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The Geometric Brownian Motion model is a continuous-time stochastic process in which a particle move according to a random fluctuations (Wiener process) and a drift term. The corresponding stochastic differential equation is:

\begin{equation} dX_{t}=\mu X_{t}dt+\sigma X_{t} dW_{t} \end{equation}

And the mean is given by: \begin{equation} E[X_{t}]=X_{0} e^{\mu t} \end{equation}

The Geometric Brownian Motion model recalls the Ornstein-Uhlbenck process, which stochastic differential equation is given by:

\begin{equation} dX_{t}=-\mu X_{t}dt+\sigma dW_{t} \end{equation}

And the mean is given by: \begin{equation} E[X_{t}]=X_{0} e^{-\mu t} \end{equation}

I have several questions concerning these two models:

  1. Could we say that the GBM model is the "reverse-in-time" of the OU process, given the change in sign in front of the drift term? If not, does exist a process equivalent to OU, but with a positive drift term?
  2. What are the consequences of having Xt in front of the σ parameter in the GBM model compared to a OU process?
  3. What are the covariance and variance expressions for a GBM model?
  4. Is the likelihood function the same as a general multivariate normal distribution?
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    $\begingroup$ One could have OU process with negative coefficient, and it would be still OE process. The real difference between the two is the $X_t$ in the noise term. $\endgroup$
    – Roger V.
    Commented Jan 23, 2023 at 19:20
  • $\begingroup$ Thanks @RogerVadim, but then do you think that I would not violate some assumption when using a negative coefficient? Is there a way to be sure about that? $\endgroup$
    – CafféSospeso
    Commented Jan 23, 2023 at 19:25
  • $\begingroup$ You will have an exponentially growing mean, which doesn't make sense in most physical problems... But there are exceptions - like escape over a potential barrier. $\endgroup$
    – Roger V.
    Commented Jan 23, 2023 at 19:34
  • $\begingroup$ In the application of my interest, it would be equivalent to a travelling wave. In other words, I am modelling spatial movements of a particle that is 'trapped' by a travelling wave, and thus exponentially travel away from its original position. It is also possible to add a limit to this exponential growing mean. Does it make sense to you? $\endgroup$
    – CafféSospeso
    Commented Jan 23, 2023 at 19:37

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