Questions on Monte Carlo methods, methods that require the repeated generation of (pseudo-, quasi-)random numbers for computing their results.

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0
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1answer
34 views

proof of Markov chain Monte Carlo

This is the first step of proof of MCMC in my notes I have a question, how come $\pi(x)\pi(x_p\mid x)=\pi(x_p)\pi(x\mid x_p)$? Is it true for any markov chains which are ergodic and aperiodic? The ...
0
votes
1answer
26 views

confused about the proof of Markov chain Monte Carlo

This is the proof from notes I'm confused about the $\pi(x_p|x)$ and $\pi(x|x_p)$ Let's say $X\sim $Bin$(10,0.3)$, so $\pi(x)=\binom{10}{x}0.3^x0.7^{(10-x)}$, so what does $\pi(x_p|x)$ or $\pi(x|...
0
votes
1answer
50 views

confused about proposal distribution in MCMC

This is a question from notes I have some questions regarding the proposal distribution which is $N(x,1)$ Is the proposal distribution symmetric i.e. $g(x_p|x)=g(x|x_p)$? I'm not sure whether it ...
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0answers
26 views

integral equation1

Given $S=X_1+X_2$ that $X_1$ and $ X_2$ are 2 arbitrary random variables. we know that the probability density function of $S$ is : $P_{S}(s)=\int_{-\infty}^\infty P_{X_1|X_2}(s-x_2|x_1)P_{X_2}(x_2)$...
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0answers
31 views

Explanation of “When two functions $a(x)$ and $b(x)$ are not correlated on domain $X$, they can be separately integrated”

Hy everyone, I was reading this paper https://hal.inria.fr/hal-00942452v1/document , and I came up with a statement that I don't fully understand, nor could I found any info on it, so decided to ask ...
2
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1answer
42 views

PDE boundary condition question regarding limits

Just as a bit of background, I'm working with the Black-Scholes PDE and I'm testing some things out by taking an initial condition for it as $\sin(S/50)$, where $S$ is the spot price (but that's ...
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0answers
45 views

How can I generate random samples from following probability density function?

Let $\mathbf{\alpha}=(\alpha_1, \ldots, \alpha_m)$. The posterior density function of $\mathbf{\alpha}$ is given by $$h_0(\mathbf{\alpha}|\mathbf{x})=‎\frac{\prod_{i=1}^{m}\alpha_i^{a_i}}{\left(1+\...
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0answers
26 views

Black scholes model for down and out European call option using Monte Carlo

I tried to implement Matlab program computing the price of the European down and out call option using Monte Carlo and Euler discretization scheme. I have initial price S0=50, strike K=50, barrier ...
0
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0answers
24 views

situation when Monte Carlo method cannot be used

This is an example from notes I don't understand why we should think about the $E(x)$ and $var(x)$ first?
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0answers
18 views

Stratified Sampling $E(E(X \mid \mu, \sigma))$

Let $X$, $\mu$ and $\sigma$ be random variables. I want to estimate $E(X)$ using Monte Carlo. I am able to sample from, and know in closed-form, both the conditional distribution of $X \mid (\mu, \...
2
votes
1answer
66 views

Can you suggest a method to generate random sample from following PDF?

‎Let‎ ${‎‎\bf{\alpha}}=(\alpha_1, \alpha_2, \ldots, \alpha_m)$ ‎and ‎‎$‎‎\textbf{b}=(b_1, b_2, \ldots, b_m, b_{m+1}).$ I intend ‎to ‎generate ‎sample ‎from PDF $$ g(\alpha_1, \alpha_2, \ldots, \...
1
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0answers
28 views

Monte Carlo Search Tree iterations

I have had this example in my exam last week and I can not figure out how to solve it. I have watched lots of tutorials on Monte Carlo Search Tree but I can't still understand this algorithm properly. ...
1
vote
0answers
46 views

Scrambled Sobol

I need to do a Monte Carlo simulation in high dimension (up to 1000) where using plain Sobol (with Kuo's direction vectors) as a random number generator is not good enough. Therefore I am ...
0
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0answers
24 views

How to compute integrals using any probability law with Monte Carlo?

