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

learn more… | top users | synonyms

10
votes
0answers
168 views

Removing deterministic discontinuities from semi-martingales

Let $X:=(X_t)_{0 \le t \le T}$ be a solution of the SDE $$ X_t = X_0 + \int_0^t \sigma(s,X_s) dW_s + \sum_{i=1}^n f_i(X_{t_i^-}) 1_{\{t > t_i\}}$$ where $t_1,\cdots,t_n \in [0,T]$ and $(f_i)_{1 \le ...
7
votes
0answers
170 views

Sufficient condition in terms of stopping times for a stochastic process to be a local supermartingale

(Question edited in response to Nate's comment) Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin ...
6
votes
0answers
499 views

Azuma's inequality to McDiarmid's inequality?

I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from ...
6
votes
0answers
150 views

Proving existence of limit by Martingale.

I'm thinking about a question: Suppose $y_n > −1$ for all $n$ and $\sum |y_n| < \infty$. Show that $\prod_{m=1}^\infty (1 + y_m)$ exists. Since $\sum |y_n| < \infty$, we must be able ...
6
votes
0answers
295 views

An application of the Optional Sampling Theorem

let $S(k), k\geq 0$ a discrete random process. Suppose $S(N)$ is with probability one either 100 or 0 and that $S(0)=50$. Suppose further there is at least a sixty percent probability that the price ...
4
votes
0answers
19 views

Harmonic functions and null recurrence

Let $(X_n)_{n \geq 0}$ be a irreducible Markov chain defined on a countable state space $S$. It is known some ways to figure out if this chain is recurrent or not looking for superharmonic functions ...
4
votes
0answers
97 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 ...
4
votes
0answers
117 views

Determine if this is a Martingale

I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$. We know that $S_t$ is a local martingale because if we stop it ...
4
votes
0answers
94 views

Submartingality of generalized stochastic exponential of a BMO martingale

I attended a talk today on BMO martingales. It was my first encounter with the subject, and this may explain my inability to solve this myself. We take a continuous local martingale $L$, and say ...
4
votes
0answers
144 views

Time scaling of Brownian motion

Let $(B_t)_{t\geq 0}$ be a standard Brownian motion and $A_t$ be an increasing continuous process adapted to the filtration generated by the Brownian Motion and $A_0 = 0$. I am trying to prove ...
4
votes
0answers
157 views

A Stopping Theorem for Right-Continuous Submartingales

Reading through the book "Brownian Motion & Stochastic Processes" by Karatzas and Shreve, I found the following problem (problem 3.24, page 20): Suppose that $ \{ X_t, \mathcal{F}_t \ | \ 0 \leq ...
3
votes
0answers
93 views

Qual Question concerning martingale

Suppose $X_n$ is a sequence of random variables that has the property that $\sup|X_n| \leq 1$ a.s. Then use Doob's decomposition to prove that $\sum_{n\geq 1} X_n$ converges a.s. iff the sum ...
3
votes
0answers
171 views

Levy's extension of the Borel-Cantelli Lemmas

Following is the statement and proof of Levy's extension of the Borel-Cantelli Lemmas, as given in Williams' "Probability with Martingales" (1991), in section 12.15 on page 124. I understand most of ...
3
votes
0answers
85 views

Absolute continuity of quadratic variation of continuous local martingales

I am interested to know if there are any simple sufficient conditions on continuous local martingale to have absolutely continuous quadratic variation. In general , we know only that quadratic ...
3
votes
0answers
48 views

Upper bounds on the sum in a Martingale process

My question is related the hitting time of not a random walk, but a more general martingale process. Suppose we start with an arbitrary $x_0=x$ with $0\leq x\leq 1$. We compute $x_{t+1}$ from $x_t$ ...
3
votes
0answers
81 views

Discontinuous local martingales with finite variations

I would like to know if a discontinuous local martingale with paths of finite variations almost surely is a martingale. I feel that it should be the case but can't find a straightforward argument. As ...
3
votes
0answers
170 views

Convergence of Martingale.

