Questions tagged [stochastic-calculus]
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68
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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 ...
- 111
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Covariance of stochastic process
I have the following stochastic differential equation
$dX_t=\kappa\left [ \theta-X_t\right ]dt + \Sigma d W_{t}$
I derived formula for $X_t$ which is in the following form
$X_{t}=\theta+e^{-\kappa t}\...
- 111
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25
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Stochastic Process Notation / Brownian Increments
I am currently reading about stochastic processes and Brownian Motion.
When books have notation such as $E[X_t] = 0$ and $Var[X_t] = \sqrt{t}$ this is considered over sample paths.
However, when we ...
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Comparison of two models with different number of parameters
I want to compare two models, which has different number of parameters. The first model is Arbitrage free Nelson-Siegel model, which has the following equation:
$y_{t}(\tau )=X_{1,t}+X_{2,t}(\frac{1-e^...
- 111
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Stochastic Calculus Algebra
Studying Brownian Motion and stochastic integrals in class, my professor rewrote this summand $$1/2*\sum_{j=0}^{n-1} (W((j+1)T/n) - W(jT/n))^2$$ as $$1/2*W^2(T) + \sum_{j=0}^{n-1} W(jT/n)(W(jT/n) - W((...
- 11
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53
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Stochastic Calculus: probability
Suppose two people want to play a game in which person A
has probability 2/3 of winning. However, the only thing that they have is a
fair coin which they can flip as many times as they want. They wish ...
- 1,204
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I don't understand Optional sampling theorem Ⅲ proof. Would any member explain this proof?
I don't understand the following proof of optional sampling theorem Ⅲ.
I followed the proofs for Optional sampling theorem Ⅰ and Ⅱ.
Referring to which statistics related books would be helpful to ...
- 1,204
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40
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Independence of Ito integrals wrt. BM
Consider the trigonometric basis:
$$
(1, \sqrt{2} \cos(2\pi t), \sqrt{2} \sin(2\pi t), \sqrt{2} \cos(4\pi t), \sqrt{2} \sin(4\pi t), ...),
$$
and define, for each $j \geq 1$,
$$
\xi_j = \int_{0}^{1} \...
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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 ...
- 187
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60
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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 ...
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26
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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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112
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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 ...
- 135
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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 ...
- 41
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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 ...
- 253
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119
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Fisher information of an Ornstein-Uhlenbeck process
I would like to compute the Fisher information of an Ornstein-Uhlenbeck process $X_t = Y_t - \beta Z_t$ where $Y_t$ and $Z_t$ are two time-series.
My log-likelihood function in this case is:
$$\...
- 349
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0
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28
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Nonparameteric Empirical Estimator For Stochastic Process
Motivation: If $X$ is a random-variable defined on some probability space $(\Omega,\Sigma,\mathbb{P})$ then Glivenko-Cantelli lemma guarantees that the empirical distribution $\frac1{N}\sum_{n=1}^N \...
- 544
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what is the sensitivity of the neural network using standardized input
Suppose I trained a neural network with standardisation of the data following (X-EX)/std(X). The input is x(t) and output is y(t). How can I calculate the sensitivity of this trained network (...
- 21
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44
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State dependent autocorrelation of a stochastic differential equation
I have an SDE of the form $$dX_t = f(X_t)dt + \sigma(X_t)dW_t$$ where f(x) is a rational expression. I need to compute the lag-1 autocorrelation of this process as a function of $X$.
Question 1: is ...
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298
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Let $X_t$ be a solution of a SDE. Does the set $\{X_t \in \{p\}\}$ has null measure?
This question was previously posted on https://math.stackexchange.com/questions/3981156/let-x-t-be-a-solution-of-a-sde-does-the-set-x-t-in-p-has-null-meas.
I think this question is easy. However, I ...
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Connection between kernel estimator of derivative and derivative of kernel estimator
Imagine $X$ has density $f$ and $X_1,\ldots,X_n$ be a given sample;
Now to estimate $f$ we can use
$$\hat f(x)=\frac{1}{nh}\sum_{k=1}^nk\left(\frac{X_i-x}{h}\right),$$
for some differentiable kernel $...
- 1
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85
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Stochastic Differential Equation with drift multiplied by everywhere discontinuous random process
(WARNING: this has been crossposted on physics.stackexchange (questions/588606). It has been suggested to post also here, let me know if it is against the rules).
Let the stochastic process $\{X_t\}$ ...
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1
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Model comparison with intractable likelihood using approximate Bayesian Computation
I have some models based on stochastic differential equations (SDEs). Because of the definition of these models, I can simulate data, but I cannot compute the likelihood function / distribution ...
- 2,641
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0
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211
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Multivariate Ornstein–Uhlenbeck process, examples with complex eigenvalues
We know that mathematically the drift matrix of the multivariate OU process can have real or complex eigenvalues. In case of real eigenvalues, the process is called non-oscillating whereas in case of ...
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Modelling negative prices [closed]
As recently oil prices were negative I was wondering if there was a possibility to model this behaviour by a certain distributional assumption of prices as the classical log normality is violated ...
- 131
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232
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Covariance of Ito Integrals
I am new to stochastic calculus and I'm still grappling with the concepts. Can someone please verify whether I am correct to solve my question this way:
In my question, I need to calculate:
\begin{...
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In predictive regressions, can we assume that the predictor follows the Ornstein-Uhlenbeck process?
A predictive regression is a regression of the form
\begin{equation}
y_t=\beta x_{t-1}+\varepsilon_t
\end{equation}
where $x_{t-1}$ is generally assumed to be a highly persistent stochastic variable, ...
