Hello Mathematicians,
I've been attempting to solve a modified SDE (Langevin equation) in which the stochastic term (Wiener Process) is raised to a power: $$(1): \frac {dX_t}{dt} = -\frac {(X_t - \alpha)}{\tau} + (\beta \frac {dW}{dt} + \kappa)^2 .$$ Where $ \alpha, \tau, \beta, \kappa$ are real valued constants and $\frac {dW}{dt}$ is standard Wiener process with zero mean and unit variance.
If we "expand" the second term and use typical Ito calculus differential notation shorthand (ie $dW^2 = dt$), I can get a Langevin equation with a few additional terms : $$(2):\frac {dX_t}{dt} = -\frac {(X_t - \alpha)}{\tau} + (\beta^2 + \kappa^2) + 2\kappa\beta\frac {dW}{dt}.$$
--Question 1: Can the above simplification be carried out or is squaring of Wiener Process not defined under Ito calculus?
If (2) above is incorrect, I can integrate (1) via an integration factor. This will yield an integral term of the form: $\int e^\frac {t}{\tau}(\frac {dW}{dt})^2dt $. Replacing the $\frac {dW}{dt}$ as Gaussian white noise, this leads to integral of square of Gaussian white noise.
--Question 2: Is the above integral defined and can its expectation(mean) and variance be computed? I have seen posts where the Wiener process isn't raised to a power (ie $\int e^\frac {t}{\tau}\frac {dW}{dt}dt $) and the calculation is relatively straightforward. However, there are very few post where the noise term is squared and integrated (possibly due to ill posed definition in the first place).
I have attempted to simulate (1) the above by using a backward Euler and substituting $\frac {dW}{dt} = \frac {\mathscr N(0,1)}{\sqrt{\delta t}} $, where $\mathscr N(0,1)$ is standard normal rv and $\sqrt{\delta t}$ is discretion time step.
--Question 3: The simulation of (1) shows that the for a "good" selection of constants and time step, the solution converges to a finite "steady state" variance and mean. I am wondering what are the convergence criteria (if any that can be analytically derived)? The simulation of (2) agrees to (1) in stead state mean and variance for suitable chosen constants, however I am finding it hard to find any mathematical formation why this is the case.
Thanks in advance, hints or suggestions welcome!