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      64 views

      Wiener isometry for semimartingales

      Suppose that $Y_t$ is a special square-integrable $\mathbb{R}$-valued semi-martingale and let $\mathcal{L}^2(Y)$ denote the set of $Y$-predictable processes satisfying $$ \mathbb{E}\left[ \int_0^{\...
      3
      votes
      1answer
      188 views

      Martingales and intersection of random walks

      Let $G=(V,E)$ be a graph with $n$ vertices. Consider a pair of independent simple random walks $(X,Y)$ on the graph, each of length $L$ starting from a node $v \in V$. We denote a length-$L$ random ...
      2
      votes
      1answer
      125 views

      Bernstein Inequality for continous local martingale

      I'm looking for a simple proof of the following fact, which is somehow Bernstein inequality in continuous time. Let $(M_t)_{t\geq 0}$ be a continuous local martingale. Then : $$P\left(\sup_{t\in [0,...
      0
      votes
      1answer
      201 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 $...
      1
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      0answers
      49 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
      91 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
      49 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
      144 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
      99 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
      64 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
      71 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
      79 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
      286 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
      93 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
      156 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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