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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.

551 questions
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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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### 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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### 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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### 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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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 $W = (W^1,W^2,... 0answers 24 views ### A modification of Kolmogorov's continuity criterion for$C_{tem}$I am wondering about how to prove a modification of Kolmogrov's continuity criterion in order to also being able to quantify the growth behaviour of the process. In particular, I am interested in the ... 0answers 35 views ### Derivative of stochastic process in$L^p$coincides with sample path derivative In the article Random ordinary differential equations, by J.L. Strand (1970) (it is available at https://core.ac.uk/download/pdf/82447522.pdf), it is stated the following result, which relates ... 2answers 140 views ### Quadratic covariation of two not independent Brownian motions Given two not independent Brownian motions,$X$and$Y$. I was wondering if we can say anything about the quadratic covariation of$X$and$Y$,$\langle X,Y \rangle_t$. I know that for two independent ... 0answers 44 views ### Conformal mappings and diffusion processes with boundary condition I have a question on a relation between conformal mappings and diffusion processes with boundary condition. Let$D_1$be a smooth simply connected domain of$\mathbb{R}^2 \cong \mathbb{C}$. This may ... 0answers 34 views ### Expected Solution of a Stochastic Differential Equation Expressed as Conditional Expectation On all you geniusses out there: this is a tough one. Preliminaries and Rigorous Technical Framework Let$T \in (0, \infty)$be fixed. Let$d \in \mathbb{N}_{\geq 1}$be fixed. Let $$(\Omega, \... 0answers 72 views ### Explicit formula for Neumann heat kernel It is well-known that u(x,y,t)=(4\pi t)^{-n/2}(e^{-|x-y|/4t}+e^{-|x-y'|/4t}), x,y\in \mathbb{R}^n_+=\{x\in \mathbb{R}^n|x_n\geq 0\}, y'=(y_1,\dots,y_{n-1},-y_n), is Neumann heat kernel of \... 1answer 150 views ### On Riemann integration of stochastic processes of order p Let x:[a,b]\times\Omega\rightarrow\mathbb{R} be a stochastic process, where \Omega is the sample space from an underlying probability space. Let L^p be the Lebesgue space of random variables on ... 1answer 127 views ### Simulation of Itô integral processes where integrand depends on terminal (Volterra process) I need to simulate a process of the form$$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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