Continuous Time Jumps In {0, 1}

Question

How would one define a continuous time stochastic process which jumps between between zero and one with intensity $\lambda$?

More specifically, I would like to define a process $X_t = Y_t + J_t$ such that $dY_t = \mu dt + \sigma dB_t$ and $J_t$ jumps between zero and one with intensity $\lambda$. I would like to be able to define $dX_t$, but am having trouble formulating the stochastic integral for $J_t$.

Answer 1

My attempt at resolving this problem (anyone confirming or disputing my intuition would be great):

Define $X_t$ and $Y_t$ as above. Define $J_t \in \{0, 1\}$ as a stochastic process which jumps with intensity $\lambda$. This implies a jump size of $\lambda$ if $J_t = 0$ and $-\lambda$ if $J_t = 1$.

The expected change in $J_t$ is $\mathbb{E}dJ_t = \lambda(1\{J_t = 0\} - 1\{J_t = 1\})dt$, where $1\{\cdot\}$ denotes the indicator function.

$dX_t = dY_t + dJ_t = \mu dt + \sigma dB_t + dJ_t$

Taking expectations gives

$\mathbb{E} dX_t = \left[\mu + \lambda(1\{J_t = 0\} - 1\{J_t = 1\})\right]dt$

Answer 2

Let $X_t$ be a Poisson process with intensity $\lambda$. Then the process $$ J_t = \left\{\begin{array}{ll} 1 & \mbox{$X_t$ is an odd integer} \\ 0 &\mbox{otherwise} \end{array}\right.$$ fits your description.

As for how to handle the process, pdevar's approach of thinking of it as a state-dependent jump size could help. So writing an SDE like $$ dJ_t = f(J_t) dX_t$$ where $X_t$ is a Poisson processes and $f(x) = 1$ when $x$ is zero and $f(x)=-1$ when $x$ is one.

Not sure writing a better-formulated SDE is gonna help with your problem. An analysis of $J_t$'s distribution and linearity will, though.