[8_markov_jump_lq]_correction

[shlff]

Hi @jstac , as I commented in #174 , there is a further issue in a Bellman equation in section Linked-Ricatti-equations-for-Markov-LQ-dynamic-programming of lecture markov_jump_lq and this PR corrects that issue:

• $-x' P_i x - \rho_i & = \max_u - [x'R_i x + u' Q_i u + 2 u' W_i x - \beta \sumj \Pi{ij}E ((A_i x + B_i u + C_i w)' P_j(A_i x + B_i u + C_i w) x + \rho_j)$ -->> $-x' P_i x - \rho_i & = \max_u - [x'R_i x + u' Q_i u + 2 u' W_i x + \beta \sumj \Pi{ij}E ((A_i x + B_i u + C_i w)' P_j(A_i x + B_i u + C_i w) x + \rho_j)$
• i.e., - -->> +

We should correct it because, in the setups above, we have

• $\min_{{ut}{t=0}^\infty} E \sum_{t=0}^{\infty} \beta^t r(x_t, s_t, u_t)$ with $r(x_t, s_t, u_t) = xt' R{s_t} x_t + ut' Q{s_t} u_t + 2 ut' W{s_t} x_t$
• the Bellman equation I modified is just obtained by turning the minimization problem in the above setting into a maximization problem
• so in the Bellman equation, we should only add a minus sign, -, in the front of the objective function from the minimization problem.

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[mmcky]

@jstac would you be able to review and confirm this change?

[jstac]

Hi @mmcky , I confirm this is correct. Please go ahead.

[mmcky]

Closing this as I have migrated this to https://github.com/QuantEcon/lecture-python-advanced.myst