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Appendix B. Computation methods

The HJB for \((y, n)\) component does not has a straightforward solution. We use false transient method to solve the ODEs concerning \((y,n)\) in this paper.

Take a general HJB that takes into consideration smooth ambiguity and brownian misspecification. Here we leave out the subscription \(m\) in \(\phi(y)\) as well as the upscription in \(\gamma_3\).

Recall that one HJB of interest for a damage specification \(\gamma_3\) is:

\[\begin{split}\begin{aligned} 0 = \max_{\tilde e} \min_{\omega^a_\ell : \sum_{\ell=1}^L \omega^a_\ell = 1} &- \delta \phi(y) + \eta \log\tilde e \\ & + \frac{1}{2} \left(\frac{d^2 \phi}{dy^2} + \frac{ (\eta - 1)}{\delta} \left(\gamma_2 + \gamma_3\mathbb{I}\{y>\bar y\} \right) \right)(\tilde e)^2 |\varsigma|^2 \\ & - \frac{1}{2\xi_b} \left[ \frac{d\phi}{dy} + \frac{(\eta -1)}{\delta}(\gamma_1 + \gamma_2 y + \gamma_3 (y-\bar y)\mathbb{I}\{y > \bar y\})\right]^2 \cdot |\varsigma|^2 (\tilde e)^2 \\ \\ & + \sum_{\ell=1}^{L} \omega_\ell^a \left(\frac{d\phi}{dy}+ \frac{(\eta -1)}{\delta}(\gamma_1 + \gamma_2 y + \gamma_3 (y - \bar y)\mathbb{I}\{y > \bar y\} ) \right)\theta_\ell \tilde e \\ \\ & + \xi_a \sum_i \omega^a_\ell(\log \omega^a_\ell - \log \pi^a_\ell) \end{aligned}\end{split}\]

The problem satisfies condition to switch max and min problem. In the code, we first compute the optimal \(\tilde e\) and then the optimizing \(\omega_\ell\), so we follow this order here.

The settup includes a tolerance level, \(tolerance\), that defines convergence and a constant step size, \(\epsilon\), for updating the value function.

We start with an initial guess of value function \(\phi_0(y)\) and initial values of \(\{ \omega_\ell\}_{\ell=1}^L\), and update the value function according to the following way: 1. For a given \(\color{blue}{\phi_i(y)}\), compute the optimizing \(\tilde e\) according to its first order condition:

\[\begin{split}\begin{aligned} 0 = &\frac{\eta}{\color{blue}{\tilde e}} + \sum_{\ell=1}^{L} \omega_\ell^a \left(\color{blue}{\frac{d\phi_i}{dy}}+ \frac{(\eta -1)}{\delta}(\gamma_1 + \gamma_2 y + \gamma_3 (y - \bar y)\mathbb{I}\{y > \bar y\} ) \right)\theta_\ell \\ & + \left(\color{blue}{\frac{d^2 \phi_i}{dy^2}} + \frac{ (\eta - 1)}{\delta} \left(\gamma_2 + \gamma_3 \mathbb{I}\{y>\bar y\} \right) - \frac{1}{\xi_b} \left[ \color{blue}{\frac{d\phi_i}{dy}} + \frac{(\eta -1)}{\delta}(\gamma_1 + \gamma_2 y + \gamma_3 (y-\bar y)\mathbb{I}\{y > \bar y\})\right]^2 \right)|\varsigma|^2 \color{blue}{\tilde e} \end{aligned}\end{split}\]

