ヴァリアンス・ガマ・プロセスの迷宮

最近確率微分計算式の一種を理解しようと思っています。
The basic formula is an adaptation of Brownian motion, setting the drift to be a gamma process. 最初の挑戦はリヴェイプロセスの定義です。 A Levy process is a stochastic process with independent, stationary increments.
The VG (Variance-Gamma) process for random variable X
X(t;sigma,v,theta) = theta*GAMMA(t;1,v)+sigma*W(GAMMA(t;1,v)). Here, W is Brownian motion with drift defined by the gamma process. The theta term indicates drift theta*t subjected to a random time change. However, the GAMMA function and alternative expressions for X are unclear.
Carr, Madan, and Chang use the following formula for a stock price at time t.
St = S0exp(mt+X(t;sigma,v,theta)+w(t))
w(t) = (1/v)ln(1-theta*v-(sigma^2)*v/2)
How to determine v and theta is unclear, but the X term is the VG process above, it seems.
Hull poses the problem of describing the convergence of the model in the limit as v goes to 0.
Strangely, the expectation of a VG variable X is theta*t. This is difficult to decipher, as the density function is an integral of g (gamma process) and the gamma function (gamma(x) = (x-1)!).
Actually, this was verifiable, as the VG variable is described in the appendix in the Madan et al paper to be X=theta*g+sigma*z, where z is Brownian motion and g is the Gamma process time increment. As the mean of the Gamma distribution is set to 1 and the interval is t, E(g)=t and E(z)=0, making E(X)=theta*t. Thus, the X and w*t terms in the St expression cancel, yielding St = S0exp(r*t), which is Geometric Brownian Motion.
The version of the Gamma Distribution with mean mu*h used in this paper does not seem to agree with the Wikipedia Gamma Distribution definition for mean theta*k.
Theta above was clarified as the drift of Gamma time increment.
この数式と確率分配はんなぜオプションに快適かまだ不明です。
ちょっと分かりにくいことはヴァリアンス・ガマの変動性に関する文です。The article states firstly that Brownian Motion has finite squared variation but infinite variation. This makes little sense, as the square of the difference with the mean should exceed the difference if degree one. Secondly, the authors state that the log price changes must be squared before they are summed. This makes no sense, unless it is a reference to the variance calculation for the the log of the price change.
What is meant by the "stochastic time step" in the variance gamma option pricing formula is unclear. It would seem that it means that the drift (expected value), rather than being a constant, is a stochastic process.
The expressions for the pdf and characteristic function of a variance gamma process are also unclear. The pdf starts with a normal distribution for the return and sets the time step to a constant g. Then the integral with respect to g of the normal pdf times the (gamma distributed) probability of this g is taken to solve for the pdf of the VG process. More confusing is the characteristic function of the VG process. The normal characteristic function is exp(i*mu*g-.5*sigma^2*g^2). However, how to translate from this to the VG characteristic function if phi(u)=(1/(1-i*theta*v*u+(sigma^2v/2)u^2)^t/v is unclear.
Several formulas for the Levy measure are then presented as well as assertions about the arrival rate of jumps. It is unclear where the arrival rate of jumps is encoded or why it is infinite. For a Poisson process, the arrival rate is lambda and the probability of k events is p= exp(-lambda)lambda^k/k!
The paper contradicts Wikipedia as, for the gamma process, it indicates a Levy measure which is finite but, citing the fact that the integral is infinite, indicates that the jump arrival rate is infinite. Wikipedia claims that the Levy measure, and not its integral, represents the jump arrival rate.