I would like a general framework for the calibration of the unknown parameters in an arbitrary stochastic differential equation. I have a proposed method that seems reasonable in theory, but is problematic in practice. Can you help?

The method

Suppose we have the SDE $dX=\mu(t,X)dt+\sigma(t,X)dW$, where $W(t)$ is a Wiener process. And suppose we know a time series realization ("historical data") $X(t_n)=X_n$ for $n=0,\ldots,N$. And further suppose that the functions $\mu$ and $\sigma$ contain unknown real parameters.

The idea is to convert the SDE into a finite difference equation (which holds approximately): $$X_{n+1}-X_{n} = \mu(t, X_n) (t_{n+1}-t_n)+ \sigma(t, X_n)\sqrt{t_{n+1}-t_n} Z_n.$$

Here: $n=0,\ldots,N-1$ and $Z_n$ are independent random variables, each with a standard normal distribution. Here, I used the heuristic fact that $dW$ is normal with mean 0 and variance $dt$.

Solving for $Z_n$ gives $Z_n$ as a function of the unknown parameters: $$Z_n =\frac{X_{n+1} - X_{n} - \mu(t, X_n)(t_{n+1}-t_n)}{\sigma(t, X_n) \sqrt{t_{n+1} - t_n}}.$$

The liklihood of the time series is given by $$\prod_{n=0}^{N-1} \phi(Z_n) = \left( \frac{1}{\sqrt{2 \pi}} \right)^N \exp \left( -\frac{1}{2} \sum Z_{n}^{2} \right).$$ Here, $\phi$ is the probability density of the standard normal. Maximizing the liklihood requires minimization of $\sum Z_n^2$. And this gives the maximum liklihood estimators.

The problem

Consider the SDE $dX = \sigma dW$, with unknown $\sigma$. Given historical data, and following the above method, we get $$\sum Z_n^2 = \frac{1}{\sigma} \sum \frac{X_{n+1} - X_{n}}{\sqrt{t_{n+1} - t_n}}.$$

However, this does not have a minimum wrt $\sigma$ (except for perhaps $\sigma = \infty$, which does not make sense).

  • 2
    $\begingroup$ I think you should use the backward pricing PDE of $V(t,X_t)$ where $V(t,X_t)$ is the price of a claim at time $t$ with current price $X_t$. $\endgroup$ – user16651 Jan 4 '17 at 15:46
  • $\begingroup$ Since you've transformed the problem into one of estimating parameters for a discrete model (as is the most common procedure), why don't you just use the Markov property and maximize $L(\Theta|X) = \prod^{T}_{t=1}p(x_t|x_{t-1}, \Theta)$? A complex reference on the subject could be arxiv.org/pdf/1408.2441v1.pdf. I might be totally off, but that is how I would do it $\endgroup$ – Tingiskhan Jan 4 '17 at 19:33
  • $\begingroup$ Or if you feel a bit more Bayesian, you could use a sequential algorithm like IBIS: academic.oup.com/biomet/article-abstract/89/3/539/251804/… $\endgroup$ – Tingiskhan Jan 4 '17 at 19:45

From the SDE $dX_t=\sigma dW_t$, \begin{align*} X_{n+1} - X_n =\sigma \sqrt{t_{n+1}-t_n} Z_n. \end{align*} That is, \begin{align*} X_{n+1}\mid X_n \sim N\left(X_n, \, \left(\sigma \sqrt{t_{n+1}-t_n}\,\right)^2 \right), \end{align*} and the conditional density function is given by \begin{align*} \phi(x\mid X_n) = \frac{1}{\sqrt{2\pi}\sigma \sqrt{t_{n+1}-t_n}}e^{-\frac{(x-X_n)^2}{2 \sigma^2 (t_{n+1}-t_n)}}. \end{align*} Then, the conditional log-likelihood function is given by \begin{align*} L(X_1, \ldots, X_{n+1}) &= \sum_{i=1}^n\ln \Big(\phi(X_{i+1}\mid X_i)\Big)\\ &=-\frac{n}{2}\ln(2\pi) -\frac{n}{2}\ln(t_{i+1}-t_i)-n\ln\sigma - \sum_{i=1}^n\frac{(X_{i+1}-X_i)^2}{2 \sigma^2 (t_{n+1}-t_n)}. \end{align*} Now, you can minimize the function to obtain the $\sigma$ estimation.


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