proof of this theorem?

Here is the proof of the theorem. I couldn't understand how to apply Ergodic theorem in this proof. Let $X=(X_t)_{t\geq0}$ be a fractional Brownian motion with self-similar parameter $H\in(0,1)$. We ...
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1answer
30 views

Martingales and variance

For a martingale $(Z_n)_{n\in \mathbb N}$ define $X_i=Z_i-Z_{i-1}$ with $Z_0=0$ Show: $$Var(Z_n)=\sum_{i=1}^nVar(X_i)$$ My attempt: We can write $Z_n=\sum_{i=1}^nX_i$, so we actually just have to ...
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Proving convergence of a martingale in $L^2$ [on hold]

I'm stuck with the following problem: Let $X$ a positive martingale bounded in $L^2$. Show that $\lim_{n\to \infty} X_n = X$ a.s. and in $L^2$.
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martingale convergence proof

This is out of Durrett 5.5.7. Let $X_n \in [0,1]$ be adapted to $\mathcal{F_n}$. Let $\alpha,\beta > 0$ such that $\alpha + \beta = 1$. Suppose that $$ P(X_{n+1} = \alpha + \beta X_n | ...
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Doob's decomposition of $X_n=\sum _{m=1}^n1_{B_m}$, where $B_m \in \mathcal{F}_m$.

$X_n=\sum _{m=1}^n1_{B_m}$, where $B_m \in \mathcal{F}_m$. I want to find the Doob's decomposition. I think $X_n=Y_n+Z_n$, where $Y_n$ is a martingale, $Z_n$ is a predicable process. Then ...
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Martingales relative to its natural filtration [on hold]

Let {$Y_n$} be a sequence of positive independent random variables with E($Y_j$) =1 for all j. Set $X_0$=1 and $X_n = \prod_{j=1}^n Y_j$, $n \geq 1$. How can I show $X_0,X_1,X_2,...$ is a martingale ...
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1answer
35 views

How to show that this process is a martingale

Consider the probability space $([0, 1), \mathcal{B}[0, 1), \lambda)$, where $B[0, 1)$ are the Borel sets on $[0, 1)$ and $\lambda$ is the Lebesgue measure. Let \begin{align*} I_k^n = \left[ ...
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1answer
29 views

A question on proving the existence of a martingle which has a deterministic square bracket

Let $g:\mathbb{R^+} \to \mathbb{R^+}$ be a non decreasing and continuous function . Show that there exists a continuous martingale M such that its square bracket $<M>_t=g(t)-g(0)?)$ I have ...
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1answer
15 views

Estimate of the expectation value

Consider a sequence $(X_n)_{n\in\mathbb{N}}$ be a sequence of i.i.d. random variables on a probability space $(\Omega,\mathcal{F}, \mathbb{P})$ with expectation $\mathbb{E}[X_1]=0$ and ...
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27 views

Exercise on quadratic variation

I am faced with the following exercise: Let $X_{1},X_{2},...$ be independent random variables satisfying $\mathbb{E}(X_{n}^{2})<\infty$ and $\mathbb{E}(X_{n})=0$ for all $n\in\mathbb{N}_{0}$. ...
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Prove that $\text{Var} \tau = \frac{1 − (p − q)^2}{(p-q)^3} $ where $\tau$-stopping time

Let $S_n = \xi_1 + \dots + \xi_n$ be asimetric random walk such that $P(\xi_i = 1) = p > \frac{1}{2}$ and $P(\xi_i = -1) = q $. Let $\sigma^2 =1-(p-q)^2$ and let $X_n=(S_n-n-(p-q)n)^2 - \sigma^2n $ ...
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Prove that $X_n = e^{(\theta S_n-n \ln \phi(\theta))}$ is martingale.

Let $\xi_1 \neq const.$ be a random variable with moment-generating function $\phi(\theta) = Ee^{(\theta \xi_1)}$. Let $S_n = \xi_1 + \dots \xi_n$. Prove that $X_n = e^{(\theta S_n-n \ln ...
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1answer
31 views

The definition of terms in Doob's decomposition theorem for submartingales

The definition of $Z_n$ in the Doob's Decomposition Theorem, I think it is a predictable submartingale starting at $0$. Is that right? Thanks for your help!:)
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Doob's decomposition Thm_ Got stuck applying induction in proving $Z_{n+1}$ is $F_n$ measurable?

