Questions tagged [stochastic-calculus]
57 questions
11
votes
0 answers
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 $\mu$ and $\sigma$ such that $X_t$ is…
adityar
- 1,267
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- 14
9
votes
2 answers
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-Uhlenbeck process as a Gaussian process with a…
Billy Smith
- 273
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- 5
7
votes
7 answers
Book recommendations for probability
I am looking for a book (English only) that I can treat as a reference text (more colloquially as a bible) about probability and is as complete - with respect to an undergraduate/graduate education in Mathematics - as possible. What I mean by that…
Spaced
- 383
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- 12
6
votes
0 answers
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 always holds:
$$
dF(X) = g(\cdot) dX
$$
i.e. the…
Frido Rolloos
- 161
- 4
6
votes
1 answer
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, $\sigma >0$ is the dispersion rate, and…
Jeff
- 385
- 1
- 12
6
votes
3 answers
Stochastic Differential Equations - A Few General Questions
I just have a few questions about stochastic differential equations. I generally did a lot of pure math but signed up for a course on probability models and stochastic differential equations because I wanted to try something different. I have really…
Islands
- 163
- 4
5
votes
1 answer
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 function. Therefore, I currently plan to use…
LiKao
- 2,329
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4
votes
1 answer
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
- 749
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- 13
4
votes
1 answer
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
$$
S_1 = 2 \sum_{j=0}^{n-1} \left[ W_\frac{j}{n} \left( W_\frac{j+1}{n} - W_\frac{j}{n} \right) \right]…
M Smith
- 145
- 5
4
votes
2 answers
Realize reducible nonstationary kernels as solution to SDEs and its extensions
I am interested in a regression application where my kernel is of the form
\begin{equation}
k(t,t^{\prime}) = k_s\left(\phi(t),\phi(t^{\prime})\right)= k_s\left(\phi(t)-\phi(t^{\prime})\right),
\end{equation}
where $k_s$ is a stationary kernel.…
jkt
- 513
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3
votes
0 answers
Continuous-time Kalman filter with no observation/measurement noise
The continuous-time (linear) state space model can be written
\begin{align*}
\text{d}\mathbf{x}_t &= \mathbf{F} \,\mathbf{x}_t \, \text{d}t +
\mathbf{G} \,\text{d} \boldsymbol{\beta}_t \\
\text{d} \mathbf{z}_t &= \mathbf{H}…
Yves
- 4,313
- 1
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- 34
3
votes
0 answers
Soft Question: What background do I need to understand Feynmann Kac Formulae by Pierre Del Moral?
I am attempting to understand Sequential Monte Carlo(SMC) deeply, but with little theoretical background on probability theory and stochastic processes. Usually, the 'statistics' perspective of markov chains and stochastics is well understood by me.…
tintinthong
- 750
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3
votes
0 answers
What is the mean and variance of a general stochastic integral?
$$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.$$
But I haven't found formulas for the rest. So,…
Kim
- 105
- 4
3
votes
2 answers
Why does a probability of 0 or 1 remain unchanged with new information, intuitively?
Related to these questions:
Prove/Disprove probability of 0 or 1 (almost surely) will never change and has never been different
Does an unconditional probability of 1 or 0 imply a conditional probability of 1 or 0 if the condition is possible?
For…
BCLC
- 2,166
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3
votes
1 answer
Distribution of stochastic integral
I would like to find the distributions of the following random variables:
$Z_k= \frac{1}{\pi} \int^{2\pi}_{0} cos(kt) dW_t$ $k=1,2,...$ and $(W_t)_{t\geq 0}$ is a Wiener process.
What is the distribution of $Z_1$, and $(Z_k) $?
I am new to…
FelB
- 95
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