# All Questions

Tagged with stochastic-processes martingales

83 questions

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124 views

### On the martingale representation theorem

I seen several sources claim that any martingale in a Brownian filtration is continuous. However while working with processes of the form $\mathbb{E}(X\mid \mathcal{F}_{s})$ for some random variable $...

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43 views

### Prior state dependent transition probability ABRACADABRA problem

The power of the martingale trick for computing the expected stopping time is amply demonstrated in this question and this answer as an advanced version of the ABRACADABRA problem. However, it seems ...

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86 views

### Concavity, martingales and stopping time

Suppose $(x_t)_t$ is a bounded $\mathbb F_t$ martingale and $f(t,x)$ is continuous, bounded, and concave in $x$. So, for any $s \ge t$, $$\mathbb E_t f(s,x_s) \le f(s,\mathbb E(x_s)) = f(s,x).$$
Does ...

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45 views

### p-Variation distance defines semi-martingales

Question
When, does the process $\tilde{X}_t$, defined path-wise by
$$
\tilde{X}_t(\omega)\triangleq \rho_{\frac1{2}}\left((y_t,\mathbb{Y}_t),(x_t(\omega),\mathbb{X}_t(\omega))\right),
$$
define a ...

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137 views

### Expectation of the exitpoint distance for the symmetric random walk

Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$.
Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\dotsb+X_n$, ...

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89 views

### Martingale representation theorem for symmetric random walk

Let $X(t)$ be a martingale w.r.t. filtration generated by Brownian motion $B(t)$. There is a well-known theorem that states that there is a unique adapted process $H(t)$ such that
$$ X(t) = \int_0^t ...

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61 views

### n-factor martingale representation theorem

Baxter & Rennie at pag. 162 state the following theorem.
Let $W$ be an $n$-dimensional $\mathbb Q$-Brownian motion and let $M_t=(M_1(t),...,M_n(t))$ be an $n$-dimensional $\mathbb Q$-martingale ...

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70 views

### Convergence of SDEs

Suppose that $\{a_n(x)\}_{n \in \mathbb{N}}$ is a sequence of real-valued Lipschitz functions with domain $\mathbb{R}^d$, which converges $m$-a.e. to a Lipschitz function $a$. Suppose that $b$ is a ...

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72 views

### Is martingale solution equivalent to weak solution for SDE driven by stable process

Consider the following SDE
$$
d X_t=b(X_t)d t+d L_t,
$$
where $L_t$ is the symmetric $\alpha$-stable process. The corresponding generator is given by
$$
L=\Delta^{\alpha/2}+b\cdot\nabla.
$$
Is the ...

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273 views

### CLT for Martingales

I posted this question originally in math stack exchange, but I got no answer.
(https://math.stackexchange.com/questions/2604591/clt-for-martingales)
In wikipedia, there is a version of a CLT for ...

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90 views

### Has there been any study of the “extreme convergence property” for martingales?

Let $(M_n)_{n \geq 1}$ be a uniformly bounded martingale over a probability space $(\Omega,\mathcal{F},\mathbb{P})$. Define the probability measure $\mu$ on $\mathbb{R}^\mathbb{N}$ to be the law of $(...

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150 views

### Non-negative martingale transforms and Radon Nikodym derivatives

Consider a filtered probability space $(\Omega, (\mathcal F_n), \mathcal F, \mathbb P)$, where $\Omega$ is the set of sequences with value in some $E \subseteq \mathbb R^d$, and $\mathcal F$ is the ...

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81 views

### Modified Pólya's Urn Process

Suppose that we have an urn that initially contains $n$ balls, partitioned into $k\geq 2$ color-classes with respect to some initial probability distribution $P=(p_1,\dots,p_k)$.
At each discrete time ...

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158 views

### Optional stopping with unbounded stopping times $\sigma\le \tau$ case

Let $M_t$ be a càdlàg martingale process. Then it is evident, by the optional stopping theorem, that for $\mathcal F_t$-stopping times $\sigma, \tau$ (not necessarily bounded) where $\sigma\le \tau$ ...

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75 views

### How can we show that the quadratic covariation of a Hilbert space valued martingale takes values in the space of nonnegative operators?

Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a complete filtration of $\mathcal A$
$H$ be a separable $\mathbb R$-Hilbert space
$(e_n)_{n\in\mathbb N}$ ...