A PyTorch implementation and empirical study of two deep BSDE formulations for optimal market-making in the Cont-Xiong (2024) dealer market model: a jump-aware formulation (FBSDEJ) preserving the discrete Poisson inventory structure, and a diffusion surrogate for comparison. Validated against a matched finite-horizon finite-difference baseline with cross-dynamics policy evaluation.
The implementation builds on Deep BSDE methods (Han, Jentzen, E, 2018) and adapts them to a jump-control setting with explicit control recovery, incorporating a moment-based mean-field proxy inspired by the fictitious play approach of Han, Hu, Long (2022). The neural network outputs the Brownian gradient Z_t and discrete jump value increments U_t^+/- at each time step; optimal bid-ask quotes are recovered from the HJB first-order conditions.
| Method | Y_0 or V(0) | Loss | Spread | Error vs FD |
|---|---|---|---|---|
| FD finite-horizon (matched T, g) | 0.457 | exact | 1.370 | -- |
| FD stationary (T = inf) | 4.555 | exact | 1.432 | -- |
| Deep BSDE surrogate (5 seeds) | 0.457 +/- 0.000 | 3.4e-05 | 1.333 | <0.1% |
| Deep BSDE jump (5 seeds) | 0.446 +/- 0.002 | 2.6e-05 | discrete | 2.4% |
Cross-dynamics evaluation: surrogate policy trained under diffusion, deployed under true Poisson execution (5000 paths, common random numbers for price).
| Metric | BSDE Policy | FD Policy |
|---|---|---|
| Mean final P&L | 0.466 | 0.469 |
| Std final P&L | 0.571 | 0.580 |
| Sharpe ratio | 0.82 | 0.81 |
| Mean |q_T| | 0.54 | 0.53 |
| Mean spread | 1.333 | 1.368 |
The surrogate-trained policy captures ~99.3% of FD-optimal mean P&L.
Deep BSDE surrogate (5 seeds, 3000 iterations, phi = 0.01) vs FD stationary reference (qualitative comparison only; stationary V(0) is not directly comparable to finite-horizon Y_0).
| Penalty | Deep BSDE Y_0 | max |Z_t| | FD V(0) | FD Spread |
|---|---|---|---|---|
| Quadratic | 0.457 +/- 0.000 | 0.097 +/- 0.018 | 4.555 | 1.432 |
| Cubic | 0.452 +/- 0.000 | 0.145 +/- 0.020 | 4.410 | 1.475 |
| Exponential | 0.448 +/- 0.000 | 0.131 +/- 0.023 | 4.384 | 1.483 |
Parameter sweeps (3000 iterations, single seed). The solver diverges at gamma ~ 3 (exponential penalty) and phi ~ 2 (quadratic penalty).
| Parameter | Value | Y_0 | max |Z_t| | Status |
|---|---|---|---|---|
| gamma | 0.5 | 0.460 | 0.05 | Stable |
| gamma | 1.0 | 0.448 | 0.12 | Stable |
| gamma | 2.0 | 0.349 | 2.22 | Marginal |
| gamma | 3.0 | -3.363 | 258 | Diverged |
| phi | 0.01 | 0.456 | 0.09 | Stable |
| phi | 0.1 | 0.376 | 0.61 | Stable |
| phi | 0.5 | 0.185 | 1.48 | Stable |
| phi | 1.0 | 0.245 | 1.39 | Marginal |
| phi | 2.0 | -6.051 | 17.7 | Diverged |
Matched FD vs Deep BSDE surrogate (1000 iterations).
| T | FD V(0,0) | BSDE Y_0 | FD Spread | Error |
|---|---|---|---|---|
| 0.5 | 0.235 | 0.236 | 1.362 | 0.3% |
| 1.0 | 0.457 | 0.457 | 1.370 | 0.1% |
| 2.0 | 0.866 | 0.864 | 1.383 | 0.2% |
| 5.0 | 1.852 | 1.172 | 1.410 | 36.7% |
Accurate for T <= 2; fails at T = 5 due to Euler discretisation error and increased training variance over long horizons with N_T = 50 steps.
Comparing Type 1 (no coupling) vs Type 3 (moment-proxy fictitious play) across 5 seeds: Y_0 = 0.4566 +/- 0.0002 vs 0.4568 +/- 0.0001. The moment-based mean-field proxy has no measurable effect at phi = 0.01. Distribution-dependent coupling is needed to make the mean-field interaction substantive.
