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

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24 views

Martingale, increasing sequence of random variables

I want to prove the following: Let $\{X_n\} _{n \in \mathbb{N}}$ be a sequence of random variables such that $P( X_{n+1} \ge X_n)=1$. Then $$\{X_n\} _{n \in \mathbb{N}} \ \text{is a martingale} ...
2
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0answers
18 views

Why do we use an exponential Martingale for the stopping time of a BIASED random walk?

The following is a passage from the lecture notes: Let a simple random move to the right with probability $p$ and to the left with probability $q = 1 − p$. We want the probability that it hits ...
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1answer
15 views

Locally lipschitz implies zero quadratic variation? [on hold]

How can I prove that a locally Lipschitz function has zero quadratic variation? Thanks.
2
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1answer
22 views

Uniform integrability of the stopped compensated Poisson process

Let $N(t)$ be a Poisson process of rate $\lambda$ and consider the compensated Poisson process $$\bar{N}(t):= N(t) - \lambda t.$$ It was already shown in another post (Is a compensated Poisson ...
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0answers
24 views

One dimensional Lazy random walk, $O(1/\sqrt{n})$?

Suppose that we have a Lazy 1-dimensional random walk $X_n$ valued in $\mathbb{Z}$, i.e. $$X_n = \sum_{i}^{n} \xi_i\;\;\;\;\;\;\;\;(\xi_i\;\text{iid}) $$ and $$\frac{1}{4}=P(\xi_1= 1)=P(\xi_1 =-1) ...
3
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0answers
37 views

Why is the black-scholes model arbitrage free when $\sigma >0$?

I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
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0answers
21 views

limit of sum of a brownian motion

Let $W_t$ be a wiener process and let $\pi$ be a partition of the segment $[0,T]:0\leq t_1\leq...\leq t_n=T$ I need to show without using the martingale property that the term below tends to $0$ in ...
2
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0answers
21 views

Proving that a local martingale given by a stochastic integral is not a martingale

Let $X_t=\int_0^t e^{W_s^2}dW_s$ for $0\leq t\leq 1$ and show that is not a martingale. I guess the reason is that the expectation is not finite, but I'm not sure how to show it precisely. In fact ...
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7 views

Logarithm of Brownian motion which is a local martingale but not a martingale

Let $W_1(t)$ and $W_2(t)$ be independent Brownian motions starting at positive points (not necessarily at the same point). Let $X_t=\log(W_1^2+W_2^2)$ and show that it is a local martingale but not a ...
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22 views

Risk-neutral (i.e. martingale) measure if density is given for a single random variable (i.e. asset)

Let $(\Omega,\mathcal F, P)$ be a probability space. And let $S : \Omega \to \mathbb R$ be a random variable, called an asset, also we are given $\pi > 0$ called a price and some $r \ge 0$ called ...
3
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3answers
48 views

How to show $M_n = X_n^2-n$ is a martingale?

Let $X_n, n = 0, 1, 2, . . .$ denote an unbiased Normal Random Walk. $X_0 = 10$, and $X_{n+1} = X_n + Y_{n+1}$, with $\{Y_n\}$ are i.i.d. $N(0, 1)$. Then how can I show that: A) $M_n = X_n^2-n$ is a ...
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0answers
28 views

Application of martingale central limit theorem

I just learned martingale central limit theorem and got a problem at hand and do not know how to form the correct martingale. Suppose we draw balls successively from a box of $2n$ balls and $n$ ...
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1answer
32 views

Gambling Game martingale

State the optional sampling theorem for martingales and bounded stopping times. You start with a capital of £100 and bet repeatedly on the toss of a coin. On each toss you may bet any whole number of ...
2
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1answer
35 views

Properties of the martingale $S_n := S_{n-1} + X_n \sqrt{1+S_{n-1}^2}$

Let $(X_n)$ be a sequence of IID random variables with $$P(X_i=1)=P(X_i=-1)=\frac{1}{2}$$ and let $(\mathcal{F}_n)$ be the natural filtration of $$\mathcal{F}_n=\sigma(X_1,\dotsc,X_n)$$ Define a ...
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0answers
18 views

