# Rewriting Diffusion Processes: Combining Independent Wiener Processes

In stochastic calculus, a rule of thumb for computations is $(dW_t)^2 = dt$ for a Wiener process $W_t$.

Say we have a diffusion process $dX_t = dW^1_t + X_t dW^2_t$, with $W_t^1, W_t^2$ independent (of each other) 1-dimensional Wiener processes. Under what conditions, using the above rule of thumb, can this computation be made rigorous:

$(dX_t)^2 = (dW^1_t + X_t dW^2_t)^2 = (1+X_t^2)dt \implies dX_t = \sqrt{1+X_t^2}dW_t^3$ for some Wiener process $W_t^3$.

$$W_t^3 := \int_0^t \frac{1}{\sqrt{1+X_s^2}} \, dW_s^1 + \int_0^t \frac{X_s}{\sqrt{1+X_s^2}} \, dW_s^2$$