Accurate value approximation does not guarantee accurate control recovery. The diffusion surrogate achieves <0.1% value error but its total spread (2/alpha = 1.333) is structurally constant -- a property fixed by the control parameterisation, not learned. The FD spread (1.370) is 2.7% wider. Despite this structural bias, the surrogate-trained policy deployed under true Poisson execution captures ~99.3% of FD-optimal mean P&L, showing that structurally biased controls can remain economically robust in low inventory-penalty regimes.
This robustness should not be expected to hold under stronger inventory penalties or models with adverse selection.
- Experiments are at d = 2 (price + inventory) after dimensional reduction. The purpose is validation and diagnosis, not computational advantage over finite differences.
- The mean-field coupling uses a moment-based proxy, not full distribution dependence.
- The diffusion surrogate solves a different control problem from the original game.
- The price process is economically inert (driftless, execution rates independent of S), reducing the value function to one effective dimension in q.
git clone https://github.com/cgarryZA/DeepBSDE-LOB.git
cd DeepBSDE-LOB
pip install torch numpy scipy matplotlibFor GPU support:
pip install torch --index-url https://download.pytorch.org/whl/cu124Train the continuous LOB solver:
python main.py --config configs/lob_d2.json --exp_name demo --log_dir ./logs --device autoTrain the jump-diffusion solver:
python main.py --config configs/lob_d2_jump.json --exp_name jump_demo --log_dir ./logs --device autoGenerate plots:
python scripts/plot_lob.py --config configs/lob_d2.json \
--result logs/demo_result.txt \
--weights logs/demo_model.pt \
--out_dir plotsRun the full experiment suite:
python scripts/run_all_experiments.py --device cuda # full (5 seeds, ~12 hours)
python scripts/run_all_experiments.py --quick --device cuda # quick (2 seeds, ~1.5 hours)Find the solver's breaking point:
python scripts/find_breaking_point.py --param gamma --lo 0.1 --hi 5.0 --penalty exponentialForward policy simulation:
python scripts/simulate_policy.py --weights logs/demo_model.ptDeepBSDE-LOB/
├── paper.pdf # Preprint (14 pages)
├── main.py # Training entry point
├── solver.py # All model + solver classes
├── config.py # Configuration dataclasses
├── registry.py # Equation registration
├── equations/
│ ├── base.py # Abstract base class
│ ├── sinebm.py # Sine-BM benchmark (Han-Hu-Long 2022)
│ ├── flocking.py # Cucker-Smale MFG benchmark
│ ├── contxiong_lob.py # Diffusion surrogate (Section 4)
│ └── contxiong_lob_jump.py # Jump-BSDE solver (Section 3)
├── configs/ # JSON experiment configs
├── scripts/
│ ├── plot_lob.py # Main visualization suite
│ ├── plot_experiments.py # Experiment result plots
│ ├── plot_grid_comparison.py # FD vs BSDE full-grid comparison
│ ├── simulate_policy.py # Forward P&L simulation
│ ├── find_breaking_point.py # Binary search for instability
│ ├── finite_difference_baseline.py # Stationary FD solver
│ ├── finite_difference_finite_horizon.py # Matched finite-horizon FD
│ ├── run_experiments.py # Basic experiment runner
│ └── run_all_experiments.py # Full experiment suite
├── plots/ # Generated figures
└── results/ # Experiment result JSON files
Key parameters in configs/lob_d2.json:
| Parameter | Default | Description |
|---|---|---|
sigma_s |
0.3 | Mid-price volatility |
lambda_a, lambda_b |
1.0 | Order arrival rates |
alpha |
1.5 | Execution probability decay |
phi |
0.01 | Inventory penalty coefficient |
discount_rate |
0.1 | Discount rate r |
penalty_type |
"quadratic" |
"quadratic", "cubic", or "exponential" |
num_time_interval |
50 | Euler-Maruyama time steps |
@misc{garry2026deepbsdelob,
author = {Christian Garry},
title = {A Deep Jump-{BSDE} Solver for Optimal Market-Making
in Limit Order Books},
year = {2026},
howpublished = {\url{https://github.com/cgarryZA/DeepBSDE-LOB}},
}- Cont, R. & Xiong, W. (2024). Dynamics of market making algorithms in dealer markets. Mathematical Finance, 34:467-521.
- Han, J., Jentzen, A. & E, W. (2018). Solving high-dimensional PDEs using deep learning. PNAS, 115(34):8505-8510.
- Han, J., Hu, R. & Long, J. (2022). Learning high-dimensional McKean-Vlasov FBSDEs. SIAM J. Numer. Anal., 60(4):2208-2232.
- Avellaneda, M. & Stoikov, S. (2008). High-frequency trading in a limit order book. Quantitative Finance, 8(3):217-224.
- Andersson, K. et al. (2023). A deep solver for BSDEs with jumps. arXiv:2211.04349.
MIT