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

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 $$L(f) = b(x) \cdot \nabla_{x} f( … asked Jan 24 by AIM_BLB 1answer Suppose that \phi(t,x):[0,\infty)\times \mathbb{R}^d\rightarrow \mathbb{R}^d is a flow. Is it possible to extend \phi to a family of stochastic flows \{\Phi(t,x,\sigma)\}_{\sigma \in [0,1]} suc … asked Feb 11 by AIM_BLB 0answers Intuitive Question Suppose I'm given a set of k time-series \{X_t^1,\dots X_t^k\}. Is there a way to determine how much of each series is dependent on the others. Formal Question More precis … asked Oct 9 '16 by AIM_BLB 1answer If X_t is a semi-martingale, \mathfrak{F}_t is the \sigma-field generated by X_t and L^2(Pred) is the set of all \mathfrak{F}_t-predictible processes. Then is it true that:$$ \mathfrak{G …
asked Nov 3 '16 by AIM_BLB
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
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
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
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 $$dX_t = \mu(t,u_t,X_t^u)dt … asked Jun 21 '17 by AIM_BLB 2answers Is there an extension of Ito's Lemma where X_t is a semi-martingale and f:\mathbb{R}^d \rightarrow \mathbb{R} is a function which is not smooth? I've been looking but have not found much, any ref … asked Jan 11 '17 by AIM_BLB 1answer Can every càdlàg semi-martingale be written as a sequence of diffusions? That is, is the set of continuous semi-martingales dense in some Skorohod space? asked Jun 6 '17 by AIM_BLB 1answer Background I've been reading this article and it keeps referring to "Grigelionis processes", which apparently generalize Levy processes. However the paper does not define these object clearly and … asked Jan 23 '16 by AIM_BLB 1answer I'm fairly familiar with the literature dealing with convergence of SDEs in \mathbb{R}^d but recently I've needed to use extended results dealing with convergence of SDEs in Hilbert Spaces. However … asked Aug 19 '16 by AIM_BLB 1answer Motivation Then the usual stochastic filtering problem says that:$$ \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
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
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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