I am intrested in providing an estimation of : $\iint C(x,y)dP_X(x)dP_Y(y)$ I am able to generate random numbers from the distribution of $P_Y$ and $P_X$. Therefore I generate a big number (n=10 000)...
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0answers
20 views

Importance sampling example

I have a question regarding importance sampling. During a lecture we were told that importance sampling is simply a shift to another PDF to improve point sampling. \begin{equation} I=\int_{a}^{b}dx\ f(...
-1
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1answer
40 views

Monte Carlo method for solving integrals [closed]

My professor gives us an intro to how to evaluate integration using Monte Carlo method. But I tried to search about it and never find the algorithm he used. Any help how can I find an explanation ...
0
votes
1answer
39 views

Loss probability and VaR

I would like to estimate Value-at-Risk analytically and through delta-gamma aproximation. I don't know if my idea is ok, but i would like to build a portfolio of European option. Suppose that in this ...
1
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0answers
37 views

Approximating Minkowski Sum of 3 dimensional Convex Polytopes by Sampling

Let $P_1,P_2...P_r$ be a set of convex polytopes with $n_r$ vertices in 3 dimensions. These polytopes basically represent uncertainties of '$r$' number of 3d-points respectively in space. The global ...
0
votes
1answer
52 views

Estimator of expectation value for standard normal distribution

In the case of a standard normal distribution, I just read that a good estimator for E[f(x)] is $\frac{1}{M}\sum_{i=1}^M f(X_i)$ (where each $X_i$ is standard normally distributed and independent). ...
1
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1answer
43 views

Log normal simulation.

I want to calculate numerically the expectation of a lognormal random variable $Y=e^X$, where $X$ is normally distributed with mean $m$ and variance $V$. The expectation is known as $e^{m+\frac{1}{2}...
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2answers
30 views

Show that this MC is ergodic?

Suppose I have a Markov Chain with States, $S = {1,2,3,4}$ and a PTM given by $P =$ $\begin{pmatrix} .25 & .25 & .25 & .25 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 &...
1
vote
1answer
30 views

Limiting Distribution of a Gibbs Distribution

I know that the Gibbs distribution at a particular state, x, is given by $\frac{e^{-\beta E_i}}{\sum_j e^{-\beta E_j}}$ with $\beta = \frac{1}{T}$, but I do not understand what a limiting distribution ...
2
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4answers
122 views

Monte Carlo double integral over a non-rectangular region (Matlab)

I want to evaluated the following integral using Monte Carlo method: $$\int_{0}^{1}\int_{0}^{y}x^2y\ dxdy $$ What I tried using Matlab: ...
0
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0answers
39 views

Gaussian processes and bias

I would like to simulate two Gaussian processes along a time grid. Ideally, I would like to see the average of the samples, for each grid point, to be close to the mean. Using the antithetic method, I ...
0
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0answers
24 views

Correlation Matrix Question

Why is this not a possible correlation matrix for any three random variables X, Y, and Z? $\begin{pmatrix} 1 & -1 & -1 \\ -1 & 1 & -1\\ -1 & -1 & 1\end{pmatrix}$
1
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1answer
32 views

Correlation Matrices proofs

(*) says that the diagonals of $R$ are $1$ and the non-diagonals are the correlation, $p$. I planned on simplifying both equations until they're equivalent, but I'm not sure how I could go about ...
0
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1answer
25 views

Generating Normals with specific means and variances

Suppose I wish to generate normals $X, Y, Z$ with the correlation matrix R but with means $0, 1, 2$, and variances $4, 16, 25$, respectively. How would you do this? The only way I know of doing ...
1
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1answer
45 views

Related problem to covering a circle with random arcs

I have a problem setup wherein we have (the following are all integers) a sequence of length $G$, and $N$ reads of length $L$. I'm interested in the problem where we consider the sequence to be ...
0
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0answers
23 views

MCMC and Metropolis-Hastings problem(s)

What does it mean for a particular state to be a "ground" state or a "stable" state? I should make clear that this is final exam review material and not homework. Also, how does one compute a Gibbs ...
0
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1answer
66 views

A Question Regarding Markov Chains and Ergodicity

Suppose the Markov chain with Probability Transition Matrix, $P$ = ($p{_x}{_y}$) is ergodic and $p{_m}(x, y) > 0$ for all states $x$ and $y$. If $n ≥ m$, show that $p_n(x, y) > 0$ for all ...
0
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0answers
9 views

Monte-Carlo estimation of Mutual Information over AWGN channel

I'm trying to solve a problem I was tasked with. Basically I have to generate a 100k 16QAM inputs and transmit them over a AWGN channel. With this I have to use the Monte-Carlo estimation to figure ...
1
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1answer
37 views

Debugging a Metropolis Hastings Algorithm Simulation

I have some questions about the Metropolis Hastings algorithm: Wikipedia says: ...choose an arbitrary probability density g(x|y) which suggests a candidate for the next sample value x, given ...
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0answers
17 views

Monte Carlo divergence

I'm running MonteCarlo simulations to compute the most likely voltage values in a grid. The process is to generate a random load from a load values distribution, and simulate the load with the power ...
2
votes
1answer
37 views

discretized Brownian motion

These are the definitions I'm working with: A (standard) Brownian motion in $\mathbb{R}$ is a stochastic process $W(t)$ $(t \geq 0)$ such that the following properties hold: $W(0) = 0$ almost ...
0
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0answers
16 views

