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      0
      votes
      1answer
      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 $...
      0
      votes
      0answers
      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 ...
      2
      votes
      1answer
      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 ...
      2
      votes
      0answers
      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 ...
      3
      votes
      2answers
      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 $S_n=X_1+\dotsb+X_n$, ...
      1
      vote
      1answer
      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 ...
      1
      vote
      1answer
      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 ...
      3
      votes
      0answers
      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 ...
      1
      vote
      0answers
      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 ...
      2
      votes
      1answer
      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 ...
      3
      votes
      0answers
      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 $(...
      2
      votes
      0answers
      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 ...
      2
      votes
      0answers
      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 ...
      1
      vote
      1answer
      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 $\sigma\le \tau$ ...
      3
      votes
      0answers
      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 $(e_n)_{n\in\mathbb N}$ ...

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