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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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        1answer
        207 views

        Is this a “contradiction” on stochastic Burgers' equation? How to understand it?

        For the stochastic Burgers' equation with linear noise, I can deduce two results. Both of them can be applied to same initial data, but the first result means the global existence with high ...
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        61 views

        The probability distribution of “derivative” of a random variable

        Disclaimer: Cross-posted in math.SE. Let me set the stage; Consider a stochastic PDE, which has to following form $$\partial_t h(x,t) = H(x,t) + \chi(x,t),$$ where $H$ is a deterministic function, ...
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        53 views

        Backward Stochastic Differential Equation

        Let $W_t$ be a standard Brownian motion. Let $T$ be the terminal date, $X_T=x$, and $$ dX_t=f_tdt+B_tdW_t $$ where $f_t$ and $B_t$ (yet to be determined) have to be adapted to the filtration generated ...
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        58 views

        Extension of probability space problem: Hilbert space valued process V.S. random field

        Maybe the question should be "Understanding the measurability: Hilbert space valued process V.S. random field" Consider the SPDE $${\rm d}u+\cdots{\rm d}t=\sigma(t,u){\rm d}W.$$ Consider the ...
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        199 views

        An application of Itô's formula to an SDE on a Lie group

        I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows. Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE $$dg(t)...
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        35 views

        Sufficient conditions for taking limits in stochastic partial differential problems

        Let's say we have a (parabolic) Cauchy problem: $$ (1) \hspace{0.5cm} u_t (x,t)+A(u) \cdot u_x (x,t))=\nu \cdot u_{xx} (x,t) + \epsilon \cdot f(u) \cdot W, $$ $$(2) \hspace{0.5cm} u(x,0)=u_0(x), $$ ...
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        49 views

        Lebesgue Integral in SDE

        In the context of proving existence of solutions of S(P)DEs, I've found that few (if any) texts offer significant mention to the deterministic drift term of the form $$ \int_0^t f(s,X(s))ds. $$ If we ...
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        59 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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        103 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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        39 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
        131 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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        57 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
        68 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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