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Formal MDP: No-Portfolio Stage (noport_stage)

no-port-cons

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

Rosetta Stone

DDSL abstract This stage YAML
Arrival state \(x_{\prec}\) \(k\) (investable assets) prestate: k
Decision state \(x\) \(k_d\) (identity) states: k_d
Continuation state \(x_{\succ}\) \(m\) (cash-on-hand) poststates: m
Control \(a\) (none) —
Pre-decision shock \(\zeta\) (none) —
Post-decision shock \(\eta\) \(\theta \sim \text{LogNormal}(\mu_\theta, \sigma_\theta)\) exogenous: θ
\(\mathrm{g}_{\prec\sim}\) identity: \(k_d = k\) arvl_to_dcsn_transition
\(\mathrm{g}_{\sim\succ}\) \(m = k_d R + \theta\) dcsn_to_cntn_transition
\(\mathbb{B}\) (decision mover) \(v(k) = \mathbb{E}_\theta[v_{\succ}(kR + \theta)]\) cntn_to_dcsn_mover
\(\mathbb{I}\) (arrival mover) \(v_{\prec} = v\) (identity) dcsn_to_arvl_mover

Model

An agent enters the stage with investable assets \(k \in \mathbb{R}_{++}\). There is no decision; the stage serves purely as a shock-realization bridge that converts an asset position into cash-on-hand.

State space. \(\mathsf X_{\prec} = \mathsf X = \mathbb{R}_{++}\) (assets), \(\mathsf X_{\succ} = \mathbb{R}_{++}\) (cash-on-hand). The action set is a singleton (no control).

Shock process. A transitory income shock \(\theta\) is realized between decision and continuation (post-decision only configuration, \(\mathsf Z_{\prec} = \{\varnothing\}\)):

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

Transitions.

\[ \mathrm{g}_{\prec\sim} \colon k \mapsto k_d = k \qquad\text{(identity)}, \qquad \mathrm{g}_{\sim\succ} \colon (k_d, \theta) \mapsto m = k_d R + \theta. \]

ADC Stage Structure

Perches

Perch Variable Space Description
\(\prec\) (arrival) \(k\) \(\mathsf X_{\prec} = \mathbb{R}_{++}\) Investable assets
\(\cdot\) (decision) \(k_d\) \(\mathsf X = \mathbb{R}_{++}\) Investable assets (identity, no choice)
\(\succ\) (continuation) \(m\) \(\mathsf X_{\succ} = \mathbb{R}_{++}\) Cash-on-hand (after shock)

Filtration

\[ \mathcal F_{\prec} = \mathcal F \subset \mathcal F_{\succ}. \]

The arrival and decision information sets coincide (no pre-decision shock). The continuation information set is strictly larger: the shock \(\theta\) is revealed.

Movers

Decision mover \(\mathbb{B}\) (cntn → dcsn). Since there is no control, \(\mathbb{B}\) is a pure conditional expectation:

\[ (\mathbb{B}\, v_{\succ})(k) = \mathbb{E}_\theta\bigl[v_{\succ}(kR + \theta)\bigr]. \]

Arrival mover \(\mathbb{I}\) (dcsn → arvl). The identity:

\[ v_{\prec}(k) = v(k). \]

Stage operator. \(\mathbb{T} = \mathbb{I} \circ \mathbb{B} = \mathbb{B}\), since \(\mathbb{I}\) is trivial:

\[ (\mathbb{T}\, v_{\succ})(k) = \mathbb{E}_\theta\bigl[v_{\succ}(kR + \theta)\bigr]. \]

This is the operator \(\mathbf{E}\) in the noport–cons conjugation.

Bellman Equation

Since there is no control, the stage Bellman equation reduces to:

\[ v_{\prec}(k) = \mathbb{E}_\theta\bigl[v_{\succ}(kR + \theta)\bigr]. \]

There is no optimisation and hence no policy function, no Euler equation, and no first-order conditions.

Forward Operator

Given a distribution \(\mu_{\prec}\) over arrival states \(k\), the forward operator pushes \(\mu_{\prec}\) through the shock realization:

\[ \mu_{\succ}(A) = \int \Pr(kR + \theta \in A) \, \mathrm{d}\mu_{\prec}(k). \]

Since the arrival mover is identity, \(\mu = \mu_{\prec}\) and the full forward map is \(\mathrm{F} = \mathrm{F}_{\succ}\).

YAML

name: noport_stage

symbols:
  spaces:
    Xk: "@def R++"
    Xm: "@def R++"
    Θ:  "@def R++"

  prestate:
    k: "@in Xk"

  exogenous:
    θ:
      - "@in Θ"
      - "@dist LogNormal(μ_θ, σ_θ)"

  states:
    k_d: "@in Xk"

  poststates:
    m: "@in Xm"

  values:
    V[<]: "@in R"
    V: "@in R"
    V[>]: "@in R"

  parameters:
    R: "@in R++"
    μ_θ: "@in R"
    σ_θ: "@in R+"

equations:
  arvl_to_dcsn_transition: |
    k_d = k

  dcsn_to_cntn_transition: |
    m = k_d*R + θ

  cntn_to_dcsn_mover:
    Bellman: |
      V = E_{θ}(V[>])

  dcsn_to_arvl_mover:
    Bellman: |
      V[<] = V

Calibration

Symbol Value Description
\(R\) 1.02 gross risk-free return
\(\mu_\theta\) 0.0 log-mean of transitory income
\(\sigma_\theta\) 0.1 log-std of transitory income

See also