$E(E(X_{n+1}| X_0…X_n)) = E(X_n)$

Using Adam's Law to prove that a martingale has constant mean, we have $$E(E(X_{n+1}| X_0...X_n)) = E(X_n)$$ Why is the left side equal to $ E(X_{n+1})$? and not also $ E(X_{n})$
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1answer
26 views

No arbitrage iff there EMM $P^*$ theorem [closed]

The definition of an arbitrage I was given: "An arbitrage strategy is an admissible strategy with zero initial value and positive probability of a positive final value." I think that an initial ...
2
votes
1answer
56 views

Almost sure convergence of the Poisson process

Let $N = \{N(t) \}_{t\geq 0 }$ be a Poisson process. I already know that $N(t)- \lambda t$ is a martingale where $\mathbb{E} [ N(t) ] = \lambda t$. I want to prove that $$ \frac{N(t)}{t} \rightarrow ...
2
votes
2answers
21 views

Product rule with stochastic differentials

I am encountering difficulty in seeing how this relationship holds: with $S_T$ being stock price at time $T$, I want to find the sde for $S_t e^{-rt}$ $$dS_t = rS_tdt + \sigma S_t d\hat B_t$$ Where ...
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0answers
10 views

de Finetti's theorem in two dimensions?

We know that for an array of exchangeable Bernoulli r.v.s $X_i, i\in \mathbb{N}$, de Finetti's theorem can be rephrased to be that $$\exists f: \mathbb{R\times \mathbb{R}}\rightarrow \{0,1\}, \; ...
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1answer
24 views

Martingale Properties

Here is a proof of a property of a martingale $X$ relative to the filtration $(F_{n})$: $n\gt m,\\$ $\\ \\ E[X_n|F_m]=E[E[X_n|F_{n-1}]|F_m]=E[X_{n-1}|F_m]=...=E[X_m|F_m]=X_m$ In the definition of a ...
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0answers
19 views

Is $Z$ a martingale?

$M$ is a continuous, strictly positive martingale. $Z$ is defined by: \begin{equation*} Z(0) = 1,~dZ = \frac{dM}{M} \end{equation*} Clearly $Z$ is a strictly positive local martingale. Is it a true ...
3
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28 views

Martingale converges to the boundary

I asked an almost same question before and it is solved by considering adjacent $Z_n$ can not be far away and obtain a contradiction. However, if the setting is altered a bit, I wonder whether it is ...
0
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1answer
11 views

Martingale: problem with conditional esperance

Let $(S_n)$ a martingale refer to $(X_n)$. Show that for all integer $k\leq l\leq m$ $$\mathbb E[(S_m-S_l)S_k]=0.$$ I don't understand the to following equality: $$\mathbb E[(S_m-S_l)S_k]=\mathbb ...
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1answer
60 views

martingale: Prouve that $(S_n=\frac{R_n}{n+2})$ is a martingale refer to $(R_n)$

A box has red balls and green balls. To each step, we take a ball and we put it back in the box with an other ball of the same color. At the beginning, the box has exactly one ball red and one ball ...
0
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1answer
12 views

Martingale: why $\mathbb E[S_{n+1}\mid R_0,…,R_n]=\frac{1}{m-1}\sum_{i=1}^{m-1}\mathbb E[X_i\mid Z_m,X_{m+1},…,X_N]$

Let $(X_k)$ a sequence i.i.d. of random variables such that $\mathbb E[|X_1|]<\infty $ and let fix $N\in\mathbb N$. We set, \begin{cases}Z_n=X_1+...+X_n\\ Y_n=\frac{1}{n}Z_n\\ R_n=Z_{N-n}\\ ...
0
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1answer
26 views

Martingale: Whay $\mathbb E[S_n]=\mathbb E[S_1]$.

