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      A stochastic process is a collection of random variables usually indexed by a totally ordered set.

      2
      votes
      0answers
      If $X_t$ is a (continuous) diffusion process solving $$ dX_t = b(t,X_t)dt + \sigma(t,X_t)dW_t , $$ then its infinitesimal generator, denoted by $L$ is of the form by $$ L(f) = b(x) \cdot \nabla_{x} f( …
      asked Jan 24 by AIM_BLB
      0
      votes
      1answer
      Suppose that $\phi(t,x):[0,\infty)\times \mathbb{R}^d\rightarrow \mathbb{R}^d$ is a flow. Is it possible to extend $\phi$ to a family of stochastic flows $\{\Phi(t,x,\sigma)\}_{\sigma \in [0,1]}$ suc …
      asked Feb 11 by AIM_BLB
      2
      votes
      0answers
      Intuitive Question Suppose I'm given a set of $k$ time-series $\{X_t^1,\dots X_t^k\}$. Is there a way to determine how much of each series is dependent on the others. Formal Question More precis …
      asked Oct 9 '16 by AIM_BLB
      1
      vote
      1answer
      If $X_t$ is a semi-martingale, $\mathfrak{F}_t$ is the $\sigma$-field generated by $X_t$ and $L^2(Pred)$ is the set of all $\mathfrak{F}_t$-predictible processes. Then is it true that: $$ \mathfrak{G …
      asked Nov 3 '16 by AIM_BLB
      0
      votes
      0answers
      The set $\mathscr{S}$, of semi-martingales is a topological vector space under the Emery topology on the space of semi-martingales. There has been some recent research on closures in this topology (f …
      asked Oct 21 '18 by AIM_BLB
      2
      votes
      1answer
      I know there are Itô formulas for cylindrical Brownian motions with values in a Hilbert space and Itô formulas for Lévy processes in $\mathbb{R}^d$. My question is: does there exist an Itô formula fo …
      asked Sep 3 '16 by AIM_BLB
      0
      votes
      0answers
      There are a variety of control problems for controlled diffusions $X_t^u$, with the terminal cost given by $$ J(u)\triangleq \mathbb{E}\left[g(X_T,u)+\int_0^t h(X_t,u_t)ds\right], $$ function $g$ and …
      asked Dec 21 '18 by AIM_BLB
      1
      vote
      0answers
      Is there any literature dealing with a stochastic control problem whose cost-functional $J_t$ is stochastic also? That is, let $X_t^u$ is the solution to a controlled SDE $$ dX_t = \mu(t,u_t,X_t^u)dt …
      asked Jun 21 '17 by AIM_BLB
      0
      votes
      2answers
      Is there an extension of Ito's Lemma where $X_t$ is a semi-martingale and $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is a function which is not smooth? I've been looking but have not found much, any ref …
      asked Jan 11 '17 by AIM_BLB
      3
      votes
      1answer
      Can every càdlàg semi-martingale be written as a sequence of diffusions? That is, is the set of continuous semi-martingales dense in some Skorohod space?
      asked Jun 6 '17 by AIM_BLB
      -1
      votes
      1answer
      Background I've been reading this article and it keeps referring to "Grigelionis processes", which apparently generalize Levy processes. However the paper does not define these object clearly and …
      asked Jan 23 '16 by AIM_BLB
      1
      vote
      1answer
      I'm fairly familiar with the literature dealing with convergence of SDEs in $\mathbb{R}^d$ but recently I've needed to use extended results dealing with convergence of SDEs in Hilbert Spaces. However …
      asked Aug 19 '16 by AIM_BLB
      1
      vote
      1answer
      Motivation Then the usual stochastic filtering problem says that: $$ \operatorname{argmin}_{Z \in L^2(\mathscr{G}_t)}\,\mathbb{E}[(Y_t-Z_t)^2], $$ where $\mathscr{G}_t$ is the $\sigma$-algebra genera …
      asked Sep 29 '16 by AIM_BLB
      3
      votes
      1answer
      Suppose I have signal process $\lambda_t$ following the dynamics \begin{equation} \begin{aligned} \zeta_t&=\mu^{\zeta}(t,{\zeta}_t)dt+\sigma^{\zeta}(t,{\zeta}_t)dW^{\zeta}_t\\ \xi_t&=\mu^{\xi}(t,\xi_t …
      asked Jul 10 '17 by AIM_BLB
      3
      votes
      0answers
      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 L …
      asked Sep 22 '18 by AIM_BLB

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