Differential form of "random walk with reset" based on Wiener process Assume such a "random walk with reset" X(t) is defined based on Wiener process (GBM)
X(t + dt) =  
   0, by probability lambda * dt; or, 
   X(t) + dW, by probability (1 - lambda * dt)

, where lambda is a constant, and
dW = W(t+dt) - W(t), 

while W is a Wiener process (GBM).
What is the differential form for X? 
I'd like to have something like
dX = ( 1 - lambda * dt) * dW + lambda * dt * ....

so that I could use this with Ito's lemma to do some calculus.
 A: Ok, let me expand my comment into an answer. Let $W_t$ be either a geometric Brownian motion, or a Brownian motion, or any Ito process - it does not matter for our case, we only will use the continuity properties of the process. 
The idea is to use the Poisson process in order to describe jumps, namely let $N_t$ be the Poisson process with a given intensity $\lambda$, so the probability of jump for $N_t$ in the interval $[t,t+dt]$ is roughly $\lambda\cdot dt$. 
The question now is: the jumps of a Poisson process is always equal to $1$ whenever it happens, so how to use it for your problem? The good thing about a Poisson process it that it has piecewise-constant paths, which are of finite variatino with probability one so the integral is easy to define: for any function $f:\mathbb R_{+}\to\mathbb R$ which has limits from the left
$$
\int\limits_0^t f(t)dN_t = \sum\limits_{s\leq t}f(s-)\Delta N_s \tag{1}
$$
where $f(s -) = \lim\limits_{r\uparrow s}f(r)$ and $\Delta N_s = N_s - N_{s-}$. Note that a Poisson process has finitely many jumps in each interval $[0,t]$ with probability $1$, so the summation in $(1)$ is finite. 
If we manage to make $X_t$ jump to zero at each moment $N_t$ has a jump, we solve the problem. How should we derive an equation, though? Well, suppose that $N_t$ has jumps at times 
$$
t_0<t_1<\dots<t_n<\dots
$$
and consider $t\in [t_{i},t_{i+1})$ - there the process $X_t$ behaves as $W_t$ since there are no jumps:
$$
X_t = (X_{t_i}-W_{t_i})+W_t\quad\text{ for }t\in[t_i,t_{i+1}).\tag{2}
$$
If we consider a jump at time $t_{i+1}$ downwards, of the absolute value $X_{t_{i+1}}$ it will make everything work. On the other hand, whenever you work with integrals w.r.t. the Poisson process you have to consider limits from the left as in $(1)$. In our case it is ok, since any of processes $W_t$ we considered above has continuous paths and so $W_{t_{i+1}-} = W_{t_i}$. 
Clearly, $-X_{t-}\cdot\Delta N_t$ is what we need since it makes jumps only together $N_t$ makes them and these jumps are of an appropriate value and direction.
As a result, the process defined by
$$
dX_t = - X_{t-}dN_t + dW_t \tag{3}
$$
solves your problem. The explicit formula for such process is given in $(2)$.
For the stochastic calculus of such process, a very nice reference is "Stochastic Calculus for Finance II" by S. Shreve, Chapter 11.
