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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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      50 views

      Defining weak solutions to infinitely many SDEs on the same probability space

      Suppose I have an SDE of the form $$dX_t=b(X_t)dt+\sigma (X_t)dB_t+\int_{\mathbb{R}}G_{t-}(y)N(dtdy)$$ which I can solve weakly if I cut off the last integral to range over the set $\{\mid{y}\mid > ...
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      Example of a “very noisy” SDE on a compact manifold with zero maximal Lyapunov exponent

      Setting: Let $M$ be a compact connected $C^\infty$ Riemannian manifold of dimension $D \geq 2$, with $\lambda$ the normalised Riemannian volume measure. Write $T_{\neq 0}M \subset TM$ for the non-...
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      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,...
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      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, \...
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      1answer
      127 views

      Simulation of Itô integral processes where integrand depends on terminal (Volterra process)

      I need to simulate a process of the form $$X_t=\int_0^t f(s,t)\mathop{dW_s}$$ where $f$ is deterministic and the integral is an It? integral. I know I can simply take finite It? sums of discrete ...
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      55 views

      Smoothness of expectation

      Suppose that $X_t$ is a strong solution to the SDE, $$dX_t = C_t \,dB_t$$ where $B_t$ is a standard Brownian motion and $C_t \ge 0$ is measurable with respect to the natural filtration generated by ...
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      87 views

      Why should we give special attention to at most polynomially growing solutions of PDEs?

      The equation \begin{gather} \frac{\partial u}{\partial t} (t,x) = \frac{1}{2} \text{Trace}[\sigma(x) \sigma(x) (\text{Hessian}_x u)(x,t)] + \langle \mu (x) , (\nabla_x u) (t,x) \rangle, \\ u(0,x) = \...
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      1answer
      66 views

      Conditioning on an irrelevant variable in a martingale control problem

      Suppose I have two independent Brownian motions $B^1_t, B^2_t$ and $\mathbb F_t$ be the natural filtration generated by them. Let $T > 0$ be a fixed finite number. Let $q_t$ be a $[-1,1]$ valued $\...
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      56 views

      Can there be a explicit expression of g as defined in the link

      This is related to the paper in the link :https://arxiv.org/pdf/1610.08468.pdf titled Algebraic normalisation of regularity structures. In the method of re- normalization the functional $g$ shown in ...
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      38 views

      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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      1answer
      134 views

      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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      87 views

      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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      1answer
      158 views

      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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      61 views

      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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      31 views

      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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