Geometric Brownian Motion

The model is defined by the following equations

$$dX^i_t = \mu^i X^i_t dt + \sigma_i X^i_t dW^i_t$$ $$\rho_{ij}dt = d<W^i, W^j>_t $$ $$C=\{\rho_{ij}\}_{1\leq i,j \leq d} = LL^T$$

with $(X^1_0,...,X^d_0) = (1,...,1)$ and the risk-free rate $r = \mu^i = 0$.

The values of $X$ are obtained through Monte Carlo simulation of the solution

$$X^i_t = \exp\left(\sigma_i W^i_t - \sigma_i^2 t/2 \right)$$

The simulation of the correlation structure is explained in correlation.

In addition to the information available in DataCarlo and AIvault, including details such as the name of each file to download, as well as the username and password, access to:

1. The neural network that generates the distribution of $X_t$ requires appending the file name to the URL

   "mlp.lpma.math.upmc.fr/" + "AIvault/BandS/Asset/"

2. The neural network that generates the derivative price value requires appending the file name to the URL

   "mlp.lpma.math.upmc.fr/" + "AIvault/BandS/Derivatives/"

3. The training dataset requires appending the file name to the URL

   "mlp.lpma.math.upmc.fr/" + "DataCarlo/BandS/Training/"

4. The validation/training dataset requires appending the file name to the URL

   "mlp.lpma.math.upmc.fr/" + "DataCarlo/BandS/Testing/"