Does anyone know the solution to the Ito integral with the scaling factor on $dW_t$ being $\frac{1}{w_t}$? In other words what is:

$\int \frac{dW_t}{W_t}$ ?

It looks dangerously close to what mathemattical finance people do when they look at geometric Brownian motion, but they start with the assumption that the price, $S_t$ follows an exponential trajectory of the form:

$S_t=S_0e^{((\mu -\frac{1}{2}\sigma^2)t+\sigma W_t)}$

so, when they take the natural log the $e^{(-)}$ falls away.

My brain may just be failing, but I think this does not help find $\int \frac{dW_t}{W_t}$.

In fact, I am curious if there is a 'standard' (cookbook) way of getting the integral of any polynomial Brownian motin, i.e., $\int (W_t)^kdW_t$, but $k=-1$ is fine for now...


On the particular case, $k=-1$, one shouldn't be surprised by this being problematic. Certainly, the hyperbola, $\frac{1}{x}$, is undefined at the origin. Therefore, it does not satisfy the usual conditions that guarantee the existence and uniqueness of a solution to the SDE, $\mathrm df(W_t) = \frac{1}{W_t}\mathrm dW_t,\ W_0=0$. This problem does not arise in finance since the stock price, $S$, cannot be zero.

On the cases of $k=0,1,2,\ldots$, a useful first step is the use of Ito's lemma to justify:

$$ \int\limits_{0}^{t} W_s^{k}\ \mathrm dW_s = \frac{1}{k+1}\left(W_t^{k+1} {}+{} {\bf 1}_{k>0}(k)\ {k+1\choose 2}\int\limits_{0}^{t}W_s^{k-1}\ \mathrm ds\right), $$

where the r.h.s. now casts the problem in terms of Riemann integrals. While, in general, determining the distributions of $\ ``\int\limits_{0}^{t}W_s^{k-1}\ \mathrm ds"$ can be involved, less ambitious goals, such as computing $\mathbb E[\int\limits_{0}^{t}W_s^{k}\ \mathrm dW_s]$, can be accomplished using this alternate form.

  • $\begingroup$ Yes, mathematically I agree on the existence and uniqueness. But my application is finance and, as you note, $S_0=0$ is non-sensical in that context. Having worked in research physics we often did not have a closed-form solution to the 'correct' PDEs, but having a solution in the neighborhood can make a problem more tractable. Given that, any idea what the solution to $\int \frac{dW_t}{W_t}$ is? $\endgroup$ – foobar Mar 7 '15 at 22:37
  • $\begingroup$ @foobar, What is "a solution in the neighbourhood of a random variable"? $\endgroup$ – ki3i Mar 7 '15 at 22:47
  • $\begingroup$ I knew you would ask...it is not a 'neighborhood' in the sense of topology. In the work I did we needed to model coupling and transmission of oscillations of a (superconducting) cavity. The geometry made simply applying Maxwell's equations too hard owing to boundary complexity. Numeric PDE could be used, but even that was a chore. In the end you solve for a simpler geometry based on idealizing and then perturbations that should have small effect, so you can then look empirically for what you want to find. If your cavity is a slight ellipsoid, solve for a sphere and work from there. $\endgroup$ – foobar Mar 7 '15 at 23:58
  • $\begingroup$ @foobar, Thanks; I think I see where you are coming from, but not where you are going. Two points: 1) The usefulness of existence and uniqueness criteria is in providing confidence that seeking SDE solutions is not a futile task. I am not convinced that $\{\int_{0}^{t}\frac{dW_s}{W_s}\}_{0\leqslant t\leqslant T}$ is well-defined. 2) Assuming it exists, can you formally specify (i.e. in terms of a stochastic process/random variables) what you would consider to be an approximate solution to a random process? $\endgroup$ – ki3i Mar 8 '15 at 1:27
  • $\begingroup$ My interest is in exploring the dynamics of a process where the notion of the process "running away" from the deterministic PDE does not occur. In geometric Brownian motion the naive solution from discrete modeling always becomes something like $e^{(\alpha + \beta)t}$ when the deterministic PDE that looks 'the same' has solution $e^{\alpha t}$. The 'Ito correction' takes out the $\beta$, as it were. I am curious about a process where it has different dynamics. In finance terms, a process where the realization under risk-neutral measure is above the deterministical (or modeled) path. $\endgroup$ – foobar Mar 8 '15 at 10:45

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