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Let $b:\mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R^d \to \mathcal M_{ d\times q} (\mathbb R)$ be Lipschitz. Let $(W_t, t\ge 0)$ be the standard $q$-dimensional Brownian motion. Then $$ d X_t = b(X_t) d t + \sigma (X_t) d W_t \quad \text{a.s.}, $$ has a unique strong solution. Now let $f:\mathbb R_{\ge 0} \to \mathcal M_{ d\times q} (\mathbb R)$ be "nice" enough. I'm interested in the existence of the solution of the SDE $$ d X_t = b(X_t) d t + f (p_t(X_t)) d W_t \quad \text{a.s.}, $$ where $p_t$ is the probability density function of $X_t$.

I would like to ask if there are references related in this direction. Thank you so much for your elaboration!

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    $\begingroup$ how about starting from considering the corresponding Fokker-Plank equation for the density eg. users.aalto.fi/~ssarkka/course_s2014/handout3.pdf? To at least see if we get a well-posed pde. So now in the FP one of the coefficients will involve the density again. $\endgroup$
    – Thomas Kojar
    Commented Feb 3, 2023 at 19:05
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    $\begingroup$ one good reference for that is "Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients" $\endgroup$
    – Thomas Kojar
    Commented Feb 3, 2023 at 19:32
  • $\begingroup$ The corresponding FP nonlinear parabolic pde is $$\partial_{t}p(x,t)=-\partial_{x}(b(x)p(x,t))+\frac{1}{2}\partial_{xx}\left((f (p(x,t)))^{2}p(x,t))\right).$$ and so for weak regularity on f we can only study this in distributional sense. $\endgroup$
    – Thomas Kojar
    Commented Feb 3, 2023 at 23:51
  • $\begingroup$ @ThomasKojar Thank you so much for your suggestion! I will check it out. $\endgroup$
    – Analyst
    Commented Feb 4, 2023 at 7:18

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