0
votes
0answers
29 views

Log normal approximation

I often read in the litterature that for small volatilities, log normal distribution can be approxiamted quite well by a normal distribution. What do you think about that ? Is there any way to ...
1
vote
0answers
42 views

Infinite discounted sum of weakly dependent Normal random variables

Say I have the expected value of a sum of weakly dependent Normal random variables of the form $\mathbb{E}\left[\sum_{n=1}^\infty a^n X_n\right]$, where $0<a<1$. I was wondering under what ...
1
vote
1answer
60 views

Showing that two Gaussian processes are independent

Say that $Z_t = (X_t, Y_t)$ is a 2-dimensional Gaussian process (by definition, it means that the random vector $(X_{t_1},Y_{t_1},...,X_{t_n},Y_{t_n})$ is a Gaussian random vector for all $t_1 ...
5
votes
1answer
207 views

$\int_0^tB_s^2\ dB_s$ - Gaussian Process and independent increments?

For $(B_t)_{t\ge0}$ a standard Brownian motion (Wiener process) define the stochastic process $X_t:=\int_0^tB_s^2\ dB_s$. I am currently trying to assess if $(X_t)_{t\ge0}$ is a Gaussian process and ...
1
vote
1answer
36 views

I want to show $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. [closed]

Let $B(t)$ be Brownian motion. Show that $e^{-\alpha t}B(e^{2\alpha t})$ is a Gaussian process. Find its mean and covariance functions. thanks .
2
votes
2answers
68 views
2
votes
1answer
124 views

determine Fisher information of $N(0,\sigma^{2})$ over $\sigma^{2}$

So far i've got that $I(\sigma^{2}) = E_{\sigma^{2}}[\frac{\delta}{\delta \sigma^{2}}\log_{\sigma^{2}}(X)]^{2}$. And i got that $\frac{\delta}{\delta \sigma^{2}}\log_{\sigma^{2}}(x) = ...
0
votes
0answers
17 views

Monte Carlo by point or by interval

Say I compute monte carlo output from input scenarios. Input are discrete time series. I choose time series as an example to make the problem more obvious - this could be also any curve. Computation ...
2
votes
1answer
69 views

Given a covarince matrix, generate a Gaussian random variable

Given a $M \times  M$ desired covariance, $R$, and a desired number of sample vectors, $N$ calculate a $N \times M$ Gaussian random vector, $X$. Not really sure what to do here. You can calculate ...
2
votes
1answer
102 views

Weighted integral of random variables

Given a random zero-mean gaussian random variable $X(t)$ with parameter $t$, such that $E [X(t) X(t^\prime)] = \sigma^2 (t) \delta_{tt^\prime}$, is it possible to produce a single gaussian random ...
4
votes
0answers
116 views

Characterization of the law of a stochastic process by its finite dimensional distributions

Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space. Let $(X_t)_{t \in [0,T]}$, $(Y_t)_{t \in [0,T]}$ (real-valued) centered Gaussian processes such that the finite dimensional distributions ...
1
vote
2answers
102 views

Standard Brownian Motion

Let $\{X_t,t\ge 0\}$ be a standard Brownian motion. Compute the density of $X_t$ conditioned by $X_{t_1}$ and $X_{t_2}$ assuming that $t_1 <t<t_2$. Can anyone give me some hint to start the ...
0
votes
1answer
155 views

Does an independent-increment Gaussian process necessarily have Gaussian increments?

Suppose a stochastic process is both independent-increment and Gaussian. Are all its increments Gaussian distributed? Thanks!
1
vote
2answers
205 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
1
vote
1answer
99 views

Variable t times a Wiener Process W(1/t)

If $W(t)$ is a Wiener process and $V(t) = t\cdot W(1/t)$ is it possible to say that Since $W(1/t)\space \sim N(0,1/t)$ that $V(t) \sim t\cdot N(0,1/t)$? And if so then is $t\cdot N(0,1/t) = ...
0
votes
1answer
98 views

Confusion regarding autoregressive process

I was reading this article related to autoregressive processes of order $1$. According to wiki it is given by $$ x_t = \phi{x_{t-1}} + \epsilon \\ |\phi| < 1 \\ x_t|x_1,\ldots,x_{t-1} \sim ...
1
vote
0answers
153 views

Composition of multi complex gaussian normal distribution

assume $w_0$, $w_1$, $w_2$, $w_3$ are circular symmetric complex Gaussian distributions, and the composite of $$ h = e^{j\theta_0}w_0 + e^{j\theta_3}w_3 - e^{j\theta_1}w_1 -e^{j\theta_2}w_2 $$ so ...