Already know $Z_0=0$, and $$Z_{n+1}=E(X_{n+1}|F_n)-X_n+Z_n$$ $X_n$ is $F_n$ measurable, $F_n$ is a filtration. How to prove $Z_{n+1}$ is $F_n$ measurable? I tried to prove by induction. Since $Z_1$ ...
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1answer
67 views

Martingale Transforms and quadratic variation

Let $M$ be a martingale with $\mathbb{E}M_{n}^2<\infty$ for all $n$. Let $C$ be a bounded predictable process and set $X=M\cdot C$. Show that $\mathbb{E}X_{n}^2<\infty$ for all $n$ and that ...
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2answers
57 views

Simple formula on $X_n^{(k)} = \sum_{1 \le i_1 < … < i_k \le n} Y_{i_1} \cdot \dots \cdot Y_{i_k}$ (to show $X_n^{(k)}$ is martingale)

Let $$X_n^{(k)} = \sum_{1 \le i_1 < ... < i_k \le n} Y_{i_1} \cdot \dots \cdot Y_{i_k}$$ If I take $k=2$ and $S_n = Y_1 + \dots + Y_n$ I have of course: $$X_n^{(2)} = \frac{1}{2} (S_n^2 - ...
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1answer
22 views

Girsanov Theorem Confusion

I'm getting completely bogged down by sign errors when using Girsanov's theorem. Keeping it simple, suppose $W_t$ is a standard Brownian motion under a probability measure $\mathbb{P}$, and we have a ...
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1answer
48 views

Prove that $S_n^2-s_n^2$ is martingale

Let $(X_i)$ be iid such that $EX_i = 0$ and $\operatorname{Var}X_i = \sigma_i^2$. Let $s_n^2 = \sum_{i=1}^n \sigma_i^2$ and $S_n = \sum_{i=1}^n X_i$. Prove that $S_n^2 - s_n^2 $ is martingale. My ...
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55 views

Martingales application

Let $X$ be a random variable, $X\in\mathcal{L}^2(\Omega,\mathcal{F},\mathbb{P})$, such that: $\mathbb{E}|X|<\infty$ and consider $\mathbb{F}$ a filtration. Define: ...
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1answer
16 views

Continuous martingale

I have a general question regarding the notion of "continuous martingale". Does this expression refer to a continuous time stochastic process that has the property of being a martingale or does it ...
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1answer
53 views

Ito Integral representation for bounded claims

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be Probability space and let $\{(X_t) : 0 \leq t \leq T \}$ be a continuous semimartingale on it. Let $\xi$ be $\mathcal{F}_T^X$ measurable and bounded. Does it ...
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Looking for resources: Generalizations of martingales to $\mathbb R^2$

In most introductory courses, a martingale $Y$ is defined as a stochastic process $$Y: T \times \Omega \to S$$ ,which satisfies certain conditions. ($\Omega$ is a probability space and a filtration ...
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47 views

Motivation behind study of martingales

Today I wanted to ask a question which I am sure has been answered in multiple places but for which I do not yet have a very clear understanding. Though martingales is a very well explored area of ...
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1answer
23 views

martingales, almost sure convergence

I am given a sequence of independent random variables $(X_n)$ with respective laws given by $P(X_n=-n^2)=\dfrac{1}{n^2}$ and $P(X_n=\dfrac{n^2}{n^2-1})=1-\dfrac{1}{n^2}$, and letting $S_n=X_1+...+X_n$ ...
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1answer
23 views

Martingale property of product of martingale and stochastic process.

