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7

You can condition on the value of $\lambda_t$. So $E[dN_t] = E[E[dN_t|\lambda_t]] = E[\lambda_t dt] = E[\lambda_t] dt$


7

We assume that the process $\{J_t, \, t\ge 0\}$ is defined at the jump times of the Poisson process $\{N_t, \, t \ge 0\}$, and all the jump sizes are independent and identically distributed. That is, \begin{align*} Q_t \equiv \int_0^t (J_t-1) dN_t = \sum_{n=1}^{N_t} (J_i-1), \end{align*} where $J_i$, for $i=1, \ldots, \infty$, are independent and $\xi_i = \...


7

Write $X_t = A_t B_t$ with $A_t = e^{(\lambda - \eta)t}$ and $B_t = \left(\frac{\eta}{\lambda} \right)^{N_t}$. Then $dX_t = A_t dB_t + B_t dA_t$ by the product rule of calculus. There are no second order terms since both $A_t$ and $B_t$ are finite variation (i.e. $\langle A_t, B_t\rangle$= 0). Next, $dA_t = (\lambda - \eta)A_t dt$, and $dB_t = B_t \cdot \...


4

I don't think you can have an explicit form. Let $Y_t= e^{at}X_t$ then : $$ Y_t -Y_0 =\sum_{i=1}^{N_t}e^{aT_i} $$ where $(T_i)_{i=1...N_t}$ are the jump times of your poisson process. then $$P(Y_t\leq x)=\sum_{n\geq 0}\frac{(mt)^n}{n!}e^{-mt}P(\sum_{i=1}^{N_t}e^{aT_i}\leq x|N_t=n)$$ $$P(\sum_{i=1}^{N_t}e^{aT_i}\leq x|N_t=n) =\int_{[0,+\infty]^n}\mathbf{1}...


4

Let $$Y_t = \int_0^t N_u du$$ where $(N_t)_{t \geq 0}$ figures a Poisson process with intensity $\lambda$. Using the stochastic Fubini theorem we have that: \begin{align} Y_T &= \int_0^T N_t dt \\ &= \int_0^T \int_0^t dN_u dt \\ &\color{lightgray}{= \int_0^T \int_0^T \mathbf{1}\{u \in [0,t]\} dN_u\ dt} \\ &\color{lightgray}{= \int_0^T \...


2

"One of the attractive features of the logistic function is the fact that it is bounded between 0 and 1, making it suitable to represent probabilities. " "The Poisson intensity model introduced in this article still has serious shortcomings despite the major advancement offered by its dynamic features. First, it is known to be unable to properly capture the ...


2

There are, as for random variables, different types of convergences for stochastic processes. Probably you mean convergence in the Skorokhod topology $J_1$. This is one convergence concept for $d$-dimensional cádlág processes. Convergence of stochastic processes $X_n \xrightarrow {\mathscr L} X $ in this sense holds if and only if the laws $\mathscr{L}(X^n)$...


1

If $X_t$ is a Poisson Counting Process with intensity $\lambda$ then the Martingale $M_t=X_t−\lambda t$ is called a Compensated Poisson Process. As $\lambda$ becomes large $M_t$ does converge to a Brownian motion with variance rate $\lambda$. This can be seen by using the "heavy arrivals" approximation of the Poisson Distribution: when the arrival rate is ...


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