Is there a uniqueness and existence result for solutions to systems of Stochastic Differential Equations of diffusion type where coefficients are $\mathcal F_t$-measurable, i.e., could depend on the entire trajectories of processes up to time $t$? $$dX_t=a_tdt+b_tdW_t, X_t \in \mathbb R^n$$ where $a_t, b_t$ are of appropriate dimensions and $\mathcal F_t$-measurable.

Most textbooks (e.g., Oksendal) provide theorems only for the cases when coefficients depend on the time $t$ values of stochastic processes, that is, $dX_t=a(t,X_t)dt+b(t,X_t)dW_t$.

  • $\begingroup$ Did you search in literatur for delayed SDEs? $\endgroup$
    – Tobsn
    Jan 11 at 18:52
  • $\begingroup$ No. But is there a general reference for such a theorem? $\endgroup$
    – Alex
    Jan 13 at 17:38
  • $\begingroup$ this could be a starting point: arxiv.org/pdf/0812.1726.pdf $\endgroup$
    – Tobsn
    Jan 13 at 20:21


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