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## The authoritative Robert Merton’s Continuous-Time Finance: sample paths of diffusion processes

Sample paths of diffusion processes have been the mainstay of mathematical finance since Louis Bachelier’s 1900 doctoral dissertation The Theory of Speculation.  This is clearly described in Chapter 3 of Robert Merton’s book from Paul Samuelson’s MIT school (merton-cont-finance).  There have been variations from the brownian motion model, often used for returns.  Stochastic calculus for brownian motion $X_t$ is extremely well-developed and fractional Brownian motions have been used as continuous time models as well $B^H_t$ which are defined as

$B^H_t = \frac{1}{\Gamma(H)}\int_0^t (t-s)^{-H} dX_t$

have been used since Mandelbrot’s work in the late 1960s.  Both of these have Ito formulae, that for $f\in C^2(\mathbf{R})$ we have

$f(B^H_t) - f(B^H_0) = \int_0^t f(B^H_s) dB^H_s + \frac{1}{2} \int_0^t f''(X_s) \phi(s) ds@ with$latex \phi(s)=s^{H-1/2}\$ which allows elegant calculations for these processes. This authoritative century-long tradition in continuous-time finance is applied usually for returns but there are many models that model volatility also by these diffusion models.

While returns are often considered the major observable for finance, volatility is the most significant object of empirical scientific study in finance if one takes the purist view that there should be an exact science of finance and that science should focus on the modeling of global financial volatility which may be helpful for applications to portfolio analytics from the inverstors’ point of view, in fact global financial volatility is an object of scientific study of independent interest because volatility of global financial markets have an impact on all human beings in the global economy and not only the investors.

We should therefore give careful scrutiny to whether these diffusion models are appropriate scientifically on empirical data or whether other types of models, such as hydrodynamic models which are superficially similar to diffusions for formally while a diffusion on $\mathbf{R}^d$ is modeled by the equation

$(\partial_t + \Delta)u(t,x) = 0$

the hydrodynamic (for Navier-Stokes) and more generally nonlinear reaction-diffusion models are modeled by

$(\partial_t + \Delta)u(t,x) = Fu(t,x)$

for some nonlinear operator $F$.  This is not merely a formal question for the latter types of models have a feature for some class of $F$ which are impossible for the former, which is the phase transition to turbulent dynamics.  Given that turbulent cascades in foreign exchange markets have been recorded in reputable science journals such as Nature since 1996, we believe it is valuable to seek fundamental models that examine which of these models are better fit to data.  To the extent that the latter types (turbulence-capable ones) fit data better we should consider this a question of paramount importance since this would provide quantitative understanding of sources of turbulence in global financial volatility, which may simply be quantitative.