Consider a GBM model with Heston volatility: $$dS_t = \mu S_t dt + \sqrt{V_t} S_t dB_t^1$$ $$dV_t = \kappa(\theta-V_t)dt+\xi \sqrt{V_t}dB_t^2,$$ where $(B_t^1, B_t^2)$ is a correlated BM. Let $$\mathscr{F}_t^S=\sigma(S_u : u\leq t),$$ and consider the filter estimate of the variance $V$ $$\hat{V}_t = \mathbb{E}(V_t | \mathscr{F}_t^S).$$

Question: Can we derive a filtering SDE for $\hat{V}_t$ analogous to the one in the Fujisaki-Kallianpur-Kunita theorem? Or is there a transformation of the above system SDE into the cases allowed in their theorem (stated below)?

To recall as stated in Theorem 8.11 of Rogers and Williams, volume 2, chapter VI.8 (in one dimension): Fujisaki, Kallianpur, and Kunita proved that in the case of an observation process $$dY_t = h_t dt + dW_t,$$ with hidden signal $X$, if we write $f_t \equiv f(X_t)$ and $\hat{f}_t \equiv \mathbb{E}(f(X_t)| \mathscr{Y}_t)$, then $\hat{f}_t$ satisfies the SDE $$d\hat{f}_t = \widehat{\mathscr{G}f_t}dt+(\widehat{f_t h_t}-\hat{f}_t\hat{h}_t+\hat{\alpha}_t)dN_t,$$ where $N$ is a $\mathscr{Y}_t$-Brownian motion (specifically $N_t=Y_t-\int_0^t \hat{h}_sds$.) Here $\alpha$ is given by the density with respect to the Lebesgue measure of $[M, W]$ where $$M_t = f(X_t)-f(X_0)-\int_0^t \mathscr{G}f_s ds,$$ is a $\mathscr{F}_t$-martingale ($\mathscr{F}_t$ being the filtration associated with $W$). We may think of the signal $X$ as solving an SDE and having infinitesimal generator $\mathscr{G}$, though the setup is stated more generally than this in the book and original paper. I have purposely left out some regularity conditions required, for brevity but if anyone is interested/concerned I can edit them in later.

I know there are other filtering approaches, like applying the Euler-Maruyama discretization and then some Bayesian analysis or particle filters, but I am interested in generalizations of this continuous-time/SDE approach to filtering.


1 Answer 1


I believe I have come across one possible method for deriving a filtering SDE but the practicality becomes questionable. The limitations of such filtering SDEs is noted in Rogers and Williams, and it seems this exact limitation is pretty severe outside of the well-known cases like the Kalman-Bucy filter. Indeed, depending on the drift $h$ of the observation process and the infinitesimal generator $\mathscr{G}$, "the integrands in the SDE for $f$ [sic] involve estimates of other functions of the signal process, and only in special cases can we expect to do anything about these", p.331, Volume 2.

If we follow the ideas in the paper by Aihara and Bagchi, we can see exactly the difficulty in the present case. First, we let $Y_t = \log S_t/S_0$. Clearly, by Ito's, $$dY_t = (\mu-\frac12 V_t)dt + \sqrt{V_t}dB_t^1.$$ Now, a random time change changes the standard GBM-Heston SDE system to $$d\tilde{Y}_t = \left(\frac{\mu}{\tilde{V}_t}-\frac12 \right)dt + dB_t^1,$$ $$d \tilde{V}_t = \kappa\left(\frac{\theta}{\tilde{V}_t}-1\right)dt+\xi dB_t^2.$$

From here on, for notational convenience, we shall drop the tildes and just write $V$ for $\tilde{V}$ but be sure, we are dealing with the time-changed system. We are essentially in the setting of the Fujisaki-Kallianpur-Kunita theorem, except that the observation's driving noise is correlated to the signal's noise. Feeling reckless, we will apply the theorem with the natural changes. Here $$h_t = \left(\frac{\mu}{v}-\frac12 \right),$$ and $$\mathscr{G}f(v) = \kappa\left(\frac{\theta}{v}-1\right)f'(v)+\frac12 \xi^2 f''(v).$$

Computing $\widehat{\mathscr{G} f}$, $\widehat{fh}$, $\hat{f}\hat{h}$, and $\alpha_t f'(V_t) dt = \rho \xi dt$ for $f(v)=v$ and simplifying the above expressions, we get the filtering SDE $$d\hat{V}_t = \kappa (\theta \widehat{V^{-1}}_t-1)dt+(\mu(1-\hat{V}_t\widehat{V^{-1}}_t)+\rho \xi)dN_t,$$ where $N$ is a $\mathscr{Y}_t$-Brownian motion.

This is as far as I could proceed. One can similarly derive an SDE for $\widehat{V^{-1}}_t$ but it, in turn, depends on higher moments of $1/V_t$, so an infinite amount of SDEs are required to be solved. Perhaps undoing the time-change would be fruitful but I have not worked it out myself yet and I do not see any other one that could simultaneously eliminate the nuisance $\widehat{V^{-1}}$ from both the drift and diffusion parts of the SDE.

In summary: (if my work here is correct, then) a filtering SDE for (Heston) volatility does exist, but is not really usable. The authors in the mentioned paper here use the Zakai equation and provide an approximation scheme, and the above is probably one reason why!

Please comment if you have any corrections or clarifications.


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