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Formal MDP: Portfolio Stage (port_stage)

no-port-cons

Source: packages/dolo/examples/models/doloplus/port-with-shocks/library/port_stage.yaml

Rosetta Stone

DDSL abstract This stage
Stage name port_stage
Arrival state \(x_{\prec}\) \(k\)
Decision state \(x\) \(k\)
Continuation state \(x_{\succ}\) \(m\)
Control \(\varsigma\)
Exogenous shocks \(\Psi, \theta\) (post-decision)

Model

A household enters the stage with investable assets \(k \in \mathbb{R}_{++}\) and chooses a risky portfolio share \(\varsigma \in [0,1]\). After the portfolio decision, a joint shock vector \(\zeta \coloneqq (\Psi, \theta)\) is realized, determining the continuation state.

State and action spaces. The state space is \(X_k \coloneqq \mathbb{R}_{++}\) and the action space is \(\Pi \coloneqq [0,1]\). The feasibility correspondence is \(\Gamma(k) = [0,1]\) for all \(k\).

Shock process. The shocks are realized after the portfolio decision (post-decision timing):

\[ \Psi \sim \text{LogNormal}(\mu_\Psi, \sigma_\Psi), \qquad \theta \sim \text{LogNormal}(\mu_\theta, \sigma_\theta). \]

A correlation parameter \(\rho_\zeta\) is declared but not yet wired.

Transitions. The arrival-to-decision transition is the identity \(k_d = k\). After the decision and shock realization, cash-on-hand is

\[ m = k \bigl(\varsigma \, \Psi + (1-\varsigma)\, R\bigr) + \theta, \]

where \(R > 0\) is the gross risk-free return.

Bellman Equation

This stage has a "max-over-expectation" structure: the agent chooses \(\varsigma\) before the shocks are realized, so the expectation is inside the max. The arrival-to-decision transition is identity, so the Bellman equation is:

\[ v(k) = \max_{\varsigma \in [0,1]} \; \mathbb{E}_{\Psi,\theta} \Bigl[ v_{\succ}\!\bigl(k(\varsigma \Psi + (1-\varsigma)R) + \theta\bigr) \Bigr], \]

where \(v_{\succ} \colon \mathbb{R}_{++} \to \mathbb{R}\) is the continuation value function. The optimal risky share is

\[ \varsigma^*(k) \coloneqq \operatorname*{arg\,max}_{\varsigma \in [0,1]} \; \mathbb{E}_{\Psi,\theta} \Bigl[ v_{\succ}\!\bigl(k(\varsigma \Psi + (1-\varsigma)R) + \theta\bigr) \Bigr]. \]

The arrival value function is the identity: \(v_{\prec}(k) = v(k)\).

First-Order Conditions

At an interior solution \(\varsigma^* \in (0,1)\), the first-order condition is

\[ \mathbb{E}_{\Psi,\theta} \Bigl[ v'_{\succ}(m) \cdot k\,(\Psi - R) \Bigr] = 0, \]

where \(m = k(\varsigma^* \Psi + (1-\varsigma^*)R) + \theta\). This is a nonlinear equation in \(\varsigma\) that typically requires numerical root-finding for each \(k\).

Forward Operator (Population Dynamics)

Given a distribution \(\mu\) over arrival states \(k\), the forward operator pushes \(\mu\) through the optimal policy and shock realization:

\[ m = k\bigl(\varsigma^*(k)\,\Psi + (1-\varsigma^*(k))\,R\bigr) + \theta, \qquad k \sim \mu, \quad (\Psi,\theta) \text{ drawn independently}. \]

The continuation-state distribution is

\[ \mu_{\succ}(A) = \int \Pr\!\bigl(k(\varsigma^*(k)\Psi + (1-\varsigma^*(k))R) + \theta \in A\bigr)\, \mathrm{d}\mu(k). \]

Calibration

Symbol Value Description
\(R\) 1.02 gross risk-free return
\(\mu_\Psi\) 0.04 log-mean of risky return
\(\sigma_\Psi\) 0.15 log-std of risky return
\(\mu_\theta\) 0.0 log-mean of transitory income
\(\sigma_\theta\) 0.1 log-std of transitory income
\(\rho_\zeta\) 0.0 shock correlation (placeholder)