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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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        Ito integral and true martingale

        Consider a twice diferentiable function $F$ on $R$ with bounded first derivative $F'$ and a Brownian motion $W$. Show that $F(W_t)-\frac{1}{2} \int_{0}^{t} F'' (W_s)ds$ is a true martingale. I tried ...
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        93 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, ...
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        134 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 ...
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        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 $S$. ...
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        140 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 ...
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        Reference for tensor multiplication and derivatives from a computational / concrete standpoint

        I am looking for a reference for some fairly elementary definitions and calculations about "tensor-valued" functions, i.e. functions of the form $A : \mathbb R^d \to \mathbb R^{d^{n\times}}$. For ...
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        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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        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 ...
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        35 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 ...
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        140 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 ...
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        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 ...
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        34 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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        Explicit formula for Neumann heat kernel

        It is well-known that $u(x,y,t)=(4\pi t)^{-n/2}(e^{-|x-y|/4t}+e^{-|x-y'|/4t})$, $x,y\in \mathbb{R}^n_+=\{x\in \mathbb{R}^n|x_n\geq 0\}$, $y'=(y_1,\dots,y_{n-1},-y_n)$, is Neumann heat kernel of $\...
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        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 $...
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        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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