I've got a theorem (without proof) that say: If $(S_n)$ is a martingale refer to $(X_n)$, then $\mathbb E[S_n]=\mathbb E[S_1]$. I don't really understand why. Is there an intuitive why to see ...
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2answers
21 views

Martingale: why $\mathbb E[S_{n+m}\mid X_1,…,X_n]= S_n$.

Let $(S_n)$ a martingale by ratio to $(X_n)$ (I'm not sure if the terme "by ratio" is correct, I hope you'll understand). A lemma of my lecture say: $$\mathbb E[S_{n+m}\mid X_1,...,X_n]= S_n,\quad ...
2
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2answers
52 views

Amazing property of martingales

let $Y_1,Y_2,..$ be a sequence of equally distributed, independent and positive random variables. Consider $X_n = Y_1…Y_n$. Under which condition is $X_n$ a (super)-martingale? Show that neglecting ...
3
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1answer
37 views

Chromatic number $\chi(G)=600$, $P(\chi(G|_S)\leq 200) \leq 2^{-10}$

I am learning martingale and Hoeffding-Azuma inequality recently but do not how to apply the those inequality or theorem here. Let $G=(V,E)$ be a graph with chromatic number 600,i.e. $\chi(G)=600$. ...
2
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1answer
79 views

Does the random variable $f(\tau)M_\tau$, where $M$ is a martingale and $\tau$ is a stopping time, have zero expectation?

Suppose that $M:=\{M_t\}_{t\geq0}$ is a martingale adapted to some filtration $\mathcal{F}:=\{\mathcal{F}_t\}_{t\geq0}$ with $M_0\equiv0$ and that $\tau$ is an $\mathcal{F}_t$-stopping time. Suppose ...
0
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1answer
16 views

Bound Involving Submartingales

Let $(X_{j})_{j \geq 1}$ be a sequence of random variables with $X_{j}$ having mean zero and a finite moment generating function $\phi_{j}(\xi) = E(e^{\xi X_{j}})$ for all $\xi$ in a neighborhood $J$ ...
2
votes
1answer
25 views

First hit of a martingale

I came across this result somewhere and I don't grasp its proof in its entirety. Let $M$ be a continuous martingale such that $M_0 = 0$. Define $\tau_x = \inf\{t\geq 0: M_t =x \}$. Then, $$P\{\tau_a ...
0
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1answer
58 views

Local martingale but not martingale

On wikipedia there is an example of a local martingale which is not a martingale, but I do not understand why it is a local martingale. We have the process $ X_t = \begin{cases} ...
8
votes
1answer
57 views

Doob-style second moment martingale inequality

Let $\{X_k\}_{k=0}^{\infty}$ be a martingale, supposing $X_0 = 0$ and $E[{X_n}^2] <\infty$. Prove that $$P\left(\max_{1\le k \le n} X_k \ge r \right) \le \frac{E[{X_n}^2]}{E[{X_n}^2] + r^2}$$ ...
0
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2answers
37 views

F measurable and conditional expectation.

(a):I found it easily cause sum of measurable sets are measurable. (b),(c): I know limsup(Sn/n) is also measurable but I can't prove that just sup(Sn/n) is measurable. (d): I solved it by using the ...
2
votes
1answer
17 views

Uncorrelated successive differences of martingale

I read somewhere that given a martingale ${X_n}$, the successive differences of the martingale series are uncorrelated, namely $X_i −X_{i−1}$ is uncorrelated with $X_j −X_{j−1}$ for $i \neq j$. I ...
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1answer
54 views

Proving a.s. convergence for martingales

Let $ε_n, n > 1$, and $V_n, n > 0$, be independent random variables, with $P(ε_n = 1) = P(ε_n = −1) = 1/2$, $P(V_n = 1) = p_n, P(V_n = 0) = 1 − p_n$, for all n. Define $X_n$ inductively by $X_0 ...
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1answer
25 views

If $B=(B_t,t\ge 0)$ is a Brownian motion and $(\mathcal{F}_t,t\ge 0$ is its generated filtration, then $X_t-X_s$ are independent of $\mathcal{A}_s$