Computing the partition-function of an exponential family member

I am working on an Monte Carlo Expectation Propagation problem. In that context I got the following property: $ I = \sum\limits_i w^{(i)} \log p_\eta(x^{(i)}) $ where $\{w^{(i)}\}_i$ are weights, $...
1
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0answers
11 views

Quasi-Monte Carlo with Conditional Distributions

I want to estimate $E(f(X))$ using quasi-Monte Carlo where $X = (X_1,\ldots,X_n)$ is a random vector and $$ X_i\sim f(\cdot; \theta), \quad \text{independent}, $$ where $\theta \in \mathbb{R}$ is some ...
0
votes
1answer
27 views

Generating correlated standard normals

Suppose I want to generate three standard normals $X, Y, Z$ with correlation matrix given by $R$= $ \begin{pmatrix} 1.0 & 0.2 & 0.2 \\ 0.2 & 1.0 & 0.2 \\ 0.2 & 0.2 & 1.0\end{...
3
votes
2answers
94 views

Error Estimates. L1 or L2 norm?

I simulate random walk on a divide difference grid to solve heat equation 1D. I want to prove numerically that this method has $Ν^{-1/2}$ error accuracy. My problem is that I don't know which norm ...
1
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0answers
19 views

Generate Correlated Normals

I want to generate normals $X,Y,Z$ with the correlation matrix $R$ but with means $0, 1, 2$ and variances $4, 16, 25$ respectively. How can I do this? Is it possible to apply Cholesky?
2
votes
1answer
29 views

Geometric Brownian motion with exponential of sum of iid's

Glasserman's "Monte Carlo Methods in Financial Engineering" on p. 265 states that the geometric Brownian motion can be modelled with : $$S(t_n)=S(0) \exp(\sum_{i=1}^n X_i)$$ where $X_i$ are iid. I ...
0
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0answers
10 views

Accuracy Rebonato Swaption Approximation Formula among Different Strikes

Can somebody explain me if the Rebonato swaption volatility approximation formula is accurate for only ATM strikes, and if yes why? Can it also be used for ITM and OTM strikes? My foundings: Let $0 &...
2
votes
2answers
56 views

References on probability theory, stochastic processes, Monte Carlo and convex optimisation, with similar writing style to Terence Tao

I learned a lot from prof Tao's notes and books because unlike many authors, he seems to prefer writing more words, explanations and intuitions rather than just mathematical formulae. His approach is ...
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0answers
22 views

Reducing sequential correlations in Metropolis Algorithm

In our last lab, we use MCMC method to simulate a walker walking in the phase space. Using the Metropolis method, a walker at its currect position will sample another point inside a cube (centered at ...
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0answers
14 views

Help with Hidden Markov model and SMC methods

So its quite a long background i don't really know where to start but here goes. The background is as follows: Background Observation model As the target is moving, it measures the signal (RSSI) ...
2
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0answers
35 views

Theoretical interpretation of simulating from a distribution

Suppose there is a random variable $X$ with marginal density $p_X$. However only the conditional densities $\{p_{X\mid\Theta}(\cdot\mid\theta):\theta \in \mathbf{T}\}$ are known directly, where $\...
2
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0answers
38 views

Can someone help me balance this game (probability question) [closed]

A team of 9 vs a team of 1. Each round each of "the 9" roll a die to "attack" and "the 1" rolls 9 dice to "defend", the nine dice are preassigned to attackers before the roll, "the 1" cannot choose ...
0
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1answer
44 views

How would you simulate Brownian motion with a die?

You can simulate Brownian motion on $[0, 1]$ for instance by splitting it into $K$ intervals and at each time $k \Delta t$ add $N(0, \Delta t)$ to your running total, where $\Delta t = 1/K$. If you ...
0
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0answers
13 views

Main differences between Monte carlo and Law of propagation of variance

I need to know the main differences/limitations between/of Monte-Carlo simulation technique and law of propagation of variance. Can someone briefly describe it or is there any good reference or link ...
0
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0answers
34 views

Ways to sample a complicated PDF on an hemisphere

I want to generate samples on the upper real unit hemisphere with the following PDF (it's not really a PDF because I can't guarantee that it integrates $1$) $$\frac{\sum_{i=0}^{n}c_i(\text{ max}(\text{...
3
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0answers
80 views

Monte Carlo with non uniform weighting

So, I just want to check if what is in my mind is in fact true. Assume, that we have are given a distribution $p_{z}(k)$ over the whole $\mathbb{Z}^+$. We are interested in approximating $p_v(v)$ over ...