10
votes
Accepted
Dealing with different definitions of the Ornstein-Uhlenbeck process
In order to have a stationary solution of the Stochastic Differential
Equation (SDE), you have to start from a random initial value $u(0)$
at the fixed time $t=0$. This value must be drawn from the ...
- 5,706
7
votes
Book recommendations for probability
I suggest you a couple of books that I admit I never had the
occasion to study. These would have been my reference
if I specialized in probability:
Ash, Dade - "Probability and Measure Theory"
...
6
votes
When can a Gaussian Process solve an SDE?
First, (to address some of the comments under the main post) it should be noted that we can easily formulate an SDE whose solution is not Gaussian at fixed times. A famous example is the geometric ...
- 71
6
votes
Accepted
Correlation between Ornstein-Uhlenbeck processes
They are not perfectly positively correlated: Even when random variables are deterministically related (which would require $X$ to be deterministic in this case), perfect correlation requires them to ...
- 133k
4
votes
Book recommendations for probability
Foundations of Modern Probability by Olav Kallenberg meets all your criteria. It is quite concise and mathematically rigorous and as one reviewer puts it "without any non-mathematical distractions".
4
votes
When can a Gaussian Process solve an SDE?
TLDR
We can assume a Gaussian as solution and insert that into the Fokker Planck equations, to get the potential forms of $\mu$ and $\sigma$ that have the Gaussian as solution.
Let's describe the ...
- 86.5k
4
votes
Accepted
Ornstein-Uhlenbeck process
Assuming that the vector valued process $X_t\in\mathbb{R}^n$ can be described via
$$
d X_t = -\Psi X_t \,dt + \sigma\, dW_t
$$
where $\Psi=(\psi_1,\ldots,\psi_n)$ and $\sigma=(\sigma_1\,\ldots,\...
- 2,284
4
votes
Accepted
Model comparison with intractable likelihood using approximate Bayesian Computation
There are examples in the ABC literature of model selection through Bayes factors. An ecological individual based model example is here:
https://doi.org/10.1016/j.ecolmodel.2017.07.017
The paper ...
- 1,118
3
votes
Accepted
Understanding policy gradient theorem - What does it mean to take gradients of reward wrt policy parameters?
Well first, you're trying to take the gradient of the return, not the reward. Also unless both the environment and the policy is deterministic, you'd be taking the gradient of the expected return. Now ...
- 26.5k
3
votes
Dealing with different definitions of the Ornstein-Uhlenbeck process
I initially believed that the mean parameter had a part to play as, in this link Rasmussen & Williams, 2006, Appendix B, equation B.27, the OU SDE is written as:
$$ dX_t = - a_0 X(t)dt + b_0 dW_t ...
- 1,502
3
votes
Realize reducible nonstationary kernels as solution to SDEs and its extensions
$\blacksquare$1.Whether this(the method of SDE solving kernel problem) is extensible to the reducible nonstationary kernels?
Let me restate your question. The method your pointed out in [Hartikainen&...
- 2,490
3
votes
Accepted
How Much And What Kind of Math for Deep and Reinforcement Learning?
I'd recommend getting an overview of the math that's currently used in deep learning architectures that are used for supervised settings (this does mean looking into approaches that involve "training ...
- 46
3
votes
Book recommendations for probability
I think Amir Dembo's notes are pretty stellar. He updates them each time he teaches the course, but even then they have really good proofs and exercises. He also has notes on stochastic processes. ...
2
votes
Book recommendations for probability
I know these are no books, but nonetheless I think these materials are quite useful:
At MIT they offer various courses for free. Some of these courses might also contain books.
Overview of free ...
2
votes
An intuitive meaning of Stochastic Differential Equation
The answer to your question depends on the type of model you are dealing with, the underlying distribution of Brownian motion, the way you have expressed it, and nature of data you are dealing with. ...
- 155
2
votes
Accepted
Why does the theoretical value of the difference between these 2 stochastic integrals differ from the observed value in r?
The Itô integral is defined using the lhs of the interval:
$$\int_0^1W_tdW_t=\lim_{n\to\infty}\sum_{j=0}^{n-1}W_{\frac{j}{n}}\left(W_{\frac{j+1}{n}}-W_{\frac{j}{n}}\right),$$
where the limit is ...
- 221
2
votes
Find stochastic differential equation which best describes time-series
Broadly I would suggest three approaches to finding stochastic differential equations to test against your data. I do not have any substantial background knowledge in the pricing of commodities, so my ...
- 9,680
1
vote
Fitting Ornstein-Uhlenbeck process in Python
I know the question is old, but just to let others find an answer: There is one package I was able to find, SdePy:
https://sdepy.readthedocs.io/en/v1.1.2/generated/sdepy.ornstein_uhlenbeck_process....
1
vote
Accepted
Expectation of $dX_t$ for $X_t$ being an Ito process
It should be pointed out that $E_t[dX_t]$ is a heuristic notation used by only non-mathematicians (e.g. empirical economists). It has no mathematical meaning.
(Nor does statements like ``$E_t dX_t = ...
- 3,378
1
vote
Understanding policy gradient theorem - What does it mean to take gradients of reward wrt policy parameters?
R is a constant used to scale the gradient. Instead of reward it could be returns, advantage, etc.
The gradient with respect to the parameters is found from the log probability of taking a specific ...
- 497
1
vote
An intuitive meaning of Stochastic Differential Equation
The most intuitive way for me to understand these equations is to simulate paths:
$$\Delta x_i\equiv x_{i+1}-x_i=\mu(x_i)\Delta t+\sigma(x_i,t_i)\xi_{t+1},\, \xi\sim\mathcal N(0,1)$$
In each ...
- 62.3k
1
vote
Book recommendations for probability
I wholeheartedly recommend Parzen's Modern Probability Theory and its Applications. This is a classic written by someone who has made enormous contributions to statistics, in e.g. non-parametric ...
- 21.1k
1
vote
How to get a uniformly distributed portfolio allocation vector?
I want to generate such vectors x and I want that they are somehow uniformly distributed on the set of possible values
There is a well-known way of generating this (I cannot seem to find a reference, ...
- 4,603
1
vote
Accepted
Request for Multi-dimensional simulation reference book
All books on simulation, like
Devroye's Random Variate Generation
Fishman's Monte Carlo
Rubinstein's and Kroese's Simulation and the Monte Carlo Method
and our own book Monte Carlo Statistical ...
- 108k
1
vote
conditional distribution of $X_t$ in a jump-diffusion model
I'll preface this response by saying I've worked with OU processes quite a bit and point processes quite a bit, but never the combination. So I could be way off and there's a much easier way...
I ...
- 11
1
vote
Realize reducible nonstationary kernels as solution to SDEs and its extensions
Denote the value of your process $u(t)$. First, note that if you consider $t_1 \neq t_2$ such that $\phi(t_1) = \phi(t_2)$ then you must have $u(t_1) = u(t_2)$. How do we know this? $u$ is Gaussian, ...
- 2,684
Only top scored, non community-wiki answers of a minimum length are eligible