I was just re-reading parts of
Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E
by the great Benoit Mandelbrot where he says
2.3. Specialists' trades
Section VI.B of M 1963b{EI4} pointed out that the discontinuities of the
process Z(t) were unlikely to be observed by examining transaction data.
They are either hidden within periods when the market is closed or the
trading interrupted, or smoothed away by specialists who, in accordance
with S.E.C. instructions, "ensure the continuity of the market" by per-
forming transactions in which they are party.
It is tempting to postulate that virtual transactions and the specialists'
transactions are identical, though the latter presumably see where the
prices are aimed and can achieve the desired llZ in less than U inde-
pendent Gaussian steps. Thus, the Bochner representation is plausible and
suggests a program of empirical research of the role of Specialists.
The method of filters. The distribution of price changes between trans-
actions has a direct bearing upon the "method of filters," discussed in
Section VI.C of M 1963b{EI4}.
which led me to dig deeper where behold this is right up my alley as its a good candidate for the next thing to implement now that i got the damned Fourier transform implemented
------====================================================================--------
Bochner's representation theorem is a fundamental result in the theory of characteristic functions, which are used to study random variables and stochastic processes. An L-stable process is a particular type of stochastic process that has a specific form of the characteristic function, exhibiting stable distributions. These processes are often used to model heavy-tailed phenomena in various fields like finance, physics, and engineering.
Bochner's representation theorem states that for any continuous positive definite function, there exists a unique Borel measure such that the function can be represented as the Fourier transform of that measure. In the context of an L-stable process, the characteristic function is given by:
Φ(t) = E[exp(i * t * X)] = exp(−c|t|^α * (1 + i * β * sign(t) * tan(πα/2)))
Here, the parameters are as follows:
Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E
by the great Benoit Mandelbrot where he says
2.3. Specialists' trades
Section VI.B of M 1963b{EI4} pointed out that the discontinuities of the
process Z(t) were unlikely to be observed by examining transaction data.
They are either hidden within periods when the market is closed or the
trading interrupted, or smoothed away by specialists who, in accordance
with S.E.C. instructions, "ensure the continuity of the market" by per-
forming transactions in which they are party.
It is tempting to postulate that virtual transactions and the specialists'
transactions are identical, though the latter presumably see where the
prices are aimed and can achieve the desired llZ in less than U inde-
pendent Gaussian steps. Thus, the Bochner representation is plausible and
suggests a program of empirical research of the role of Specialists.
The method of filters. The distribution of price changes between trans-
actions has a direct bearing upon the "method of filters," discussed in
Section VI.C of M 1963b{EI4}.
which led me to dig deeper where behold this is right up my alley as its a good candidate for the next thing to implement now that i got the damned Fourier transform implemented
------====================================================================--------
Bochner's representation theorem is a fundamental result in the theory of characteristic functions, which are used to study random variables and stochastic processes. An L-stable process is a particular type of stochastic process that has a specific form of the characteristic function, exhibiting stable distributions. These processes are often used to model heavy-tailed phenomena in various fields like finance, physics, and engineering.
Bochner's representation theorem states that for any continuous positive definite function, there exists a unique Borel measure such that the function can be represented as the Fourier transform of that measure. In the context of an L-stable process, the characteristic function is given by:
Φ(t) = E[exp(i * t * X)] = exp(−c|t|^α * (1 + i * β * sign(t) * tan(πα/2)))
Here, the parameters are as follows:
- t is a scalar argument representing time.
- X is the random variable associated with the L-stable process.
- α is the stability parameter (0 < α ≤ 2), which characterizes the tail behavior of the distribution. When α = 2, the process reduces to a Gaussian process.
- β is the skewness parameter (-1 ≤ β ≤ 1), which measures the asymmetry of the distribution.
- c is a scale parameter (c > 0), which influences the overall scale of the distribution.
- E denotes the expectation operator, and exp(x) is the exponential function e^x.
- i is the imaginary unit (√-1), and sign(t) is the sign function.
