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        Questions tagged [stochastic-calculus]

        Stochastic calculus provides a consistent theory of integration for stochastic processes and is used to model random systems. Its applications range from statistical physics to quantitative finance.

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        About a class of expectations

        Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...
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        Reference: Stochastic Filtering Infinite Dimensions

        I've come across these Hilbert Space Signal Finite Dimensional Measurements and Linear Gaussian Hilbert space signal and measurements. Is there any literature solving the Zakai equation when both ...
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        Why control a continuous approximation of stochastic gradient descent instead of just the SGD?

        In "Stochastic modified equations and adaptive stochastic gradient algorithms" (Li et. al 2015) the authors approximate stochastic gradient descent, as in $$x_{k+1} = x_k - \eta u_k \nabla f_{\...
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        Domain of the Generator of a Bessel process

        Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$ \begin{align} \rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t} \end{align} where $(W_{t})_{t\geq ...
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        Divergence form degenerate pde and Feynman Kac

        Consider $$ Au:=\operatorname{div}\left(y^{\beta}\nabla u\right) \text{ for } (x,y)\in \mathbb{H} $$ and $u|_{\mathbb{R}}(x,0)=\phi(x)$ and some $\beta\in (0,1)$. For $\phi\in L^{2}(\mathbb{R},dx)$ (...
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        Expectation of random variables coincides

        Let $Y_1:=(X_i)_{i \in \mathbb Z}$ be a family of random variables that are identically distributed but not necessarily independent. We can then also define the shifted sequence $Y_2:=(X_{i+1})_{i \...
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        How to calculate the probability of 2 events happening in time series under only cdf information?

        In time domain $0\rightarrow T$, there are two independent events $A$ and $B$. $B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...
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        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 ...
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        Time discretization in the Feynman-Kac formula with boundary conditions

        I am applying the Feynman-Kac theory for solving a PDE with boundary conditions. For the SDE simulation I use the Euler-approximation, which introduces a time-step $h$ for the Brownian Motion, and ...
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        Is there solution to a backward stochastic differential equation with $yz$ in the generator?

        Please consider the following backward stochastic differential equation: $$ Y(s)=\xi+\int_{s}^{T}a(u)Y(u)+b(u)Y(u)Z(u)du-\int_{s}^{T}Z(u)dW(u)$$ Here $a(s)$, $b(s)$ are square-integrable stochastic ...
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        When is $f(t,W_t)$ an Ito process?

        Consider a Brownian motion $(W_t)_{t\in[0;T]}$. If $f\colon [0;T] \times \mathbb R \to \mathbb R$ is $C^{1,2}$, we know that $(f(t,W_t))_{t\in[0;T]}$ is an Ito process and we can directly write down ...
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        Reference: Stochastic Optimal Control with cost functional

        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 ...
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        Hitting probability of a transient diffusion process

        I have a question about properties of transient diffusion process. In the case of $d$-dimensional Brownian motion $B=(B_t,P_x)$ ($d \ge 3$), we can prove that \begin{align} (1)&\quad 0<P_{x}(\...
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        Matching Stochastic Flows

        Let $\nu = (\nu_t)_{t \in [0,T]} \in C( [0,T], \mathcal{P}_2(\mathbb{R}) ) ,$ where $\mathcal{P}_2(\mathbb{R})$ denotes the space of probability measures with finite moment equipped with the ...
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        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 ...

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