A real-valued stochastic process $B=(B_t,t\ge 0)$ is called Brownian motion $:\Leftrightarrow$ $B_0=0$ $B$ has independent and stationary increments, i.e. ...
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1answer
30 views

Application of martingale representation theorem

I am reading a proof that uses the following fact without proof (a bit strange): Let $W$ be a real Brownian motion generating the right-continuous, completed filtration $\{\mathcal{F}_t \}_{t \geq ...
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1answer
31 views

Show that $((N_t-t)^2-t)_{t \geq 0}$ is a martingale for a Poisson process $(N_t)_{t \geq 0}$

I am asked to show that if $N$ is a poisson process of intensity $1$, then: $X_t=N_t-t$ is a martingale. $X_t^2-t$ is a martingale. I have done the first part easily, using independence of ...
2
votes
1answer
16 views

Proving that a process is a positive martingale

Let $X$ be the strong solution to the SDE $$ dX_t = \tanh X_t \,dt + dW_t, $$ where $W$ is a scalar Brownian motion defined on a probability space $(\Omega, \mathcal{F} ,\mathbb{P})$. (Such solution ...
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0answers
17 views

Application of Doob's Optional Stopping Time Theorem on new stopping time

Consider a random walk on a line starting at 0. On each step the probability of moving in either direction (right or left) is 1/2. There are two particular points on the line -a, and b. If $\tau$ is ...
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1answer
66 views

Exponential of a uniform integrable martingale is a submartingale

For reference I want to prove this Lemma: Let $M$ be a uniformly integrable martingale with the additional property that $\mathbb{E}[ \exp(M_\infty)] < 1$. Then $\exp(M)$ is a uniformly ...
2
votes
1answer
38 views

Density of running supremum of Brownian motion until a stopping time

I am stuck on an exercise in my book: The question relies on the following fact: Let $M$ be a continuous, non-negative local martingale such that $M_0=1$ and $M_t \rightarrow 0$ almost surely as ...
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2answers
57 views

A variation of Lévy's characterization of Brownian motion

It is shown here, without using stochastic calculus, that if $W_t$ is a standard Brownian motion, then $$ f(W_t)-\frac{1}{2}\int_0^t f''(W_s)ds $$ is a martingale, where $f\in C^2$ and compactly ...
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0answers
22 views

Markov chain converges to boundary

I am learning martingale and related concepts recently and come across the following problem. Suppose $D$ is a bounded, connected, open subset of $\mathbb{R}^2$ with boundary $\partial D$. ...
2
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1answer
35 views

Examples of Wiener Martingales

$(X_t,\mathcal{F}_t)$ is called a Weiner martignale if i) $X_t$ is a Wiener Process ii) $(X_t,\mathcal{F}_t)$ is a martingale. (Here $\mathcal{F}_t$ is an increasing $\sigma$-field family). Let ...
2
votes
1answer
32 views

Checking if $B_t^3 $ and $3tB_t$ are martingales?

$$\mathbb{E}[ B_t^3 - 3tB_t + 3B_t | \mathcal{F}_s]$$ $$\mathbb{E}[B_t^3 | \mathcal{F}_s] - 3\mathbb{E}[t B_t | \mathcal{F}_s\}$$ $$\mathbb{E}[(B_t^3 - B_s^3 + B_s^3) | \mathcal{F}_s] + [ not \space ...
2
votes
1answer
27 views

Finding $b$ such that $e^{5B_t - bt}$ is a martingale

I have $X_t = e^{5B_t}$ and Where $B_t$ is brownian motion at time $t$. $M_t = X_t \cdot e^{-bt}$ I need to find a value for $b$ such that $M_t$ is a martingale. I am encountering difficulty, ...
0
votes
1answer
55 views

Proving a sequence forms a martingale

Let $\Omega = \mathbb N = \{1,2,3,\cdots\}$ and $\mathscr F_n$ be the $\sigma$-field generated by the sets $\{1\},\{2\},\cdots,\{[n+1,\infty)\}$ Define a probability on $\mathbb N$ by setting ...