Dolo+ Primitives¶

Built-in symbols, functions, distributions, and reserved keywords available in Dolo+/DDSL without explicit definition.

Brackets: Intervals vs Tuples/Lists¶

Context	Meaning	Example
After `@def`	Interval (continuous range)	`x @def [0, 1]` means \(x\in[0,1]\)
After `@in`	Element of a tuple/vector	`[m, b] @in Xa` means \((m,b)\in\mathsf X_{\prec}\)
Perch tag	Measurability declaration	`V`, `c[>]`
Plain list	Collection of items	`parameters: [β, ρ, R]`

Mathematical Domains¶

Symbol	Meaning	Usage Example	Notes
`R`	Real numbers	`x @def R`	All reals
`R+`	Positive real numbers	`assets @def R+`	\(x\geq 0\)
`R-`	Negative real numbers	`debt @def R-`	\(x\leq 0\)
`R++`	Strictly positive reals	`consumption @def R++`	\(x>0\)
`N`	Natural numbers	`periods @def N`	\(0,1,2,\ldots\)
`Z`	Integers	`count @def Z`	\(\ldots,-1,0,1,\ldots\)
`[a, b]`	Closed interval	`probability @def [0, 1]`	\(a\leq x\leq b\)
`(a, b)`	Open interval	`growth_rate @def (-1, inf)`	\(a<x<b\)

Distribution Functions (IID)¶

Function	Parameters	Description	Example
`Normal(Σ, μ)`	Σ: covariance, μ: mean	Multivariate normal	`W @def Normal(Sigma_w, Mu_w)`
`UNormal(σ, μ)`	σ: std dev, μ: mean	Univariate normal	`w @def UNormal(sigma_w)`
`Uniform(a, b)`	a: lower, b: upper	Uniform	`w @def Uniform(-0.5, 0.5)`
`LogNormal(σ, μ)`	σ, μ: of underlying normal	Log-normal	`w @def LogNormal(sigma_w, mu_w)`
`Beta(α, β)`	α, β: shape	Beta	`w @def Beta(alpha_w, beta_w)`
`Bernoulli(π)`	π: probability of 1	Bernoulli	`success @def Bernoulli(0.7)`
`Binomial(n, π)`	n: trials, π: success prob	Binomial	`successes @def Binomial(10, 0.5)`

Joint Distribution Declarations¶

When exogenous shocks are correlated, dolo-plus uses a joint vector declaration that mirrors vanilla dolo's !Normal syntax. The joint vector entry uses the same Μ/Σ keys as dolo's Normal class. The individual component variables are the other exogenous entries (declared with @in space).

Syntax:

exogenous:
    # Individual component spaces
    Ψ: "@in Rp"
    θ: "@in Θ"

    # Joint distribution (mirrors dolo !Normal structure)
    ζ:
      "@dist": Normal
      Μ: [μ_Ψ, μ_θ]
      Σ: [[σ_Ψ^2, ρ_ζ*σ_Ψ*σ_θ], [ρ_ζ*σ_Ψ*σ_θ, σ_θ^2]]

Key	Meaning	Notes
`"@dist": Normal`	Distribution family of the underlying vector	Mirrors dolo's `!Normal` YAML tag
`Μ`	Mean vector of the underlying distribution	Same key as dolo's `Normal` class (capital mu U+039C)
`Σ`	Covariance matrix of the underlying distribution	Same key as dolo's `Normal` class (capital sigma U+03A3). Entries can be parameter expressions (e.g. `σ_Ψ^2`)

The components are inferred: all other exogenous entries (those with @in declarations) are the variables generated by the joint distribution, in order. The mapping from the underlying Normal to the economic variables is determined by their @in spaces — e.g. Ψ: "@in Rp" (R++) with an underlying Normal implies \(\Psi = \exp(\xi_\Psi)\) (LogNormal).

Transform to vanilla dolo: The Μ and Σ copy directly to the output !Normal block. Component variables become log-space exogenous symbols, and equations use exp() to recover level-space values:

# Vanilla dolo output
symbols:
    exogenous: [logΨ, logθ]

exogenous: !Normal
    Μ: [μ_Ψ, μ_θ]
    Σ: [[σ_Ψ^2, ρ_ζ*σ_Ψ*σ_θ], [ρ_ζ*σ_Ψ*σ_θ, σ_θ^2]]

# Transition uses exp() to recover level-space:
#   k_d[t] = k_d[t-1]*(ς[t-1]*exp(logΨ[t]) + ...) + exp(logθ[t])

Stochastic Processes¶

Function	Parameters	Description	Example
`AR1(ρ, σ, μ)`	ρ: autocorr, σ: std dev, μ: mean	AR(1)	`z @def AR1(rho_z, sigma_z, mu_z)`
`VAR1(ρ, Σ, μ)`	ρ: autocorr matrix, Σ: cov, μ: mean	VAR(1)	`Z @def VAR1(Rho_z, Sigma_z, Mu_z)`
`MarkovChain(P, S)`	P: transition matrix, S: states	Discrete Markov chain	`s @def MarkovChain(P_s, S_s)`
`ConstantProcess(μ)`	μ: constant value	Deterministic	`c @def ConstantProcess(mu_c)`
`AggregateProcess(μ)`	μ: aggregate state	Aggregate shock	`agg @def AggregateProcess(mu_agg)`
`Product(...)`	Variable args	Product of processes	`shocks @def Product(AR1(...), UNormal(...))`

