Volterra equations with affine drift: looking for stationarity
arxiv(2024)
摘要
We investigate the properties of the solutions of scaled Volterra equations
(i.e. with an affine mean-reverting drift) in terms of stationarity at both a
finite horizon and on the long run. In particular we prove that such an
equation never has a stationary regime, except if the kernel is constant (i.e.
the equation is a standard Brownian diffusion) or in some fully degenerate
pathological settings. We introduce a deterministic stabilizer ς
associated to the kernel which produces a fake stationary regime in the
sense that all the marginals share the same expectation and variance. We also
show that the marginals of such a process starting from when starting various
initial values are confluent in L^2 as time goes to infinity. We establish
that for some classes of diffusion coefficients (square root of positive
quadratic polynomials) the time shifted solutions of such Volterra equations
weakly functionally converges toward a family of L^2-stationary processes
sharing the same covariance function. We apply these results to (stabilized)
rough volatility models (when the kernel K(t)= t^H-1/2, 0更多
查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要