Limit theorems for additive functionals of stochastic functional differential equations with infinite delay
For a given \(r > 0\), define \({C_r} = \{ \phi \in C(( - \infty ,0];{\mathbb R^d}):\mathop {\sup }\limits_{ - \infty < \theta \le 0} {e^{r\theta }}\left| {\phi (\theta )} \right| < \infty \} \) with norm \({\left\| \phi \right\|_r} = \mathop {\sup }\limits_{ - \infty < \theta \le 0} {e^{r\theta }}\left| {\phi (\theta )} \right|\), where \(C(( - \infty ,0];{\mathbb R^d})\) denotes the family of continuous functions from \(( - \infty ,0]\) to \({\mathbb R^d}\). Choosing \({C_r}\) as the phase space, the paper considers the stochastic functional differential equations (SFDEs) with infinite delay \(dX(t) = b({X_t})dt + \sigma ({X_t})dW(t),{X_0} = \xi \in {C_r}\), where \(b:{C_r} \to {\mathbb R^d}\) and \(\sigma :{C_r} \to {\mathbb R^{d \times m}}\) are continuous functionals, \(W(t)\) is an \(m\)-dimensional Wiener process on a compete filtered probability space and \({X_t}(\theta ):\theta \mapsto X(t + \theta ) \in {\mathbb R^d},\theta \in ( - \infty ,0]\) denotes the segment process. The paper aims to investigate the large-time behavior of the functional \({t^{ - 1}}\int_0^t {f({X_s}(\xi )ds} \) as \(t \to \infty \). It establishes the strong law of large numbers and the central limit theorem for SFDEs with non-Lipschitz coefficients and infinite delay. The paper also establishes the law of iterated logarithm for SFDEs with infinite delays under some dissipative condition.
stochastic functional differential equations with infinite delay
scientific article; zbMATH DE number 7436217
scientific article