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Transformed Ornstein Uhlenbeck process

Say I have ๐‘‹ that follows an Ornstein-Uhlenbeck process: $๐‘‘๐‘‹_๐‘ก=๐œ™X_t๐‘‘๐‘ก+๐œŽ๐‘‘๐‘Š_๐‘ก$ Let $๐‘Œ_๐‘ก=exp(๐‘‹_๐‘ก)$. How can I calculate the autocorrelation function of $Y_t$?
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Estimating jump diffusion parameters from first passage time data?

I know that there is a literature on approximating the first passage time distribution of jump diffusion processes. I know there is also a literature on estimating parameters of jump diffusion ...
ThinkConnect's user avatar
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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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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^...
Shelley's user avatar
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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((...
MathStudent's user avatar
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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 ...
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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 ...
Christian Prince's user avatar
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61 views

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 ...
Thomas's user avatar
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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 ...
donut's user avatar
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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: $$\...
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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 \...
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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 (...
Xu Shan's user avatar
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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 ...
Ville's user avatar
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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 ...
Matheus Manzatto's user avatar
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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 $...
Johannes's user avatar
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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 $\{X_t\}$ ...
ernst's user avatar
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1 answer
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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 ...
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Multivariate Ornsteinโ€“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 ...
Question Anxiety's user avatar
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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{...
nymeria's user avatar
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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, ...
Carl's user avatar
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1 vote
1 answer
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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 ...
Demebleeee's user avatar
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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 ...
Soran's user avatar
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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 ...
Frido's user avatar
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1 answer
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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?
TrungDung's user avatar
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2 answers
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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 $E_t[dX_t]$?
econmajorr's user avatar
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367 views

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{\...
Yves's user avatar
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2 votes
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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 = \...
sci9's user avatar
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2 votes
2 answers
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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. ...
figs_and_nuts's user avatar
1 vote
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282 views

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?
2 votes
0 answers
80 views

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 ...
bardwell's user avatar
23 votes
2 answers
1k views

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 $...
adityar's user avatar
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10 votes
2 answers
3k views

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-...
Billy Smith's user avatar
1 vote
0 answers
71 views

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) = ...
user2696565's user avatar
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2 votes
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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 $Pโ€ฒ(t)=rP(t)$, where $P(...
Kasthuri's user avatar
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1 answer
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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, $...
Jeff's user avatar
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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 ...
Edv Beq's user avatar
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3 votes
1 answer
322 views

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 ...
Hamall's user avatar
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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 ...
user32141's user avatar
3 votes
0 answers
70 views

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 ...
tintinthong's user avatar
3 votes
0 answers
167 views

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....
Kim's user avatar
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0 answers
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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 $V_{T_{i+1}}$ ...
financegrad's user avatar
2 votes
1 answer
35 views

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 ...
financegrad's user avatar
4 votes
1 answer
52 views

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_\...
M Smith's user avatar
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257 views

Conditional Expectation for Geometric Brownian Motion

Given a geometric Brownian motion: $\frac{dZ}{Z} = \mu dt + \sigma dW$ Is there a closed-form solution to $\mathbb{E}[z_s | (z_s > a)\cap(z_t > b)]$ for $t \geq s$?
R. Thomson's user avatar
0 votes
1 answer
75 views

Basic Stochastic Calculus: Integration of wiener proces

How do one compute the following integral: $$\int_{a}^{b}dw_u$$when $W_t$ is a wiener process. My initially guess will be $W_b-W_a$, but I cannot argue for that because I am not sure wether I can use "...
Emil Frank Soerensen's user avatar
1 vote
0 answers
159 views

Question on variance and expectation of Brownian Motion related things

In a mathematical finance text by Ubbo F Wiersema, I came across the following Say $\Delta t$ is very small. $\Delta B(t)$ denotes $\textit{brownian motion increment}$. Then $E[\Delta t\Delta B(t)]=0$...
Bernhard Listing's user avatar