Notation and Symbols - Deep Learning for Solving and Estimating Dynamic Models in Economics and Finance

Symbol	Meaning
$\x \in \R^d$	Input or state vector, depending on context
$p(\x)$	Policy/control associated with state $\x$
$G(\x, p(\x))$	Equilibrium or residual operator (e.g., Euler equation)
$\mathcal{N}_\rho$	Neural network with parameters $\rho$
$\ell$	Loss function (supervised or residual-based)
$\eta$	Learning rate
$\beta_1, \beta_2$	Adam momentum coefficients
$\mathcal{D}$	Dataset or collocation set
$\theta$	Generic model or structural parameter vector; when structural parameters and neural-network weights must be distinguished, $\rho$ denotes network weights
$A$	Number of OLG cohorts (Ch. 5)
$G_t(k_i, \varepsilon_j)$	Histogram mass at grid point $k_i$ , shock $\varepsilon_j$ (Ch. 6)
$\mu_t$	Cross-sectional wealth distribution (Ch. 6--8)
$V(a,z)$	Value function in continuous-time HJB (Ch. 7--8)
$g(a,z)$	Stationary density from the KFE / Fokker--Planck equation (Ch. 7--8)
$\mathrm{SCC}_t$	Social cost of carbon (Ch. 11)

Where necessary, chapter-specific notation (e.g., HJB/PDE operators, kernel functions) is introduced locally to avoid ambiguity. In a few places, the script intentionally reuses symbols such as $\eta$ when that is standard in the underlying literature; in those cases, the local chapter definition takes precedence.

Symbols with conflicting uses across chapters. Several symbols below are reused with different meanings depending on the chapter, because each chapter inherits the convention of its primary source. This table collects the conflicts in one place; chapters that introduce a new local meaning also add a one-line warning at first use.

Symbol	Meanings (by chapter)
$\gamma$	IES $=1/\text{CRRA}$ in Ch. Chapter 3 (IRBC); CRRA in Ch. Chapter 7 (cake-eating) and Ch. Chapter 8 (continuous-time HA); reused as $\sigma_u$ in Ch. Chapter 11 (OLG-IAM) to free $\sigma_t$ for emissions intensity; RL discount factor $\gamma\in[0,1)$ and BatchNorm scale parameter in Ch. Chapter 1; Hyperband / Successive-Halving reduction factor in Ch. Chapter 4.
$\eta$	Learning rate (Ch. Chapter 1--Chapter 2); TFP shock in OLG (Ch. Chapter 5); idiosyncratic productivity in Krusell--Smith (Ch. Chapter 6); OU mean-reversion (Ch. Chapter 8); small numerical shift in I-spline basis; normalized temperature costate (Ch. Chapter 11); normalized spatial coordinate in PINN bilinear BC construction (Ch. Chapter 7); functional-derivative test perturbation in KFE adjoint argument (Ch. Chapter 8).
$\alpha$	Capital share in Cobb--Douglas production (Ch. Chapter 2, Chapter 5, Chapter 6, Chapter 8, Chapter 11); ReLoBRaLo smoothing parameter (Ch. Chapter 4); boundary MPC head in I-spline (Ch. Chapter 7).
$\zeta$ vs. $\alpha$	Capital share is denoted $\zeta$ in Ch. Chapter 3 (Azinovic et al. convention) and $\alpha$ everywhere else.
$T$	ReLoBRaLo softmax temperature (Ch. Chapter 4); time horizon (Ch. Chapter 7, Chapter 11); atmospheric temperature $T^{\mathrm{AT}}_t$ in DICE (Ch. Chapter 11); data sample size in SMM (Ch. Chapter 10).
$\delta$	Capital depreciation rate (most chapters); Dirac measure in master equation (Ch. Chapter 6); Huber-loss threshold in robust regression (Ch. Chapter 1).
$\psi$	IES in Ch. Chapter 11 Epstein--Zin preferences (paired with $\gamma_u$ for risk aversion); Cobb--Douglas capital exponent in the GP-VFI growth model used to demonstrate active-subspace scaling (Ch. Chapter 9, Section 9.6).
$\mu$	Cross-sectional wealth distribution $\mu_t$ (Ch. Chapter 6--Chapter 8); SGD momentum coefficient (Ch. Chapter 1); Lagrange / KKT multiplier on investment irreversibility (Ch. Chapter 3); emissions abatement rate $\mu_t\in[0,1]$ (Ch. Chapter 11).
$\rho$	Network parameters $\mathcal{N}_\rho$ (most chapters); RMSprop decay coefficient and recurrence spectral radius (Ch. Chapter 1); discount rate in continuous-time HJB (Ch. Chapter 7, Chapter 8); ReLoBRaLo baseline-mix coefficient (Ch. Chapter 4). The variant $\varrho$ is reserved for shock persistence in Ch. Chapter 2. TFP persistence is denoted $\rho_z$ in Ch. Chapter 3 (Azinovic et al. convention) and $\varrho$ elsewhere.
$\sigma$	Logistic activation $\sigma(z)=1/(1+e^{-z})$ in Ch. Chapter 1 (single-output and final-layer use); the same symbol is used as a generic non-linearity in the RNN recurrence and as the LSTM gate non-linearity later in the same chapter, so the meaning is always logistic but the typographic role (specific vs. generic) shifts. Ch. Chapter 11 (climate) reserves $\sigma_t$ for emissions intensity, $\sigma_u$ for household CRRA.
$\tau$	Bounded time index $\tau_t$ used as a network input in the deterministic CDICE-DEQN derivation (Ch. Chapter 11, Section 11.11); separately, the per-period carbon tax rate (also written $\tau_t$ ) later in the same chapter, in the OLG-IAM and Pareto-improving-tax discussion. The notation is reset locally at each first use; readers should rely on the surrounding sentence rather than on the symbol alone.