  1. After compute the optimizing \(\tilde e\) from above, we compute the optimizing \(\omega_\ell\) according to its first order condition:

\[\color{blue}{\omega_\ell} = \frac{\pi_\ell^a \exp\left( -\frac{1}{\xi_a}\left[ \color{blue}{\frac{d\phi_i}{dy}} + \frac{(\eta -1)}{\delta}(\gamma_1 + \gamma_2 y + \gamma_3 (y - \bar y)\mathbb{I}\{y > \bar y\} )\right] \color{blue}{\tilde e} \cdot \theta_\ell \right)}{\sum_{\ell=1}^L \pi_\ell^a \exp\left( -\frac{1}{\xi_a}\left[ \color{blue}{\frac{d\phi_i}{dy}}+ \frac{(\eta -1)}{\delta}(\gamma_1 + \gamma_2 y + \gamma_3 (y - \bar y)\mathbb{I}\{y > \bar y\} )\right]\color{blue}{\tilde e} \cdot \theta_\ell \right)}, \quad \ell = 1,2,\dots,L\]

  1. Plug the above computed \(\tilde e\) and \(\{\omega_\ell\}_{\ell=1}^L\) back into the above HJB. Update \(\phi_i(y)\) to \(\phi_{i+1}(y)\) by solving the following ODE:

    \[\begin{split}\begin{aligned} \frac{\color{red}{\phi_{i+1}(y)} - \color{blue}{\phi_i(y)}}{\epsilon} = &- \delta \color{red}{\phi_{i+1}(y)} + \eta \log\color{blue}{\tilde e} \\ & + \frac{1}{2} \left(\color{red}{\frac{d^2 \phi_{i+1}}{dy^2}} + \frac{ (\eta - 1)}{\delta} \left(\gamma_2 + \gamma_3\mathbb{I}\{y>\bar y\} \right) \right)(\color{blue}{\tilde e})^2 |\varsigma|^2 \\ & - \frac{1}{2\xi_b} \left[ \color{red}{\frac{d\phi_{i+1}}{dy}} + \frac{(\eta -1)}{\delta}(\gamma_1 + \gamma_2 y + \gamma_3 (y-\bar y)\mathbb{I}\{y > \bar y\})\right]^2 \cdot |\varsigma|^2 (\color{blue}{\tilde e})^2 \\ \\ & + \sum_{\ell=1}^{L} \color{blue}{\omega_\ell^a} \left(\color{red}{\frac{d\phi_{i+1}}{dy}} + \frac{(\eta -1)}{\delta}(\gamma_1 + \gamma_2 y + \gamma_3 (y - \bar y)\mathbb{I}\{y > \bar y\} ) \right)\theta_\ell \color{blue}{\tilde e} \\ \\ & + \xi_a \sum_i \color{blue}{\omega^a_\ell}(\log\color{blue}{\omega^a_\ell} - \log \pi^a_\ell) \end{aligned}\end{split}\]

    Blued \(\color{blue}{\tilde e}\) and \(\color{blue}{\omega_\ell}\) indicate they are computed using \(\color{blue}{\phi_i(y)}\).

    The method we use to solve the ODE is biconjugate-gradient method. Use ?scipy.sparse.linalg.bicg for document. See also wiki page for biconjugate gradient method.

  2. Check whether the convergence condition is satisfied. We call left-hand side formula left-hand side error. Set a tolerance level, \(tolerance\). We say that the algorithm converges, if:

    \[\frac{|\color{red}{\phi_{i+1}(y)} - \color{blue}{\phi_i(y)}| }{\epsilon} < tolerance\]

    and we get the solution \(\phi(y) = \phi_{i+1}(y)\). Otherwise, assign \(\phi_{i+1}(y)\) to \(\phi_i(y)\), and repeat step 1-4.

# core loop in functions in `source/` can be described as follows
An initial guess: ϕ
Intial values of distorted probabibity of ω_ℓ: πc_o
constant step size: ϵ
tolerance level: tol
left hand error = 1 # random value larger than tol
report numbers of iteration: episode = 0
while left hand side error > tol:
    compute  dϕdy # first crder derivative
    compute dϕdyy # second order derivative
    compute e_tilde
    compute optimizing ω_ℓ: πc
    solve the ODE by conjugate gradient to get ϕ_new
    update left hand error
    compute right hand error
    ϕ = ϕ_new
    episode += 1