bellgrid

GPU-native backward-induction for continuous-state stochastic dynamic programs.

bellgrid solves Bellman equations exactly (up to interpolation error) across the entire state space. It's opinionated about backward induction, vectorization, and constraints; unopinionated about your application domain. Composes K continuous states with discrete-state primitives (DiscreteState, MarkovChain) and any mix of continuous and discrete actions. Supports asinh/log-warped grids, scalar / multivariate Gauss-Hermite shock quadrature, and a JIT-compiled multilinear kernel that runs on CPU or CUDA.

Quick start

pip install bellgrid                                                # CPU-only torch from PyPI
pip install bellgrid --extra-index-url https://download.pytorch.org/whl/cu126   # GPU

A minimal Merton consumption-portfolio (log utility, single risky asset, lognormal returns):

import math, torch
from bellgrid import Problem, ContinuousState, ContinuousAction, solve
from bellgrid.grids import WarpedGrid, RegularGrid
from bellgrid.shocks import Normal
from bellgrid.solvers import BackwardInduction

beta, mu, sigma = 0.96, 0.04, 0.15
# closed-form coefficients for V(w) = A + B log(w)
B = 1.0 / (1.0 - beta)
A = math.log(1 - beta) / (1 - beta) + (beta / (1 - beta) ** 2) * (math.log(beta) + mu)

def transition(state, action, shock, t):
    return {"wealth": (state["wealth"] - action["consume"]) * torch.exp(mu + sigma * shock["z"])}

def reward(state, action, shock, t):
    return torch.log(action["consume"])

problem = Problem(
    states=[ContinuousState("wealth", warp="asinh", range=(1e-3, 200.0))],
    actions=[ContinuousAction("consume", bounds=(1e-6, "wealth"))],
    transition=transition,
    reward=reward,
    shocks=[Normal("z", sigma=1.0)],
    horizon=range(0, 20),
    discount=beta,
    terminal_reward=lambda state: A + B * torch.log(state["wealth"]),
)

policy, value = solve(
    problem,
    state_grid={"wealth": WarpedGrid(n=128)},
    action_grid={"consume": RegularGrid(n=500)},
    solver=BackwardInduction(n_quad=7),
)

# Optimal consumption rate at any wealth ≈ 1 - β = 0.040
w = torch.tensor([2.0, 10.0, 25.0, 50.0])
policy({"wealth": w}, t=10)["consume"] / w
# → tensor([0.0401, 0.0401, 0.0401, 0.0401])  (closed-form: 0.04 at every point)

Reward is whatever scalar callable matches your problem: utility maximization, cost minimization, profit, payoff. Sign convention: bellgrid maximizes — negate costs.

Examples

Eight canonical problems, each validated against an analytical or numerical reference. Open the .ipynb files in JupyterLab or view them on GitHub.

Notebook	Problem	Validates against
01_merton	Log-utility Merton consumption-portfolio	Closed form V = A + B log w, c/w = 1 − β
02_carroll_deaton	CRRA lifecycle savings with a borrowing constraint	Endogenous Grid Method (Carroll 2006)
03_american_option	American put on GBM	CRR binomial tree (n=2000), agreement within ~1e-4
04_lqg	2-D linear-quadratic-Gaussian control	Discrete-time Riccati recursion
05_two_asset_merton	2-asset Merton with correlated returns (MultivariateNormal)	Numerical FOC for the optimal portfolio share
06_regime_switching_option	American put under regime-switching vol (MarkovChain)	Bracketed by constant-vol references at σ_low, σ_high, σ_stationary
07_retirement_decision	Lifecycle work vs retire decision (DiscreteState, irreversible)	Qualitative — boundary falls with age, accumulate → retire → decumulate dynamics
08_jump_diffusion_option	American put under Merton (1976) jump-diffusion (Jump + Normal, multi-shock)	European case validated against Merton 1976 series expansion (agreement within ~1e-3); American case shows the jump premium and lower exercise boundary

Each notebook opens with the problem statement and equations, then runs bellgrid against the reference side-by-side.

What's built

• States: ContinuousState (with optional asinh / log warp), DiscreteState, MarkovChain.
• Actions: ContinuousAction (with optional state-dependent bounds), DiscreteAction.
• Shocks: Normal, Lognormal, MultivariateNormal (Gauss-Hermite); Uniform on [low, high] (Gauss-Legendre); Categorical with finite support (exact); Jump (Bernoulli-approximated Poisson with Normal log-magnitudes). Multiple independent shocks per problem are supported via tensor-product quadrature.
• Grids: RegularGrid, WarpedGrid.
• Solvers: BackwardInduction for finite-horizon problems, PolicyIteration (value iteration under the hood) for infinite-horizon stationary problems. CPU or CUDA, JIT-compiled K-D multilinear interpolation.
• Simulator: simulate() shares the user's transition and reward with the solver, so they can't drift apart.

Planned

• Implicit differentiation of policy / value wrt model parameters.
• Local action search instead of grid enumeration (for problems with many continuous actions).

When to use bellgrid (vs. RL)

You have (or can write) a transition model, state is roughly 1–6 continuous dims plus discrete, and you need a policy that is correct across the entire support — including tails the agent rarely visits.

	bellgrid	RL
State dim sweet spot	1–6 continuous + discrete	thousands
Correctness	Exact across the full grid	Approximate, on-distribution
Tail / edge-case behavior	By construction	Only if explored in training
Constraints / kinks	First-class	Hard to encode
Off-policy what-ifs	Cheap recompute	Full retrain
You don't have a model	Doesn't apply	Where RL wins

Longer version with concrete borderline cases in docs/when_to_use.md; full API surface in docs/api.md.

License

MIT.