3  Model Dynamics

The state at a time \(t\) is \(W_t\) and is non-negative.

The transitions are

\[ W_{t+1} = s(W_t - w_1 - w_2) + w_t^T r_t \]

where \(w_{1_t},w_{2_t}\) are the weights of risky assets (Nvidia and Warner Bros, respectively) at time \(t\),\(w_t = (w_{1_t},w_{2_t})^T\), \(r_t=(r_{1_t},r{2_t})^T\) are the return vector for risky assets at time \(t\), and \(s\) is the fixed rate of return for the risk-free asset.

3.1 Probabilistic transitions

The probabilistic transitions depend on the returns \(r_t\) where \(\mu\) is the mean returns of the risky assets.

\[ \mu = (\mu_1,\mu_2)^T . \]

The covariance matrix of returns, describing the variances and correlations between assets is represent by \(\Sigma\).

Important note: We are not taking in count a transitions cost in this problem.