Combining two stochastic processes, one Wiener and one Bernoulli process?

Questions
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1

Hello everybody,

What I am trying to do is simulating a random walk with certain fixed probability resetting to its origin.

I am thinking about combining two stochastic processes together, one is Wiener process and the other one is Bernoulli process.

diffusionwithresetting = TransformedProcess[diff*reset, {diff \[Distributed] WienerProcess[], reset \[Distributed] BernoulliProcess[1/4]}, t];
data = RandomFunction[diffusionwithresetting, {0, 10, 0.1}, 1]

But the problem is that the time step for those two processes is different. Any one know how to do this?

Jay