BSDE Solver

Backward Stochastic Differential Equation solver using least-squares Monte Carlo for high-dimensional HJB problems.

Approach

1. Forward simulation: simulate N state trajectories from t=0 to T using SimulationRunner.
2. Backward regression: starting from the known terminal condition at T, step backward. At each time step, regress the continuation value onto polynomial basis functions of the state variables.
3. Control optimization: at each regression point, evaluate the control (bid/ask intensities) that maximizes the Hamiltonian.

Example

#![allow(unused)]
fn main() {
use solver::numeric::bsde::BsdeSolver;
use solver::models::heston_hawkes::HestonHawkesModel;
use solver::numeric::basis::PolynomialBasis;

// Heston-Hawkes: N=3 model (inventory, variance, intensity)
let model = HestonHawkesModel::new(/* ... */);
let basis = PolynomialBasis::new(2, PolynomialType::Power);
let solver = BsdeSolver::new(10_000, 100, // paths, time steps
    solver::numeric::bsde::InitializationMode::Local(5.0));
let solution = solver.solve(&model, [0.0, 0.04, 0.5], 1.0, &basis);
// solution contains bid/ask intensities and value function
}

Dimension-agnostic design

The solver uses Rust const generics and is fully generic over the state dimension N. The same code works for:

The polynomial basis function count grows as O(N^2) for degree-2 polynomials:

Configuration

• Number of paths and time steps.
• Basis function type (polynomial, scaled).
• Regularization parameter for the regression.
• Convergence tolerance.

When to use

BSDE is preferred when:

• The state dimension is 3 or higher.
• Grid-based methods would be intractable.
• You can tolerate Monte Carlo noise in the solution.
• You need to scale to models with more factors in the future.