2014/12/05 by Suprio Bhar, Bhar, Suprio
Economics, Econometrics and Finance · #Stochastic processes and financial applications #Financial Risk and Volatility Modeling #Complex Systems and Time Series Analysis
paper · pdf · doi:10.48550/arxiv.1412.1912
In Rajeev (2013), 'Translation invariant diffusion in the space of tempered\ndistributions', it was shown that there is an one to one correspondence between\nsolutions of a class of finite dimensional SDEs and solutions of a class of\nSPDEs in \S', the space of tempered distributions, driven by the\nsame Brownian motion. There the coefficients \\σ, \b of the\nfinite dimensional SDEs were related to the coefficients of the SPDEs in\n\S' in a special way, viz. through convolution with the initial\nvalue y of the SPDEs.\n In this paper, we consider the situation where the solutions of the finite\ndimensional SDEs are stationary and ask whether the corresponding solutions of\nthe equations in \S' are also stationary. We provide an affirmative\nanswer, when the initial random variable takes value in a certain set\n\C, which ensures that the coefficients of the finite dimensional\nSDEs are related to the coefficients of the SPDEs in the above `special'\nmanner.\n