Regularity conditions for existence of solutions to the path-dependent BBSM SDE

Determine the regularity conditions that guarantee the existence of a solution to the stochastic differential equation dA_t^{(h)} = φ_t dt + ψ_t dB_t + γ_t h(B_t) dB_t for t ∈ [0,T] with initial condition A_0^{(h)} > 0, where h is a Cherny–Shiryaev–Yor piecewise continuous function and φ_t, ψ_t, γ_t are adapted processes in the unified Bachelier–Black–Scholes–Merton framework. The purpose is to establish existence of the continuous-time limit of the binary path-dependent pricing model derived via the Cherny–Shiryaev–Yor invariance principle.

Background

The paper develops a binary, path-dependent option pricing model within the unified Bachelier–Black–Scholes–Merton (BBSM) framework by applying the Cherny–Shiryaev–Yor invariance principle (CSYIP). In the weak limit, the discrete model leads to a continuous-time risky asset dynamics given by dA_t^{(h)} = φ_t dt + ψ_t dB_t + γ_t h(B_t) dB_t, where h(·) is a CSY piecewise continuous function of the driving Brownian motion.

Because the diffusion coefficient depends directly on the Brownian motion B_t rather than on the state A_t, the resulting dynamics fall outside the class of Markovian Itô SDEs typically covered in standard treatments (e.g., Duffie, 2001). The authors therefore identify the need to delineate the precise regularity conditions under which solutions exist for this non-standard SDE.