The question is: 5.2.11. Let $X_n$ and $Y_n$ be positive integrable and adapted to Fn. Suppose $\mathbb E(X_{n+1}|\mathcal F_n) ≤ (1 + Y_n )X_n$ with $ \sum Y_n < \infty$ a.s. Prove that ...
3
votes
0answers
302 views

Superharmonic function and supermartingale

If $f$ is a nonnegative superharmonic function in dimension 2, how to prove that $f$ is constant? There is an exercise in R.Durrett's probability book, which gives out a method to prove it by ...
3
votes
0answers
152 views

An identity about a continuous local martingale

Let $M_t$ be a continuous local martingale with $M_0=0$ and define $I_t^0=1$ and $I_t^n= \int_0^t I_s^{n-1}\; dM_s$. Prove that we have $$n I_t^n= I_t^{n-1}- \int_0^t I_s^{n-2} \;d((M)_t) $$ where ...
2
votes
0answers
37 views

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 ...
2
votes
0answers
67 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 ...
2
votes
0answers
26 views

Is there a Burkholder-Davis-Gundy inequality for martingale increments?

is there a Burkholder-Davis-Gundy inequality for martingale increments? More specifically, I would like to find a finite bound of order $h^{p/2}$ for the expectation $$\operatorname{E} \left[ \sup_{t ...
2
votes
0answers
45 views

Rate of convergence of a martingale

I have a question related to convergence rates of martingales: Assume that there is a sequence of maximized likelihood ratios: $ \frac{f_{\hat{\theta}_{n}} \left ( Y_{1},Y_{2},\dots,Y_{n} \right ) ...
2
votes
0answers
45 views

Second (centered) moment for martingales

Take the process ${x}_t$ following geometric Brownian motion (GBM) $$x_t=\mu x_t \,dt+\sigma x_t \,dW_t$$ with $x_0>0$ known. It has first moment equal to $$\text{E}[x_t]=x_0 e^{\mu t}$$ and second ...
2
votes
0answers
32 views

Finding a pre-visible process

Question: Let $W_t$ be a standard brownian motion under P with filtration $\mathscr F_t$. Let: $$ M_t=\mathbb E[W_T^2|\mathscr F_t] $$ Show that $M$ is a P martingale. This is simple enough using ...
2
votes
0answers
208 views

Is the absolute value of Brownian motion a super martingale?Is it a sub martingale? Is it a Markov process?

I've just started to study random processes and I'm trying to solve the following problem: Let $W(t)$ be a Brownian motion with filtration $F(t)$ generated by $ W(t)$ (i.e., $F(t)=\sigma \left( ...
2
votes
0answers
54 views

Lower bound for stochastic process

Suppose the non-negative stochastic process $(X_t,Y_t)$ is such that $E\{X_t - X_a | Y_u \in A \,\,\forall u \in [a,t] \} \geq Z(A)(t-a)$. Let $T_{A}$ be the time of a visit to $A$. Assuming that the ...
2
votes
0answers
146 views

Checking proof that a given process is a martingale

I am interested in justify the well known result about the process $M^\lambda _t =\exp\left(\lambda B_t - \frac{\lambda^2}{2} t\right)$ being $\mathcal F_t$-martingale in the filtered probability ...
2
votes
0answers
59 views

Dimension free Concentration bounds for Martingales

Consider the following random process which is defined on $n$ numbers $0\leq x_1,\ldots,x_n\leq 1$: At each step, pick an arbitrary number, say $x_i$. Then randomly (and independently) change its ...
2
votes
0answers
83 views

Azuma's inequality with high probabilistic bounds

Let $(X_n)_{n \geq 0}$ be a super-martingale, that is $\mathbb{E}[X_{n+1} \mid X_1, \dots, X_n] \leq X_n$. Let's further assume that $\Pr[|X_n - X_{n-1}| < c_n] \geq 1-\delta$. Does there exist any ...
2
votes
0answers
162 views

What are the conditions for when a martingale has a continuous modification?

