# Tagged Questions

The martingales tag has no wiki summary.

**8**

votes

**3**answers

762 views

### Compactness of the set of densities of equivalent martingale measures

Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S_t)_{t\in[0,T]}$. Under the no free lunch under vanishing risk (NFLVR) assumption, the set $\mathcal ...

**1**

vote

**0**answers

122 views

### Second Equality of Wald [closed]

Hi,
I'm doing an exercice about the second equality of Wald.
Let $(X_i)_{i\ge 1}$ be a sequence of integrable random variables. Let $F = (F_i)$ be a filtration such as $X$ is adapted. We suppose ...

**4**

votes

**1**answer

261 views

### What is the optimal growth of the constant in BDG?

Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} ...

**7**

votes

**5**answers

785 views

### Properties preserved under passage to augmented filtration

Dear all,
generally speaking, my question is about which properties of a stochastic process are preserved when I skip from the original to the augmented filtration.
Recall that if ...

**6**

votes

**2**answers

754 views

### De Finetti's theorem, the pointwise ergodic theorem, and reverse martingales

De Finetti's theorem says that an exchangeable sequence of random variables $X_i$ is a mixture of i.i.d. random variables. In other words, if $\mu$ is a measure on $\mathbb{R}^\infty$ that is ...

**4**

votes

**1**answer

193 views

### Reference request: Martingale decompositions (positive/negative and u.i./singular)

For a paper I am writing, I need these two facts. The proofs are fairly short, but I would rather just cite them. This is for martingales index by natural numbers. Also, I call a martingale which ...

**3**

votes

**4**answers

801 views

### Men in a bar - stoch. processes

Hello everyone,
I'm trying to solve a applied stochastic process problem and even the example is beautiful, I don't know how to approach it.
Here the problem:
10 men want to get out of a bar, they do ...

**1**

vote

**0**answers

522 views

### When are the limits of Martingales are Martingales?

Suppose I have a sequence of continuous time random variables $X_n(t)$ where $t \in [0,1]$, adapted to a filtration $F_t$, that are martingales with respect to this filtration and that $\sup_n ...

**2**

votes

**1**answer

515 views

### Stochastic integrals as honest martingales — exponential damping

We have a given positive martingale ρt, with the dynamics:
$$\textrm{d}\rho_t = \lambda_t \rho_t \textrm{d}W_t$$
where $W_t$ is a standard Brownian motion. Now we have an "exponentially dampened" ...

**3**

votes

**1**answer

388 views

### Stochastic integrals as honest martingales — comparison criterion

We have a given positive martingale $\rho_t$, with the dynamics:
$$\textrm{d} \rho_t = \lambda_t \rho_t \textrm{d} W_t$$
where $W_t$ is a standard Brownian motion. Now we have a "dumped" process p_t:
...

**7**

votes

**2**answers

554 views

### Is this ergodic inequality true?

Is anything similar to the following inequality true,
$\displaystyle P\{\max_{n \leq k \leq m} |A_k f - A_n f| > \epsilon\} \leq C \frac{||A_m f - A_n f||_1}{\epsilon}$
where $A_n f = ...

**3**

votes

**10**answers

3k views

### Best introduction to probability spaces, convergence, spectral analysis

I'm not sure if this stuff all falls under what most would just term "probability", but I'm researching applied macroeconomics and need to get a handle on the following concepts ASAP:
probability ...

**3**

votes

**1**answer

591 views

### Lower-semicomputable supermartingales with bounded increments

I'm interested in whether Levin and Solomonoff's results on "universal semimeasures" can be extended to other settings. One case that especially interests me is finding "universal" strategies in the ...

**1**

vote

**2**answers

271 views

### Martingale part of the discontinuous put payoff

I need the martingale part of the put payoff (not $C^2$..). Where $S_t=exp(\sigma W_t -\frac{\sigma^2t}{2})$
$d[(S_t -K)^+ ]$ ??
I guess I need to use local times but how?

