  There are a large variety of hybrid stochastic systems that couple a
continuous process with some form of stochastic switching mechanism. In many
cases the system switches between different discrete internal states according
to a finite-state Markov chain, and the continuous dynamics depends on the
current internal state. The resulting hybrid stochastic differential equation
(hSDE) could describe the evolution of a neuron's membrane potential, the
concentration of proteins synthesized by a gene network, or the position of an
active particle. Another major class of switching system is a search process
with stochastic resetting, where the position of a diffusing or active particle
is reset to a fixed position at a random sequence of times. In this case the
system switches between a search phase and a reset phase, where the latter may
be instantaneous. In this paper, we investigate how the behavior of a
stochastically switching system is modified when the maximum number of
switching (or reset) events in a given time interval is fixed. This is
motivated by the idea that each time the system switches there is an additive
energy cost. We first show that in the case of an hSDE, restricting the number
of switching events is equivalent to truncating a Volterra series expansion of
the particle propagator. Such a truncation significantly modifies the moments
of the resulting renormalized propagator. We then investigate how restricting
the number of reset events affects the diffusive search for an absorbing
target. In particular, truncating a Volterra series expansion of the survival
probability, we calculate the splitting probabilities and conditional MFPTs for
the particle to be absorbed by the target or to exceed a given number of
resets, respectively.
