CPU/GPU-Accelerated Jump Diffusion HJB Equations: A Comparative Study for Low-Latency Crypto Market Making

Author: Frankline Misango Oyolo

Date: 2025

Institution: Arithmax Research

Market Making HJB Equations GPU Computing HFT Cryptocurrency

Summary

We present a comparative analysis of CPU and GPU implementations for solving jump-diffusion Hamilton-Jacobi-Bellman (HJB) equations in high-frequency cryptocurrency market making. By formulating the market maker's decision problem as a stochastic optimal control problem, we derive optimal quoting strategies through HJB partial differential equations.

Our key innovation is the integration of jump-diffusion processes that explicitly capture the discontinuous price movements characteristic of cryptocurrency markets, implemented with accurate Gauss-Hermite quadrature for numerical stability.

Key Results

Up to 43% higher profitability in volatile market conditions
Sub-second latency with GPU implementation
62% reduction in inventory risk vs traditional strategies
Real-time order flow toxicity tracking

Key Mathematical Equations

Jump-Diffusion Price Process:

$$dS_t = \mu S_t dt + \sigma S_t dW_t + S_t \int_{\mathbb{R}} (e^y - 1) \tilde{N}(dt, dy)$$
where:
• $\mu S_t dt$: deterministic drift
• $\sigma S_t dW_t$: continuous diffusion
• Jump term: sudden price movements

Hamilton-Jacobi-Bellman Equation:

$$\frac{\partial V}{\partial t} + \max_\delta \left[ \mathcal{L}^\delta V + f(x,\delta) \right] = 0$$
where:
• $V(t,x)$: value function
• $\delta$: control (bid/ask spreads)
• $\mathcal{L}^\delta$: infinitesimal generator
• $f(x,\delta)$: instantaneous reward

Optimal Bid/Ask Spreads:

$$\delta_{\text{bid}}^* = \arg\max \left[ \lambda_{\text{bid}}(\delta) \cdot (\delta - c) - \frac{\partial V}{\partial q} \right]$$ $$\delta_{\text{ask}}^* = \arg\max \left[ \lambda_{\text{ask}}(\delta) \cdot (\delta - c) + \frac{\partial V}{\partial q} \right]$$
where:
• $\lambda(\delta)$: execution intensity
• $c$: transaction cost
• $q$: inventory position

GPU Implementation Algorithm

Parallel HJB Solver

Initialize value function V on (S,q,t) grid (201×201×T points)
For each time step (backward in time):
Launch GPU kernel with thread per grid point
Each thread computes optimal spreads δ_bid*, δ_ask*
Apply finite difference scheme for spatial derivatives
Integrate jump term using Gauss-Hermite quadrature
Update V(t-1) using implicit scheme
Extract optimal policy from converged value function
Deploy to real-time trading system with <1ms latency

CPU/GPU-Accelerated Jump Diffusion HJB Equations: A Comparative Study for Low-Latency Crypto Market Making

Summary

Key Results

Key Mathematical Equations

GPU Implementation Algorithm

Full Paper