Default reading. When in doubt, default to the most common usage: $\rho$ is the neural-network parameter vector, $\eta$ is the learning rate, $\alpha$ is the Cobb--Douglas capital share, and $\sigma$ is the logistic activation. Chapter-specific reuses always override locally, and the chapter introductions flag a divergent meaning at first use. The conflict table above is a forward-reference for non-linear readers; a cover-to-cover reader will see each meaning introduced once and need not consult the table on a first pass.

Abbreviations and acronyms. The following acronyms appear throughout the script. They are introduced in full at first use within each chapter; this list serves as a quick reference.

ABC	Approximate Bayesian Computation	KFE	Kolmogorov forward (Fokker--Planck) Eq.
ACE	Analytic Climate Economy (Traeger)	KKT	Karush--Kuhn--Tucker conditions
AD	Automatic Differentiation	KS	Krusell--Smith (1998) economy
AdamW	Adam with decoupled weight decay	LSTM	Long Short-Term Memory net
AS	Active Subspace	MAGICC	Reduced-complexity climate emulator
BAL	Bayesian Active Learning	MC	Monte Carlo
BC	Boundary Condition	MFG	Mean Field Game
BSDE	Backward Stochastic Differential Eq.	ML	Machine Learning
CDICE	Calibrated DICE (Folini 2024)	MLE	Maximum Likelihood Estimator
CRN	Common Random Numbers	MLP	Multi-Layer Perceptron
CRRA	Constant Relative Risk Aversion	MMW	Maliar--Maliar--Winant (2021)
DEQN	Deep Equilibrium Net	MPC	Marginal Propensity to Consume
DGM	Deep Galerkin Method	NAS	Neural Architecture Search
DICE	Dyn. Integ. Climate-Econ. model	NTK	Neural Tangent Kernel
DKL	Deep Kernel Learning	OLG	Overlapping Generations
DL	Deep Learning	PDE	Partial Differential Equation
DNN	Deep Neural Network	PINN	Physics-Informed Neural Net
ECS	Equilibrium Climate Sensitivity	QMC	Quasi-Monte Carlo
EGM	Endogenous Grid Method	ReLoBRaLo	Relative Loss Balancing
ELU	Exponential Linear Unit	RL	Reinforcement Learning
EMINN	Economic Model Informed NN	RNN	Recurrent Neural Network
FaIR	Reduced-complexity climate emulator	SBI	Simulation-Based Inference
FB	Fischer--Burmeister loss	SCC	Social Cost of Carbon
FD	Finite Differences	SDE	Stochastic Differential Equation
FNO	Fourier Neural Operator	SGD	Stochastic Gradient Descent
FOC	First-Order Condition	SMM	Simulated Method of Moments
GE	General Equilibrium	TF / TF2	TensorFlow / TensorFlow 2
GMM	Generalized Method of Moments	UQ	Uncertainty Quantification
GP	Gaussian Process	VFI	Value Function Iteration
HA	Heterogeneous Agent	ZLB	Zero Lower Bound
HJB	Hamilton--Jacobi--Bellman Eq.	XLA	Accelerated Linear Algebra (TF/JAX)
IRBC	Internat. Real Business Cycle	DeepONet	Deep Operator Network
JAX	JAX autodiff library (Google)	DeepHAM	Deep Heterogeneous-Agent Model