Mathematical Functions¶

Function	Usage	Description
`exp(x)`	Exponential	`exp(z)`
`log(x)`	Natural logarithm	`log(c)`
`log10(x)`	Base-10 logarithm	`log10(y)`
`sqrt(x)`	Square root	`sqrt(variance)`
`abs(x)`	Absolute value	`abs(profit)`
`sign(x)`	Sign function (-1, 0, 1)	`sign(excess_demand)`
`^` or `**`	Exponentiation	`k^alpha` or `k**alpha`

Trigonometric Functions¶

Function	Usage	Description
`sin(x)`	Sine	`sin(theta)`
`cos(x)`	Cosine	`cos(theta)`
`tan(x)`	Tangent	`tan(angle)`
`asin(x)`	Arcsine	`asin(ratio)`
`acos(x)`	Arccosine	`acos(ratio)`
`atan(x)`	Arctangent	`atan(slope)`

Comparison and Aggregation Functions¶

Function	Usage	Description
`min(a, b, ...)`	Minimum	`min(c, c_bar)`
`max(a, b, ...)`	Maximum	`max(0, profit)`
`floor(x)`	Floor	`floor(n_periods)`
`ceil(x)`	Ceiling	`ceil(n_goods)`
`round(x)`	Rounding	`round(price, 2)`

Operations (Special Computational Methods Required)¶

Operations differ from simple function compositions in that they require specific computational methods. All operators below use the _{...} convention: the content inside _{...} specifies with respect to what the operator acts. See symbols-conventions.md for disambiguation rules.

Differentiation Operators¶

Operator	Mathematical	Usage	Description
`d_{x}`	\(\partial_x\)	`d_{a}V`	Partial derivative of \(V\) wrt \(a\) (holding other arguments fixed)
`D_{x}`	\(\mathrm{D}_x\)	`D_{a}V`	Total derivative (Jacobian / Fréchet) wrt \(a\) (includes indirect dependence)

These are used in values_marginal declarations and in mover sub-equations (MarginalBellman, ShadowBellman). Differentiation variables must be declared symbols. See open design question on whether d_{.} is a naming convention or a first-class operator.

Expectation Operations¶

Operator	Mathematical	Usage	Description
`E_{shock}(·)`	\(\mathbb{E}_\theta[\cdot]\)	`E_{θ}(V[>])`	Expectation over named shock. Requires integration method.
`E[·\\|·]`	\(\mathbb{E}[\cdot \mid \cdot]\)	`E[V \\| state]`	Conditional expectation. Requires conditional integration.
`Var[·]`	\(\mathrm{Var}[\cdot]\)	`Var[returns]`	Variance operator. Requires second moment computation.

Optimization Operations¶

Operator	Mathematical	Usage	Description
`max_{c}{·}`	\(\sup_c\{\cdot\}\)	`max_{c}{u(c) + β*V[>]}`	Maximum value over control. Requires optimization algorithm.
`argmax_{c}{·}`	\(\arg\sup_c\{\cdot\}\)	`c = argmax_{c}{...}`	Argument maximiser. Returns optimal control.

Implicit System Operator¶

Operator	Mathematical	Usage	Description
`solve_{x1, x2, ...}{·}`	Find \((x_1, x_2, \ldots)\) satisfying conditions	`solve_{c[>], h[>]}{...}`	Solve a system of equations for the listed unknowns. Requires root-finding or analytical method.

The solve_{...} operator declares that the enclosed conditions implicitly define the listed unknowns. The number of conditions must equal the number of unknowns.

operators #operators/solve #invEuler¶

Example (two FOCs defining consumption and housing choice):

InvEuler: |
  solve_{c[>], h_choice[>]}{
    alpha * c[>]^(-gamma_c) - beta * d_{a}V[>] = 0,
    (1 + tau) * alpha * c[>]^(-gamma_c)
      - (1 - alpha) * kappa^(1 - gamma_h) * h_prime^(-gamma_h)
      - beta * d_{h}V[>] = 0
  }

Single-control shorthand: when the system is a single explicit inversion, solve_{} can be written as a direct assignment. These are equivalent:

# Explicit (standard EGM shorthand):
c[>] = (beta * dV[>] / alpha)^(-1/gamma)

# Equivalent solve form:
solve_{c[>]}{alpha * c[>]^(-gamma) - beta * dV[>] = 0}

The methodization layer determines the solution method (analytical inversion, Newton, partial-EGM, etc.).

Grid Discretization Methods (as decorators)¶

Method	Usage	Best For
`@method: gauss-hermite`	`@n_points: 7`	Normal distributions
`@method: tauchen`	`@n_points: 5, @m: 2`	AR(1) with low persistence
`@method: rouwenhorst`	`@n_points: 5`	AR(1) with high persistence
`@method: equiprobable`	`@n_points: 10`	General distributions
`@method: custom`	With `values` and `probabilities`	User-defined discretization

Reserved Keywords¶

Keyword	Context	Meaning
`symbols`	Top-level section	Variable declarations
`equations`	Top-level section	Model equations
`calibration`	Top-level section	Parameter values
`domain`	Top-level section	State space bounds
`grids`	Top-level section	Discretization specs
`exogenous`	In symbols or top-level	Shock specification
`prestate`	In symbols	Arrival state variables
`states`	In symbols	Decision state variables
`poststates`	In symbols	Continuation state variables
`controls`	In symbols	Control/choice variables
`parameters`	In symbols	Model parameters (Υ-level)
`settings`	In symbols	Numerical configuration (Ρ-level)
`values`	In symbols	Value function objects
`values_marginal`	In symbols	Marginal value objects