# 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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### I have a question regarding calculus, more specifically integral? [on hold]

I don't understand how RT1 integral(from V1 to V2) dV/V = RT2*ln(V2/V1), I don't understand how this becomes: -R*T2*ln(V1/V2). I have a picture right here of it and website link (answered by Andrew ...

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### Book recommendations beyond an introduction [on hold]

So I’ve scraped the surface of many topics, but I would like to go further. Can anyone recommend some continuations to the following introductory books? It’s okay if necessarily it needs to be a ...

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### 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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### 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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### 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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### Simulation of Itô integral processes where integrand depends on terminal

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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### Stochastic Integral with Time-Dependent Integrand

The Ito integral $\int_0^t H_s dX_s$ is typically defined for predictable, locally bounded processes $H$ and continuous semimartingales $X$. I'm wondering whether one can make sense of a "stochastic ...

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### Merton Problem and Equivalent Martingale Measure

Why can one not take the Equivalent Martingale Measure approach to solve the Merton problem? Are there any real-world problems combined with that approach?

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### About a class of expectations

Consider being given a $n-$dimensional random vector with a distribution ${\cal D}$, vectors $a \in \mathbb{R}^k$, $\{ b_i \in \mathbb{R}^n \}_{i=1}^k$ and non-linear Lipschitz functions, $f_1,f_2 : \...

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### Reference: Stochastic Filtering Infinite Dimensions

I've come across these Hilbert Space Signal Finite Dimensional Measurements and Linear Gaussian Hilbert space signal and measurements.
Is there any literature solving the Zakai equation when both ...

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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 ...