Stability and convergence of numerical methods for the LOB SPDE with rough volatility

Establish stability and convergence of numerical approximations for the stochastic partial differential equation du(t,x) = [η u_xx(t,x) − β sgn(x) [u_x(t,x)]^- − ζ u(t,x) + J(x,u(t,x)) + G(x,ℓ(t))] dt + c u(t,x) √Y(t) dW(t) (equation (eq20241113_1)), in which the multiplicative diffusion coefficient c u(t,x) √Y(t) is not globally Lipschitz and fails linear-growth conditions and Y(t) is temporally irregular due to its definition via a singular stochastic integral equation.

Background

The paper proposes an SPDE for the centered limit order book density with a multiplicative noise term c u(t,x) √Y(t), where Y(t) is defined by a fractional-type singular stochastic integral equation derived from Hawkes-process scaling limits. Existing stability and convergence results for numerical methods typically assume globally Lipschitz diffusion coefficients and linear growth, assumptions violated here.

The authors explicitly note that these standard assumptions fail for their diffusion term and therefore the stability and convergence behavior of numerical schemes for this SPDE are not established.