$M_t$ is a martingale with respect to $\mathcal{F _t}$ for $t \geq 0$ and $Z$ is a bounded $\mathcal{F_r}$ measurable random variable. $0\leq r < s <\infty$. I want to show that $Z( M_{s\wedge ...
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Arbitrage-free market for continuous distribution

Is it true, that a one-period market say $(0,t)$ is arbitrage-free if $S_t$ is continuously distributed on $\mathbb{R}$? I.e., for continuous distributions on $\mathbb{R}$, there always exists a ...
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1answer
32 views

Paley Wiener stochastic integral

Sorry for the stupid question, no answers necessary anymore! let $(B_t)_{t\in [0,1]}$ be a standard Brownian motion and $F\in C[0,1]$ differentiable. Then the sequence (which is an easy version of ...
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Can't see how implication on definition of Martingale was arrived at

A Martingale is a discrete time stochastic process $Z_1, Z_2, ..., Z_n$ for any time $n$ that satisfies $E[|Z_n|] < \infty$ $E[Z_{n+1}| Z_0, Z_1, ..., Z_n] = Z_n$ By the linearity of expectation ...
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Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
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24 views

Equivalent Stopping Times for Brownian Motions

For standard Brownian motion $B$, define stopping time $T_1:=\inf\{t>0: B_t = 3\}$ and $T_2:=\inf\{t>0: B_t = -3\}$ and $T_3 := \min\{T_1, T_2\}$. Can I say that $T_3 = \inf\{t>0, B_t \in ...
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68 views

Let us assume that m balls are thrown independently at random into n bins.

Let us assume that $m$ balls are thrown independently at random into $n$ bins. Let $X$ denote the number of bins that contain afterwards exactly one ball. I want calculate $Pr(X=0) $ In literature I ...
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2answers
35 views

Sub-Martingale and Martingale

An integrable sub-martingale $S_t$ with $\mathbb E(S_t)$ being a constant is a martingale. Is this statement true, please? I think so.
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Is this Stochastic integral a martingale ?

Let $(B_t)$ be a Brownian motion and set $X_t = \int_0^t B_t^2 dB_s$. Is $X_t$ martingale? My idea is to rewrite $X_t$ in terms of Ito's Formula $(f(x) = \frac{1}{3}x^3)$ $X_t = \int_0^t B_t^2 dB_s ...
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1answer
56 views

How to show the following definition gives Wiener measure

On the first page of Ustunel's lecture notes, he defines the Wiener measure in the following way: Let $W = C_0([0,1]), \omega \in W, t\in [0,1]$, define $W_t(\omega) = \omega(t)$. If we denote by ...
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1answer
30 views

Prove X is a martingale

Prove $X = (X_n)_{n \geq 0}$ is a martingale w/rt $\mathscr{F}$ where X is given by: $X_0 = 1$ and for $n \geq 1$ $X_{n+1} = 2X_n$ w/ prob 1/2 $X_{n+1} = 0$ w/ prob 1/2 and $\mathscr{F} = ...
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33 views

Brownian and Brackets

A continuous martingale with deterministic bracket must be a Brownian motion. Is this statement ture or not, please? If true, how to show it? If not, what is a counter example?
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1answer
24 views

A question on semi-martingale and its variations

With probability one, paths of semimartingales have unbounded variation. What I know is that a martingale is also a semi-martingale, for example, Brownian motion. Hence, Brownian is an example of ...
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2answers
57 views

Zero mean but not a martingale

I am looking for a simple stochastic process which has zero mean for all $t\geq0$ but it is not a martingale. I been looking in to local martingales but having trouble keeping the mean zero for all ...
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1answer
17 views

Martingale property of negative Brownian motion

Let $B_t$ be Brownian motion, with $B_0=0$. Next define $M_t=-B_t$. Have I understood it correctly that $M_t$ is not a Martingale? $E[M_t]=0$ $E[M_{t+1}|M_t]=-M_t$ and therefore not a Martingale? ...
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89 views

Dose “optional stopping theorem” imply “optional sampling theorem”?

Suppose $X$ is a martingale,$\tau$ and $\sigma$ are two stopping times which satisfy (a)$\sigma\le\tau$ and (b)the "optional stopping theorem" holds,that is to say: $$\mathbb E[X_\sigma]=\mathbb ...
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1answer
19 views

measurability in backwards martingales

$X$ is a backwards martingale with $X_0\in L^1 $ According to the convergence theorem:$X_{-n}\to X_{-\infty} $ a.s. But how to get the conclusion that $X_{-\infty}$ is $\mathcal F_{-\infty} $ ...