I'm trying to prove the uniqueness and existence of the solution for the following Stochastic Wave Equation in one-spatial dimension as follows:
$ \frac{d^{2}u}{dt^{2}} - \frac{d^{2}u}{dt^{2}} = \sigma(u(t,x))\dot{W}(t,x),\ t \in [0,T]$
(where $\sigma$ satisfies the Lipschitz-condition and $\dot{W}$ is space-time white noise, i.e gaussian martingale measure whose covariance is Dirac delta),
with vanishing initial conditions, which means all the boundary terms are simply equal to zero.
The usual way of the proof is: First we define some suitable Banachspace $M$ with some map $\Phi: M \rightarrow M$ and Picard iteration step, then show the $L^{2}$-convergence via Gronwall inequality and somehow it turns out at the end that the map $\Phi$ has a fixed point. (by Banach theorem probabily..)
This is all I know vaguely so far, and since I have dealt with only SDE or stochastic heat eqn previously, I am struggling with how to define necessary terms precisely. Especially, it is also still unclear where to use which lemma, for example: it is said that Burkholder ineq. or Cauchy-Schwarz ineq. should somehow be involved in the proof..
Can somebody help me to find out how? I would be grateful if there is an answer at least roughly how I should start. ( for instance: is there some reference/similar proof which I can imitate such that the right solution comes almost automatically into my mind?)