Gamma 過程模擬（增量分佈）

gamma 過程是 Levy 過程 $ X $ ， 在哪裡 $ X_t $ 具有帶參數的伽馬分佈 $ at,b>0 $ 和密度 $$ f\left(x\right)=\frac{b^{at}}{\Gamma\left(at\right)}x^{at-1}e^{-bx} $$

我想通過增量來模擬伽馬過程，但是分佈是什麼 $ X_t - X_s $ ? 當然是伽馬，但有什麼參數？

Lévy 過程定義為（Lévy 過程和隨機微積分，David Applebaum）：

假設給定一個機率空間 $ (\Omega, \mathcal{F}, P) $ . 一個 Lévy 過程 $ X = (X (t), t \geq 0) $ 取值 $ \mathbb{R}^d $ is essentially a stochastic process having stationary and independent increments; we always assume that $ X (0) = 0 $ with probability 1. So:

• each $ X (t) : \Omega \to \mathbb{R}^d $ ;
• given any selection of distinct time-points $ 0 \leq t_1 < t_2 < \ldots < t_n $ , the random vectors $ X(t_1), X(t_2) − X(t_1), X(t_3) − X(t_2), \ldots, X (t_n) − X(t_{n−1}) $ are all independent;
• given any two distinct times $ 0 \leq s < t < \infty $ , the probability distribution of $ X(t) − X(s) $ coincides with that of $ X(t − s) $ .

The Gamma distribution is scale invariant under summation, i.e. $$ \sum_{i=1}^N X_i = \mathrm{Gamma}\left(\sum_{i=1}^N k_i, \theta\right) $$ so thanks to the third property $ X_t - X_s $ has parameters $ a (t - s), b $ .

引用自：https://quant.stackexchange.com/questions/61946