Update: This question has been moved to Mathoverflow. Please answer it there. There is a theorem as follows: Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. ...
2
votes
0answers
83 views

Doob Decomposition of American Option

I am trying to figure out the Doob decomposition of an American put option in a discrete time binomial model. I know how to price the American put, but I'm having trouble expressing it as the sum of ...
2
votes
0answers
162 views

how to show that the price process is a martingale

Suppose I have an $d$-dimensional semimartingale $S=\{S_t\}$ with $t\in[0,T]$ under $P$. $S $ need not to be continuous (RCLL can be assumed). Suppose $Q$ is an equivalent measure w.r.t. $P$ such that ...
2
votes
0answers
306 views

strong and weakly orthogonality for martingales

If I have two RCLL martingales $X,Y$, both bounded in $L^2$, hence uniformly integrable. Then we call $X,Y$ weakly orthogonal if $E[X_\infty Y_\infty]=0$ and we call $X,Y$ strongly orthogonal if $XY$ ...
2
votes
0answers
246 views

Doob inequality for continuous martingales

In our class we have proven Doob's inequality for discrete martingale, i.e. Let $(M_n)_{n \in \mathbb{N}}$ a martingale, then $$ \| \max_{0\le k\le n} M_k\|_p \le C_p \|X_n\|_p$$ for $p\in ...
2
votes
0answers
243 views

Show that the stopped sequence is a martingale

This is a series problem and I'm struggling with the last part. I assume that the last part has nothing to do with previous ones, so i won't put up the other parts. Question is : Let $\tau$ be a ...
2
votes
0answers
74 views

Optional stopping theorem for Hilbert valued martingales

Suppose $X_n$ is a Hilbert-valued martingale. Does the optional sampling theorem apply in this case? Does anyone know where I can find a proof? Thank you.
1
vote
0answers
109 views

Condition for a process to be a supermartingale

I am struggling in this question: Let $W$ denote a Brownian motion. Given that $ X_t = e^{- \lambda t} X_0 + \int_0^t \sigma e^{- \lambda (t-s)} \,dW_s$ solves the SDE \begin{equation} dX_t = - ...
1
vote
0answers
16 views

Interpreting a sequence and showing that it is a martingale.

I saw that one guy already asked this question, but he did not get an answer and I wasn't able to comment his thread. So, hopefully this is allowed. I am wondering about the following problem: The ...
1
vote
0answers
46 views

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 ...
1
vote
0answers
32 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 | ...
1
vote
0answers
32 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 $n\in\mathbb{N}_{0}$. ...
1
vote
0answers
27 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
vote
0answers
35 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
vote
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 ...
1
vote
0answers
21 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 ...
1
vote
0answers
49 views

A problem on super/sub martingale

Let $(X_n, \mathscr{F_n}), n \geq 0$ be a super martingale and $T$ an $\{F_n\}$-stopping time a.s. bounded by $N \lt \infty$. Show that $$ E[|X_T|] \leq 3 \max_{n \leq N} E[|X_n|]$$ I can prove that ...
1
vote
0answers
27 views

If a random integral has moments of all orders, is the same true for its kernel?

Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| ...
1
vote
0answers
72 views

Occupancy distribution bounds for $k$ balls in $m$ bins

Suppose we throw $k$ (distinct) balls into $m$ (distinct) bins, and let $B$ count the number of non-empty bins. I am interested in lower bounds on $B$. More precisely, I wish to bound from above the ...
1
vote
0answers
65 views

Applying Ito lemma to multi dimensional semimartingales

Let $X_t=(X^1,\dots,X^d$) be a d-dimensional semimartingale. Then the formula for Ito's lemma I have found in several places, including wikipedia, is: $f(X_T)-f(X_0)=\sum_{i=1}^d\int_0^T ...