# Questions tagged [stochastic-calculus]

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### Is it allowed to pull a stochastic variable/process outside the integral?

If we encounter an integral of the form: $I=\int\limits_0^t X(s)y(s)ds$, where $X(s)$ is a stochastic (variable or process) and $y(s)$ is a deterministic function, say $e^{as}$. I want to know under ...
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### Cannot obtain empirical Laplace distribution for increments of a laplace motion

Consider the Laplace motion (a special type of Levy process where the stationary and indepedent increments are Laplace distributed). One representation of the Laplace Motion is through Brownian ...
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### I want to calculate $\int f(X_t) dB_t$ where $B(t)$ is Brownian motion and $X_t$ satisfies $d X_t = \mu dt + \sigma dB_t$

Let $B_t$ be Brownian motion, and $X_t$ satisfies the following Ito SDE: $$d X_t = \mu\, dt + \sigma\, d B_t,$$ and $f$ is a function over $X_T$. I want to calculate $\mathbb{E}[f(X_t)dB_t]$. It ...
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### What are the deep learning books covering stochastic differential equations only?

I want to solve a simple Stochastic Differential Equation say $$dY=Y^2 dt+\sigma Y^2 dW$$ and then make future predictions. I am conversant with MATLAB and LSTMs in python. Is there a book that can ...
1 vote
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### Deriving a Stochastic Equation

Edit: I'm currently reading a paper (The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing, American Journal of Operations Research, ...
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### How can one find a system of SDE's from a probability density function?

Suppose I have a joint distribution function say $p(x,y,z)=f_{X, Y, Z}(x,y,z)$. Is it possible to find a system of stochastic differential equations or a single stochastic differential equation from ...
1 vote
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### Sample path boundedness of Gaussian Processes

Consider a Gaussian Process $GP(0,k(x,x'))$ with zero mean and bounded, continues covariance function $k(x,x')<c,\quad \forall x,x\in\mathbb{R}^n$. Are the sample paths of this process (almost ...
1 vote
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### Find stochastic differential equation which best describes time-series

I have a time series with daily observations in a time span of 20 years describing the price of commodities. Given that this time series is non-stationary, is it possible to find a Stochastic ...
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### Understanding policy gradient theorem - What does it mean to take gradients of reward wrt policy parameters?

I am looking for a little clarity on what the policy gradient theorem means. My confusion lies in the fact that the reward $R$ in reinforcement learning is non-differentiable in the policy parameters. ...
1 vote
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### What are the prerequisites for stochastic calculus?

I am considering learning stochastic calculus myself, but do not have math background. Could you please suggest a list of books which will help to understand stochastic calculus?
Let $R = (R_1, \dots , R_M)'$ denote a vector of excess returns of $M$ assets observed at $n$ time points, $0 < t_1 < t_2 < \cdots < t_n < T$, within a time span $T > 0$. We wish ...