Stochastic differential equations

[Chenmath]

Hello to all, I am a new member, but I've been reading and getting help from this forum for a year!
I recently started to study about stochastic calculus because I am considering risk management/ actuarial/ finance job.
I would appreciate your help.

If we have Poisson $(\lambda)$ and $W$ Standard Brownian motion
How can I solve these?

a) $dX_t=-\lambda X_t dt+X_{t-}dN_t$, $t\geq 0,X_0=1$.

b)$dX_t=X_{t-}\cdot(-\lambda dt+dW_t+dN_t)$, $t\geq 0,X_0=1$.

I think the first one could be solved using Girsanov transformation but I am not sure if this will work. As for the second one I am completely lost.

Thank you

 

[Valkarie]

for your help!To solve the first equation, we can use the solution formula for a stochastic differential equation with Poisson arrivals. The solution is given by:$X_t = X_0 \exp(-\lambda t)+\int_0^t X_{s-} dN_s$.To solve the second equation, we can use the Ito's lemma. The solution is given by:$X_t = X_0 \exp(-\lambda t + \frac{1}{2}\int_0^t dW_s^2) + \int_0^t X_{s-}dW_s$.