Hey in my script is the following stated and i wanted to understand why this is true.
Let $T \sim Exp(\lambda)$ be an exponential random variable on a probability space $(\Omega,\mathcal{A},\mathbb{P})$. Define the process $$ X_t(\omega) = (t-T(\omega))\mathbb{1}_{\{t > T(\omega) \}} $$
and let $(\mathcal{F}_t)_{t \geq 0} $ be the natural filtration. Then its stated that this process fullfills the simple markov property and has transition function: $$P_t(0,A)= e^{-\lambda t }\mathbb{1}_A (0) + \int_0^t \mathbb{1}_A(t-s) \lambda e^{-\lambda s} ds \\ P_t(x,A)=\mathbb{1}_A(x+t).$$ I am having a hard time to verify this statements. In my script i have that a markov process is said to have a transition fucntion, if $\mathbb{P}(X_{t+h} \in A \vert X_t ) = P_h(X_t,A)$. I interprete this as follows: $$ P_h(x,A)=\mathbb{P}(X_{t+h} \in A \vert X_t =x ) $$ with this i started to calculate the transition function: $$P_h(0, (-\infty,y])=\mathbb{P} (X_{t+h} \in (-\infty,y] \vert X_t=0)= \mathbb{P} (X_{t+h} \leq y \vert X_t)= \mathbb{P}((t+h-T) \mathbb{1}_{\{t+h>T\}} \leq y \vert X_t=0 )= \mathbb{P}((t+h-T) \mathbb{1}_{\{t+h>T\}} \leq y \vert t<T )=(1)$$ Now i wanted to use the fact that we condition on the fact that $\{t<T\}$, but i dont see how to continue. Even if i write $$(1)= \frac{ \mathbb{P}((t+h-T) \mathbb{1}_{\{t+h>T\}} \leq y , t<T )}{\mathbb{P}(T>t)} $$ i dont see how i can calculate the numerator.
For some $x \neq 0 ( i.e. x>0 )$ i have: $$ P_h(x,(-\infty,y])=\mathbb{P}(X_{t+h} \leq y \vert X_t=x) = \mathbb{P}((t+h-T) \mathbb{1}_{\{t+h>T\}} \leq y \vert X_t=x )= \mathbb{P}((t+h-T) \mathbb{1}_{\{t+h>T\}} \leq y \vert t-T=x>0 )= \mathbb{P}((x+h) \mathbb{1}_{\{x+h>0\}} \leq y \vert t-T=x>0 )= (2) $$ since $x>0$ and $h \geq 0$ we get rid of the indicator and have $$ (2)= \mathbb{P}((x+h)\leq y \vert t-T=x>0 )= \frac{\mathbb{P}((x+h)\leq y , t-T=x>0 )}{\mathbb{P}((t-T>0 )}= \mathbb{P}(x+h \leq y) \frac{\mathbb{P}((t-T>0 )}{\mathbb{P}((t-T>0 )} = \mathbb{1}_{\{x+h \leq y\} } $$ Hence for $x \neq 0 $ i get $ P_t(x,A)=\mathbb{1}_A(x+t)$.
Maybe someone can help me with the other statement, or is there a other usefull property how to calculate the transition functions? Thanks.