0
votes
1answer
31 views

Characteristic function of a stochastic process with stationary and independent increments

Let $(X_t)_{t\geq 0}$ be a stochastic process with independent and stationary increments. I have to show that $E[e^{itX_1}]=\phi^n(t)$ Since increments are independent, I can write ...
2
votes
0answers
51 views

Characteristic function of compound Poisson process

It is widely known that the characteristic function of a compound Poisson process is $$ \phi_X(u) = \exp \left(t\lambda \int_{\mathbb{R}} (e^{iux}-1) F(dx) \right). $$ But if I try to derive it via ...
3
votes
1answer
86 views

Characteristic function under risk neutral measure

I am trying to derive a characteristic function (in Levy-Khintchine form) of a compound Poisson process $X_T$ under a risk neutral measure $\mathbb{Q}$, using the Esscher transfrom to change the ...
0
votes
0answers
16 views

Charateristic function evaluation

I have a signal given by the following equation: $y_k = X_k S_0 + \sum_{l=0 \& l\neq k}^{N-1}S_{l-k}+n_k$ where $X_k$ are independent and identically distributed random variables. $n_k$ is a ...
0
votes
1answer
109 views

Using characteristic function to deduce convergence of Bernoulli random variables

Let $Y_1, Y_2,...$ be a sequence of independent Bernoulli(0.5) random variables and $X_n = \sum_{i=1}^{n} Y_i 2^{-i}$ I need to use the characteristic function to deduce that $X_n$ converges in ...