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The Gyongy's theorem:

Let $X_t$ be a stochastic process satisfying $$dX_t = \mu_t dt+\sigma_tdW_t$$ where $\mu_t, \sigma_t$ are bounded stochastic process adapted to the filtration $\mathcal{F}_t$. Then there exists a stochastic differential equation $$dY_t = b(t,Y_t) +s(t,Y_t)dW_t$$ such that $Y_t$ and $X_t$ have the same probability distribution for every $t$. In addition, the two functions $b(t,y)$ and $s(t,y)$ satisfy $$b(t,y) =\mathbb{E}(\mu_t|X_t=y)$$ $$s^2(t,y) = \mathbb{E}(\sigma_t^2|X_t=y)$$

This beautiful theorem gives us a method to construct an equivalent local volatility model (i.e. the model of the process $Y_t$) from a given stochastic volatility model (i.e. the model of the process $X_t$).

I would like to know whether there exists a method to construct an equivalent stochastic volatility model from a given local volatility model. In other words, given the process $Y_t$, can we construct the process $X_t$ such that $X_t$ and $Y_t$ have the same probability distribution for every $t$?

If there are LVMs that can not be generated from any SVM, I would like to know their characteristics?

PS: we may have $(\mu_t, \sigma_t) = (b(t,X_t), s(t,X_t))$, but this result is too trivial. Is there a way to contruct $(\mu_t, \sigma_t)$ such that they follow some stochastic differential equations that are not directly related to $X_t$? (for example, $\sigma_t$ follows a CIR model)

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    $\begingroup$ Great question, have often wondered this myself. The answer I suspect is no, but would love to see somebody show / prove this. $\endgroup$
    – Frido
    Jun 15 at 22:36
  • $\begingroup$ Why is what you describe in PS too trivial? What you are trying to construct instead boils down to matching the distribution of $s(t,X_t)$ with an SDE where another $\sigma_t$ and $X_t$ are only correlated and not related by a possibly nonlinear function. Unlikely in my opinion. There is also a lot of literature about the different smile dynamics of local volatility vs. stochastic vol models. These issues would not exist if your proposal worked. $\endgroup$
    – Kurt G.
    Jun 16 at 10:01
  • $\begingroup$ @KurtG. If what you said is true, the class of LVM must be larger than the class of SVM. We can split the LVM class into two sub-classes: the first sub-class contains LVM that are generated by some SVM (via the Gyongy's theorem), the second sub-class contains the remaing LVM (that cannot be constructed from any SVM). $$$$ In this case, my question is "what are the characteristics of these two sub-classes of LVM"? In other words, given a LVM, how do we know whether it is generated by a SVM? $\endgroup$
    – NN2
    Jun 18 at 6:21
  • $\begingroup$ I said mainly the same what you said yourself: for the reverse LVM$\to$SVM the solution we are seeking is $\sigma_t=s(t,X_t)\,.$ It is trivial and imho not too trivial to be true. $\endgroup$
    – Kurt G.
    Jun 18 at 12:34
  • $\begingroup$ @KurtG. no, what I’m seeking is $\sigma_t = \sigma(t, B_t,…)$ which is not directy a function of $X_t$ (whether the solution $\sigma_t=s(t,X_t)$ which depends directly on $X_t$ is trivial or not, it is not important here and so not in the question). $$$$ Besides, I modified the question by asking whether there exists any conditions such that the reverse LVM to SVM does ( or does not) have solution. $\endgroup$
    – NN2
    Jun 18 at 13:03

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