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        Questions tagged [stochastic-differential-equations]

        Stochastic ordinary and partial differential equations generalize the concepts of ordinary and partial differential equations to the setting where the unknown is a stochastic process.

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        Characterization of Time-homogeneous flows for conditional expectation

        Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...
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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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        2D Stochastic Navier Stokes equations with Navier boundary condition

        For the 2D Stochastic Navier Stokes equations with Navier boundary condition $$du = (\Delta u - u\cdot \nabla u - \nabla p)dt + \Phi dW$$ where we consider additive white noise here. I want to use the ...
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        Show that the transition semigroup of the strong solution to a Langevin-type SDE is immediately differentiable

        Let $\varrho\in C^1(\mathbb R)$ with $\varrho>0$ $\lambda$ denote the Lebesgue measure on $\mathcal B(\mathbb R)$ $\mu$ denote the measure with density $\varrho$ with respect to $\lambda$ $b:=2^{-...
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        Stability of the Langevin semigroup under $C_c^\infty(\mathbb R)$

        Let $h\in C^2(\mathbb R)$ $(X^x_t)_{(t,\:x)\in[0,\:\infty)\times\mathbb R}$ be a continuous process on a probability space $(\Omega,\mathcal A,\operatorname P)$ with $$X^x_t=x-\frac12\int_0^th'(X^x_s)...
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        Gradient bound for the Markov semigroup generated by the solution to an Langevin SDE

        Let $h\in C^2(\mathbb R)$ with $$h''\ge\rho\tag1$$ for some $\rho>0$ and $$\int\underbrace{e^{-h}}_{=:\:\varrho}\:{\rm d}\lambda=1$$ $\mu$ be the measure with density $\varrho$ with respect to the ...
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        How is the dominated convergence theorem applied in the proof of Lyapunov’s criterion?

        Let $$\Gamma(f,g):=\frac12f'g'\;\;\;\text{for }f,g\in C^1(\mathbb R),$$ $\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$ with a continuously differentiable and positive density $\...
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        58 views

        How is the Cauchy-Schwarz inequality used in the proof of Lyapunov's criterion in the book “Analysis and Geometry of Markov Diffusion Operators”

        Let $(E,\mu,\Gamma)$ be a full Markov triple (see definition below), $J\in\mathcal A$ with $J\ge1$ and $g\in\mathcal A_0$. In the proof of Theorem 4.6.2 of the book "Analysis and Geometry of Markov ...
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        1answer
        192 views

        Existence of a Lyapunov function for a log-concave measure

        Let $d\in\mathbb N$, $f:\mathbb R^d\to\mathbb R$ be convex with $$\int e^{-f(x)}\:{\rm d}x<\infty\tag1$$ and $\mu$ denote the measure with density $e^{-f}$ with respect to the Lebesgue measure on $\...
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        85 views

        Existence of a Lyapunov function for $-h'\varphi'+\varphi''$ where $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz

        Let $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and $$L\varphi:=-h'\varphi'+\varphi''\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ The formal adjoint of $L$ is $$L^\ast\psi:=\psi''+(h'...
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        Bayesian parameter estimation

        I am generally not that knowledgeable for math, so if my question is too broad or inaccurate, please let me know. I am currently reading a paragraph of one paper (https://www.fil.ion.ucl.ac.uk/spm/...
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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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        25 views

        Application of Gram-Charlier expansion for Swaption pricing with drift extension

        I am doing a project for university and I'm stuck at some point. The aim is to implement the Gram-Charlier expansion into the 2-factor Hull-White model with Drift extension. I already found out how ...

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