Question

How do I prove that the distribution of $x_t$ in $x_t=x_{t-1}e^{-μt}+ θ(1-e^{-μΔt}) + \int_{t-1}^t e^{-μ(t-s)}\sigma dB_s $ is a normal distribution?

score 89 · Answer 1 · 2022-05-04 13:36:42

89

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How do I prove that the distribution of $x_t$ in $x_t=x_{t-1}e^{-μt}+ θ(1-e^{-μΔt}) + \int_{t-1}^t e^{-μ(t-s)}\sigma dB_s $ is a normal distribution?

Published on 04 May 2022 - 13:36

score 63 · Answer 2 · 2022-05-08 17:54:50

63

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score 26 · Answer 3 · 2022-05-10 19:45:06

26

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score 203 · Answer 4 · 2022-05-11 07:26:28

203

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score 43 · Answer 5 · 2022-05-11 15:01:00

43

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score 108 · Answer 6 · 2022-05-16 20:16:21

108

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202

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77

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score 53 · Answer 9 · 2022-05-25 13:59:27

53

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score 28 · Answer 10 · 2022-05-25 18:30:07

28

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score 112 · Answer 11 · 2022-05-30 03:24:06

112

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score 120 · Answer 12 · 2022-05-31 19:23:00

120

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score 1.6k · Answer 13 · 2022-06-08 02:49:43

1.6k

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score 168 · Answer 14 · 2022-06-08 18:52:34

168

Views

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score 116 · Answer 15 · 2022-06-19 00:40:50

116

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Published on 19 Jun 2022 - 0:40

List Question

How do I prove that the distribution of $x_t$ in $x_t=x_{t-1}e^{-μt}+ θ(1-e^{-μΔt}) + \int_{t-1}^t e^{-μ(t-s)}\sigma dB_s $ is a normal distribution?

projection of a non-zero mean Gaussian vector into a Ball

Modeling a sequential Random Vector as a Random Processes

Bound KL divergence between two distributions by KL divergence of two Gaussian mixture models

Property $E[\mathbf{xx}^T] = \mathbf{\Sigma}-\mathbf{\mu\mu}^T$.

Two definitions of a Stochastic Process?

Sampling from a Poisson distribution

Evaluating integral $\int_0^\infty x^4e^{-2a x^2}dx $

Showing Markov property for "Brownian Motion"

Is this Markov process Gaussian ? $Y_t= X_t - (1-t) X_0 - tX_1, \, t \in [0,1]$

How to evaluate the following Gaussian integral by completing the summation of squares

Generate singular values of a matrix(Gaussian Random) using Quadrant law in python.

Does a Markov chain with Gaussian transitions $p(x_t|x_{t-1})=\mathcal N(\sqrt{1-\beta_t}x_{t-1},\beta_tI)$ tend to $\mathcal N(0,I)$?

Help with a subgaussian proposition

Why is the covariance matrix inverted in the multivariate Gaussian distribution?

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