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I don’t really understand the notion of “choosing” a probability space and Brownian motion when it comes to weak solutions to an SDE. What does it mean to “choose”? Do we have free reign over the BMs, and if not, which BMs are allowed to be “chosen”, and which cannot? Is there a concrete example of a weak solution that explains the concept of “choosing” well? (I know that the integral $\int_0^T\textrm{sgn}(W_s)\textrm{d}W_s$ has a weak solution only but I don’t see how or why. Edit: apparently this is wrong, my bad!)

Likewise, with a strong solution, what does it mean by a “given” probability space/BM? What is an example of a strong solution which using a different BM causes the solution to fail? (I always felt like any augmented sigma algebra generated by the BM would work.)

user107224
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  • https://math.stackexchange.com/questions/2987743/difference-between-weak-or-martingale-and-strong-solutions-to-sdes – Tobsn Apr 17 '21 at 08:05
  • @Tobsn that was exactly the question I was reading; I feel like it didn’t really help me grasp the concept of “choosing” and “given – user107224 Apr 17 '21 at 16:33
  • "$\int_{0}^{T}\operatorname{sgn}(W_{s})dW_{s}$ has a weak solution" ... this is nonsense. You should read carefully again through that post, since what saz wrote is actually quite instructive. In particular his example 3 on Tanaka's equation. – Tobsn Apr 17 '21 at 18:09

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