Stochastic selection problem for a Stratonovich SDE with power non-linearity
Анотація
In our paper (Bernoulli 26 (2020) 1381–1409), we found all strong Markov solutions that spend zero time at 0 of the Stratonovich stochastic differential equation dX=|X|α∘dB, α∈(0,1). These solutions have the form Xtθ=F(Btθ), where F(x)=11−α|x|1∕(1−α)signx and Bθ is the skew Brownian motion with skewness parameter θ∈[−1,1] starting at F−1(X0). In this paper we show how an addition of small external additive noise εW restores uniqueness. In the limit as ε→0, we recover heterogeneous diffusion corresponding to the physically symmetric case θ=0.
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