| Abstract: |
| We consider a semistochastic continuous-time continuous-state space
random process $\{X(t)\}_{t\geq 0}$ that models
deterministic growths (governed by an autonomous ODE) that is subject
to (downward) jumps due to random severity events occurring at random
times. The times of occurrence of the disturbances
are modeled by a Poisson process whose rate $\Lambda$ is allowed to
depend on the value of $X(t)$. At each time $t$ a random event occurs
the value $X(t^-)$ is multiplied by a continuous random variable
(``severity``) supported on $[0,1]$, which gives the value of $X(t)$.
An important example of such a process is the dynamics of the carbon
content of a forest whose deterministic growth is interrupted by natural
disasters (fires, droughts, insect outbreaks, etc.).
The time-dependent distribution of the process
$\{X(t)\}_{t\geq 0}$ satisfies an integro-differential PDE.
We derive explicit expressions for the stationary distribution of the
random process, and we develop a method for giving an upper bound of
the rate at which the distribution of $X(t)$ approaches the stationary
distribution. This is a collaboration with James Broda and Nikola Petrov. |
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