Identifying nonlinear covariate effects in semimartingale regression models

Let X_t be a semimartingale which is either continuous or of counting process type and which satisfies the stochastic differential equation dX_t=Y_tα(t, Z_t) dt+dM_t, where Y and Z are predictable covariate processes, M is a martingale and α is an unknown, nonrandom function. We study inference for α by introducing an estimator for {Mathematical expression} and deriving a functional central limit theorem for the estimator. The asymptotic distribution turns out to be given by a Gaussian random field that admits a representation as a stochastic integral with respect to a multiparameter Wiener process. This result is used to develop a test for independence of X from the covariate Z, a test for time-homogeneity of α, and a goodness-of-fit test for the proportional hazards model α(t, z)=α₁(t)a₂(z) used in survival analysis. © 1990 Springer-Verlag.