Tagged Questions

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

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-3
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0answers
14 views

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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0answers
16 views

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 | ...
0
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0answers
14 views

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 ...
-1
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0answers
27 views

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 ...
1
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1answer
33 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[ ...
1
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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 ...
0
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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 ...
1
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0answers
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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0answers
24 views

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 $ ...
1
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0answers
29 views

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 ...
1
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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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0answers
31 views

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 $Z_1$ ...
2
votes
1answer
66 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 ...
0
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2answers
56 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 - ...
3
votes
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 ...
2
votes
1answer
45 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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2answers
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: ...
0
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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 ...
2
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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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0answers
16 views

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 ...
2
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2answers
46 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 $S_n=X_1+...+X_n$ ...
2
votes
1answer
22 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 ...
0
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0answers
10 views

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 ...
1
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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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2answers
19 views

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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0answers
18 views

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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1answer
23 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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0answers
64 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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1answer
52 views

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 ...
1
vote
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 ...
2
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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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2answers
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 ...
2
votes
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 ...
0
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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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0answers
87 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 ...
0
votes
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} $ ...
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1answer
37 views

Seeking for counterexample in closed martingale.

In the martingale convergent theorem: $X_n$ is a martingale with $\sup_n\mathbb E[|X_n|]<\infty$. Then there exist a $X_\infty\in L^1$ such that $X_n\to X_\infty\text{ a.s.}$ I want to find a ...
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0answers
23 views

Problems about the upcrossing lemma.

The following pictures comes from "Probability with Martingales" which denotes a stochastic integral(discrete): $$Y:=H\cdot X$$ Here $H$ is previsible.According to the gambling strategy ,$H=0$ in the ...
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1answer
56 views

Show independence of stochastic integral and stochastic process

Let $ M_t $ and $ N_t$ be two continuous local martingales with respect to a filtration $ \mathcal{F}_t $. Suppose that $ M_t $ and $ N_t$ are independent and set $X_t = \int_0^t M_s^4 \mathrm{d} M_s ...
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1answer
39 views

Why can $\int_0^t f''(X_s) \, d\langle X \rangle_s$ not be a local martingale?

We know from Itos formula, if $X$ is a continuous local martingale and $f$ has two continuous derivatives, we can write $f(X_t)$ as $$ f(X_t) = \int_0^t f'(X_s) dX_s + \frac{1}{2} \int_0^t ...
1
vote
1answer
14 views

Remove drift from exponential Weiner process

I have the following problem: let $X_t$ solve $$ dX_t = b X_t \, dt + \sigma X_t \, dW_t$$ where $W_t$ is a Weiner process. Find $s(\cdot)$ such that $Y_t = s(X_t)$ is a martingale. We can see by ...
0
votes
1answer
32 views

If $\{X_n\}$ is a martingale, then $E[X_n-X_{n-1}]=0$

Apparently this should be quite simple, but I have been trying for a while and can't seem to get this. Let $\{X_n\}$ be a martingale, then we have: $$E[X_n-X_{n-1}]=0$$ According to some notes I found ...
3
votes
1answer
27 views

A simple question about the definition of martingales

The definition of Martingale denotes that $E(M_{n+1}\mid\mathcal{F}_n)=M_n$. This implies $E(E(M_{n+1}\mid\mathcal{F}_n))=E(M_n)$. Then does it mean that $E(M_{n+1})=E(M_n)$ using the tower property? ...
0
votes
1answer
54 views

Prove $A_t := W_t^3-3t W_t$ a martingale

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let $(A_t)_{t \geq 0}$ where $A_t = W_t^3 - 3tW_t$. ...
2
votes
1answer
38 views

What is the difference between $\mathbb E[Z|\mathcal G]=Y$ and $\mathbb E[Z|\mathcal G]\stackrel{\text{a.s.}}{=}Y$?

I'm somewhat confused by the definition of martingale: Let $(\Omega, \mathcal F, \mathcal F_n, \mathbb P)$ be a filtered probability space. We call $(X_n)_{n\in\mathbb N}$ martingale if for all ...
2
votes
0answers
65 views

Martingale Can't be Strictly Increasing

If the sample paths of a martingale are almost surely continuous and not constant on any interval, is it true that they are almost surely not increasing on any interval? Edit for clarity: Let ...
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0answers
47 views

Why this is a martingale?

Setup: $W$ probability space $Z_i : W \to L_i $ random variables ($L_i$ finite, for example $\{0,1\}$) $f: Z_1 \times \ldots \times Z_n \to \mathbb{R}$ $X_i := \mathbb{E}[f \mid Z_1,..,Z_i]$ Why ...