Questions tagged [stochastic-calculus]

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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 ...
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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 ...
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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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Geometric Brownian motion drift estimation problem

Given: $$ d \ln{S_t} = \left( \mu - \frac{\sigma^2}{2} \right)dt+ \sigma dW_t $$ We have that: $$ \mathbb{E}_t[d \ln(S)]=\left( \mu - \frac{\sigma^2}{2} \right)dt $$ $$ \mathbb{V}_t[d \ln(S)]=\sigma^2 ...
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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 (...
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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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277 views

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 $...
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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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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 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 ...
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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, ...
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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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4 votes
1 answer
168 views

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?
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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{\...
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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 = \...
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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. ...
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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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11 votes
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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 $...
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9 votes
2 answers
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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-...
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1 vote
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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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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 $P′(t)=rP(t)$, where $P(...
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6 votes
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, $...
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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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2 votes
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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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Stochastic Integral - Standard Brownian Motion

I was just running through practice exams, we didn't really cover this in class so any guidance (reference etc) would be appreciated. (I'm struggling to use the integral with MathJax so I'm pasting it ...
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3 votes
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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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3 votes
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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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2 votes
1 answer
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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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4 votes
1 answer
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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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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 $t \geq s$?
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1 answer
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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 "...
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1 vote
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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 $E[\Delta t\Delta B(t)]=0$...
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1 vote
1 answer
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How Much And What Kind of Math for Deep and Reinforcement Learning?

I have a lot of books on measure theoretic probability theory, functional analysis, graduate level topology, convex optimization, stochastic calculus, numerical analysis using a functional analysis ...
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2 votes
1 answer
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Distribution of a slice of a line with normally distributed slope

Give the random variable $X \sim N(\mu,\sigma)$ consider the process $Y_t = tX$ with $t \in I = [t_1,t_2]$, informally a line with a random slope. I would like to find the distribution of the points ...
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1 vote
1 answer
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How to get a uniformly distributed portfolio allocation vector? [duplicate]

Imagine we have a portfolio allocation vector $(x_1, ..., x_n)$ with $x_1+...+x_n=1$; Also we assume that the vector has only elements $\geq 0$, so we have $|x_1|+...+|x_n|=1$, so no short selling; ...
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1 vote
1 answer
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Request for Multi-dimensional simulation reference book

Do you have any reference book about multi-dimensional simulation? The random variables are not identically distributed and are correlated. I understand the procedure to generate correlated normal ...
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1 vote
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223 views

How to calculate the covariance of a stochastic integral and a Riemann integral?

Right now, I want to figure out the covariance of a stochastic integral and a Riemann integral in the following form: $$\mathbb{E}\left(\int_{0}^{t}\exp[B(t)-B(s)]ds \cdot \int_{0}^{t}\exp[B(t)-B(s)]...
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2 votes
1 answer
262 views

conditional distribution of $X_t$ in a jump-diffusion model

I'm working with a variant of the Ornstein Uhlenbeck model that includes Lévy-type jumps, (i.e. a jump-diffusion model with mean reversion). I want the distribution of $X_t$ (given the parameters of ...
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1 vote
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The pdf of last interval of a Poisson process in $[0,T]$

Assume a Poisson point process with rate $\lambda$ in time $[0,T]$. Supoose $X$ is the random variable representing the time between the last arrival and $T$. What is the probability density function ...
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