**2**

votes

**2**answers

583 views

### Supermartingales and convergence

These feel like basic enough questions, but I don't know where to find the answer.
Let $X_1,X_2,X_3,\dots$ be a supermartingale such that $|X_{n+1} - X_n| < K$ for all $n$ ($K$ fixed). Does the ...

**10**

votes

**1**answer

2k views

### Martingales in both discrete and continuous setting

I am wondering, polynomials like
$S_n^4-6n S_n^2+3n^2+2n$ for $$S_n=\sum_{i=1}^n{X_i}$$ where $$\mathbb{P}(X_i=1)=\mathbb{P}(X_i=-1)=\frac{1}{2}$$ is a martingale (under the conventional filtration). ...

**0**

votes

**0**answers

686 views

### Is stopped brownian motion not a martingale ?

In page 45 of the book "Financial Derivatives In Theory and Practice by P.J.Hunt and J.E.Kennedy, it seems to me that the author says the stopped Brownian Motion is not a martingale as follows.
...

**6**

votes

**1**answer

531 views

### Change of space-time in Walsh's stochastic integral

One can read about Walsh's construction of martingale integral in the paper (pp.16-23)
www.math.utah.edu/~davar/ps-pdf-files/SPDEBookDK.pdf
For $U,V\in \mathcal{B}(\mathbb{R}\times \mathbb{R}^+), ...

**5**

votes

**1**answer

330 views

### Stieltjes integrals of predictable processes

I am looking for a direct proof of the fact that, roughly speaking, if $S=S_0+A+M$ is an $L^2$ semimartingale, and $M$ (the martingale part) has the martingale representation property, then for any ...

**11**

votes

**5**answers

2k views

### Brownian motion, martingales, Markov Chains - Rosetta Stone

What are the most
fundamental/useful/interesting ways in
which the concepts of Brownian motion,
martingales and markov chains are
related?
I'm a graduate student doing a crash course in ...

**1**

vote

**0**answers

300 views

### Change of Time in Stochastic Integral

Hi everyone,
Let's be given $I(0,t)$ a Stochastic Integral with respect to a local martingale $ M_t$ of the form :
$I(0,t)=\int_0^t h(s_-)dM_s$ with $h\in L(M)$ (for example $h$ is an adapted ...

**4**

votes

**1**answer

854 views

### Distribution of running maximum of a local martingale

Let $(\Omega, \mathcal{F}, \mathbb{P}, \mathcal{F}_t)$ be a given
probability space with usual conditions, on which $W$ is a standard
Brownian motion. For $x \ge 0$, consider
$$X(t) = x + \int_0^t ...

**4**

votes

**2**answers

409 views

### Path continuity for (closed) martingales?

Take a time interval $[0,T]$, and a filtered probability space $(\Omega,P,\mathcal{F},\mathcal{F}_t)$. If $X \in L^1(\mathcal{F}_T)$, then $M_t = E [X \ | \ \mathcal{F}_t]$ is a martingale. If I ...

**3**

votes

**2**answers

816 views

### Is the truncated Brownian motion of the class DL?

Let $W$ be a standard Brownian motion under given probability space.
For a given constant $a$, $W^a$ is a truncated Brownian motion by stopping time
$T^a = \inf(t>0:W(t) = a)$. That is, $W^a(t) = ...

**3**

votes

**1**answer

319 views

### initial condition of a diffusion approximation

I am trying to prove that a certain sequence of Markov chains $x^N_k$ converges towards a diffusion process. The invariant measure of $x^N$ is $\pi^N$ and the Markov chain $x^N$ is started in ...

**1**

vote

**5**answers

1k views

### Martingales and Betting Strategies

Does anyone know of a good introduction to the theory of martingales and betting strategies from the point of view of statistics and/or probability theory? I'm looking for something basic, with lots ...