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      All Questions

      1
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      1answer
      101 views

      Continuity of green functions

      I have a technical question on a continuity of green function. Setting Let $E$ be a locally compact separable metric space and $m$ a locally finite measure on $E$. Let $X=(\{X_t\}_{t \ge 0},\{P_x\}...
      1
      vote
      1answer
      63 views

      Filtration exercise

      I am struggling with 1.7 exercise from the Karatzas, Shreve "Brownian motion and stoch. calulus". Denote by $\mathcal{F}^X_{t_0}$ the natural filtration corresponding to a process $X:[0,\infty)\times ...
      6
      votes
      1answer
      80 views

      The distribution of the area of a region cut out by chordal SLE?

      Let $\mathbb{D}$ be the unit disc. Let $a,b \in \partial \mathbb{D}$. Let $\gamma$ be a chordal $SLE_{k}$ from $a$ to $b$. For $k \leq 4$, $\gamma$ is a simple curve, and so $\mathbb{D} \setminus \...
      0
      votes
      0answers
      103 views

      An application of Girsanov's Theorem

      Let $(W,H,i)$ be the classical Wiener space where $W=C_0([0,1])$, $H$ is the Cameron-Martin space. Let $A= I_{W}+a$ such that $A:W \rightarrow W$ and $a \in L^{0}(\mu,H)$, $a$ has adapted derivative, ...
      6
      votes
      1answer
      150 views

      Maxima of Brownian motion

      It is well-known that Brownian motion attains infinitely many maxima in each time interval $[0,T]$ a.s.. From a physics perspective it seems reasonable that when the disorder of the path of a ...
      2
      votes
      0answers
      86 views

      Conditioning on future events, strong Markov property, independence

      I have a question on an argument appearing in this article P. Setting Let $S=(1,\infty) \times (-1,1) \subset \mathbb{R}^2$ and let $X=(\{X_t\},\{P_x\}_{x \in S})$ be a diffusion process on $S$. ...
      1
      vote
      1answer
      142 views

      Inequalities for moments of a certain integral

      Let $X(t)$ be a stationary Gaussian process, $EX(t)=0$, the correlation function $R(\tau)$ is given. What bounds from above can be given for the $p$-th moment ($p>0, p \in \mathbb{R}$) of the ...
      1
      vote
      0answers
      91 views

      How to judge the solution process of an SDE to lie on the sphere?

      Consider the following SDE on $\mathbf R^d$: \begin{equation}\tag{*} dX_t^i = -\frac{d-1}{2}X_t^i dt + \sum_{j=1}^d(\delta^{ij}-X_t^iX_t^j)dW_t^j, \quad i=1,2,...,d, \end{equation} where $W = (W^1,W^2,...
      0
      votes
      0answers
      26 views

      A modification of Kolmogorov's continuity criterion for $C_{tem}$

      I am wondering about how to prove a modification of Kolmogrov's continuity criterion in order to also being able to quantify the growth behaviour of the process. In particular, I am interested in the ...
      0
      votes
      0answers
      37 views

      Derivative of stochastic process in $L^p$ coincides with sample path derivative

      In the article Random ordinary differential equations, by J.L. Strand (1970) (it is available at https://core.ac.uk/download/pdf/82447522.pdf), it is stated the following result, which relates ...
      0
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      2answers
      162 views

      Quadratic covariation of two not independent Brownian motions

      Given two not independent Brownian motions, $X$ and $Y$. I was wondering if we can say anything about the quadratic covariation of $X$ and $Y$, $\langle X,Y \rangle_t$. I know that for two independent ...
      2
      votes
      0answers
      46 views

      Conformal mappings and diffusion processes with boundary condition

      I have a question on a relation between conformal mappings and diffusion processes with boundary condition. Let $D_1$ be a smooth simply connected domain of $\mathbb{R}^2 \cong \mathbb{C}$. This may ...
      0
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      0answers
      37 views

      Expected Solution of a Stochastic Differential Equation Expressed as Conditional Expectation

      On all you geniusses out there: this is a tough one. Preliminaries and Rigorous Technical Framework Let $T \in (0, \infty)$ be fixed. Let $d \in \mathbb{N}_{\geq 1}$ be fixed. Let $$(\Omega, \...
      1
      vote
      1answer
      152 views

      On Riemann integration of stochastic processes of order $p$

      Let $x:[a,b]\times\Omega\rightarrow\mathbb{R}$ be a stochastic process, where $\Omega$ is the sample space from an underlying probability space. Let $L^p$ be the Lebesgue space of random variables on $...
      1
      vote
      1answer
      126 views

      Stochastic Integral with Time-Dependent Integrand

      The Ito integral $\int_0^t H_s dX_s$ is typically defined for predictable, locally bounded processes $H$ and continuous semimartingales $X$. I'm wondering whether one can make sense of a "stochastic ...

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