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

learn more… | top users | synonyms

0
votes
1answer
24 views

Proving existence of Itō Integral

Here's an extract from some Continuous Martingales notes I can see how K-W implies the blue box inequality but how does that inequality give continuity? Also what is the functional theorem that ...
0
votes
1answer
14 views

Questions on proving a stochastic process to be a martingale

I need to prove that a stochastic process $M_{t} $to be a martingale, is it necessary and sufficient to prove that $E[M_{t}]=M_{0}$ and if so, can it be proved rigorously? Thank you!
0
votes
1answer
26 views

Calculate a differenciation

$$a>0,$$ $$b>0,$$ $$\sigma >0$$ $X$ is the solution of : $$dX_t=aX_t(b-X_t)\,dt+\sigma X_t \, dB_t,\quad X_{0}=1 $$ I have also shown before that $$L_t=e^{(ab-\sigma^2/2)t+\sigma B_t}$$ Now ...
5
votes
1answer
52 views

martigale convergence theorems

Let $S_n = X_{1}+\cdots + X_{n}$ be a martingale satisfying $E[X_{k}^{2}]\leq k<\infty$, for all $k$. Show that $S_{n}$ obeys the weak law of large numbers: ...
0
votes
0answers
22 views

Girsanov theorem [duplicate]

I work on an exercice and I have to calculate: $$E(W_{t}^2e^{(\int_{0}^{T}\theta_{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta_{s}^2ds)})$$ $$\theta$$ is deterministic function I don't know how to ...
2
votes
2answers
46 views

Martingale definition

To prove that one process is Martingale, generally we prove 3 things : 1) X is adapted. 2)$$ \mathbf{E} ( \vert X_n \vert )< \infty $$ 3) $$\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n $$ I ...
3
votes
1answer
26 views

Can this sequence be a martingale?

Consider the following sequence of random variables: $X_1$ has only values $0$ and $1$ with positive probability $X_2$ only $0,1,2$ $X_3$ only $0,1,2,3$. Let's stop here. Can this sequence be a ...
2
votes
0answers
29 views

Branching Process in simple random walk

Suppose we have a simple random walk on $\mathbb{Z}$ which stars at $1$, i.e. we have iid increment $X_n$ valued in $+1,-1$ with probability $\frac{1}{2}$ each and the sum $S_n=\sum_{i=1}^{n}X_n+S_0$ ...
0
votes
1answer
30 views

positiv Martingale process

I would to like to prove that the process: $$e^{\int_{0}^{T}\theta _{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2ds}$$ is a martingale which is positiv and has a mean=1 $$\theta is continuous ...
-1
votes
1answer
36 views

Lemme itô and Martingale [on hold]

I want to to find values of $a$, $b$ such that the process: $$e^{W_{t}^2+at+b\int_\limits{0}^{t}W_{s}^2\,ds}$$ be a martingale Could you please help me do that Thank you
2
votes
0answers
21 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 ...
0
votes
1answer
23 views

Locally lipschitz implies zero quadratic variation? [closed]

How can I prove that a locally Lipschitz function has zero quadratic variation? Thanks.
2
votes
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 ...
1
vote
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
votes
0answers
41 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 $ ...
-3
votes
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
votes
0answers
22 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 ...
0
votes
0answers
8 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 ...
0
votes
0answers
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
votes
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 ...
1
vote
0answers
29 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 $n$ ...
1
vote
1answer
34 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
votes
1answer
37 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 ...
0
votes
0answers
21 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})$
0
votes
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 ...
0
votes
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\}, \; ...
1
vote
1answer
25 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 ...
1
vote
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
votes
0answers
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
votes
1answer
12 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 ...
1
vote
1answer
61 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
votes
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
votes
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 ...
0
votes
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
votes
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
votes
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
votes
1answer
80 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
votes
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
votes
1answer
61 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
62 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
votes
2answers
38 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
21 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 ...
1
vote
1answer
61 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 ...
1
vote
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. ...
1
vote
1answer
31 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 ...
1
vote
1answer
34 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
17 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 ...