- 1,144
1
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1
answer
7k
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Fitting Ornstein-Uhlenbeck process in Python
Hi~ I am wondering that are there some packages in python for the users to fit an OU process? I know that we can convert this problem into a regression problem or an AR(1) fitting problem and back out ...
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What is the volatility surface of a parameter that depends on a Stochastic process?
Hi this question takes a few steps to setup so please bear with me :)
Short version:
Does a parameter that depends on the output of a stochastic model inherently have variance? If so why?
Long ...
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affine function of random variable
If $X$ is a random variable, then $F(X) = a + bX$ where $a,b$ are constants is an affine function.
What is the technical/formal name of a function which is not affine, but for which the following ...
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1
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249
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Ornstein-Uhlenbeck process
Given a multivariate Ornstein-Uhlenbeck process that is a stochastic process, is it correct that each component of this process is a univariate Ornstein-Uhlenbeck process?
- 832
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2
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Expectation of $dX_t$ for $X_t$ being an Ito process
Let $X_t$ be an Ito Process:
$$dX_t = f(t, X_t)dt + g(t, X_t)dW_t$$
What is $E_t[dX_t]$?
How can we compute it and importantly what is the intuitive explanation of $E_t[dX_t]$?
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Continuous-time Kalman filter with no observation/measurement noise
The continuous-time (linear) state space model can be written
\begin{align*}
\text{d}\mathbf{x}_t &= \mathbf{F} \,\mathbf{x}_t \, \text{d}t +
\mathbf{G} \,\text{d} \boldsymbol{\...
- 4,808
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Wiener process definition as Gaussian summation [duplicate]
In this lectures Wiener process is defined by summing white Gaussian random variables and then limit them when sample time go to zero.
$$ {\bf{w}}(t) = \int_0^t {{\bf{\tilde q}}(\tau )} d\tau = \...
- 337
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2
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273
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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. ...
- 2,263
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255
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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?
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Stochastic Differential equation: CAPM
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 ...
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When can a Gaussian Process solve an SDE?
Considering an SDE of the form
$$dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t ,$$
where $W_t$ is a Wiener process, is there a set of necessary and sufficient conditions on the structure of the functions $...
- 1,472
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Dealing with different definitions of the Ornstein-Uhlenbeck process
I've run up against a wall in reconciling two different definitions of the Ornstein-Uhlenbeck process, and would appreciate some help.
On the one hand, as discussed here, we can define an Ornstein-...
- 293
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69
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Correlation and Covariance of lognormal random variables
Let $X_i$ be lognormally distributed stochastic variable and $j$ < $i$ then is this the correct formula to calculate their correlation (please specifically note the index of t):
$$ Correl(i,j) = ...
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155
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Is this inhomogeneous Poisson process?
I asked this question in math stack exchange but since it is more related to probability and statistics, I thought I can ask here.
Consider the population growth model where $P′(t)=rP(t)$, where $P(...
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Correlation between Ornstein-Uhlenbeck processes
Consider the Ornstein-Uhlenbeck process, $U(t)$, whose evolution follows:
$$
\mathrm{d}U(t) = -\theta U(t) \mathrm{d}t + \sigma \mathrm{d}W(t),
$$
where $\theta \in (0,2)$ is the mean-reversion rate, $...
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Is an ITO diffusion time slice always Normally distributed?
As the title says, if we take a time slice on any Ito diffusion - are we guaranteed that the data is always Normally distributed?
This seems like a useful property for computer generalization and ...
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260
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An intuitive meaning of Stochastic Differential Equation
I'm trying to approach for the first time Ito's calculus and SDE, maybe this is a trivial question. If the following is a generic SDE:
$ dx = \mu(x)dt + \sigma(x)dB_t$
Can i consider the $dx$ as a ...
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Is this short rate really constant?
Suppose that a financial instrument has a constant short-term rate $r$ and its price $S$ is driven by the equation
$$S_t = \mu_t S_t \, {\rm d}t,$$
where $(\mu_t$) is a process adapted to the ...
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Soft Question: What background do I need to understand Feynmann Kac Formulae by Pierre Del Moral?
I am attempting to understand Sequential Monte Carlo(SMC) deeply, but with little theoretical background on probability theory and stochastic processes. Usually, the 'statistics' perspective of markov ...
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What is the mean and variance of a general stochastic integral?
$$X_T = x_t+\int_t^T\mu(s,X_s)ds+\int_t^T\sigma(s,X_s)dW_s$$
where $W$ is a Wiener process. What is the variance and mean of this process?
It is well known $$E\left[\int_t^T\sigma(s,X_s)dW_s\right]=0....
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Expectation and variance of a stochastic time process conditioned on its past
$$dV_t=-k(V_t-1)dt+ \epsilon\sqrt{V_t}dW_t$$
$W_t$ is wiener process and the rest is just some parameters.
For $T_{i+1}>T_{i}$ how do I find the expectation and variance of $V_{T_{i+1}}$ ...
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How to transform Gaussian variables so certain criterias are satisfied?
Given two standard normal distributed variables $Z_1$ and $Z_2$ then $N$ is defined as $$N=\alpha Z_1 + \beta Z_2$$
How do I derive $\alpha$ and $\beta$ such that $N$ is standard normally distributed ...
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Why does the theoretical value of the difference between these 2 stochastic integrals differ from the observed value in r?
Consider the stochastic integral
$$
2 \int_0^1 W_t \hspace{2mm} dW_t
$$
Using r, this may be evaluated using one of the following summations
$$
S_1 = 2 \sum_{j=0}^{n-1} \left[ W_\frac{j}{n} \left( W_\...
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