# Preservation of martingale property when changing to a product space.

Let the process $M=(M_t, t\ge 0)$ be a martingale on the probability space $(\Omega_1, \mathcal F_1, P_1)$ with respect to the natural filtration of $M$.

Let $X=(X_t, t\ge 0)$ be a process on the probability space $(\Omega_2, \mathcal F_2, P_2)$.

Let $W=(M,X)$ be the coupled stochastic process on the probability space $(\Omega_1\times\Omega_2, \mathcal F_1\times\mathcal F_2, P_1\times P_2)$ (let us call it 'the product space').

Questions:

1. Does it make sense to only consider the process $M$ on the product space. That is, the first element in $W$ alone?

2. If 1. does makes sense, will $M$ and $X$ be the same stochastic processes on the the product space as they where in their original probability spaces respectively? Independence is assumed. That is do $M$ and $X$ follow the same laws/distributions on the product space as their original probability spaces from which they originally where defined, if $M$ and $X$ are independent of each other?

3. If 1. does makes sense. Will $M$ be a martingale with respect to its natural filtration in the product space? That is, will $M$ be a $(P_1\times P_2)$-martingale?

4. Could there be some lack of completeness in the product space? If so, is it necessary to avoid the lack of completeness to make 1., 2., 3. and 5. valid? If it is necessary, could this always be done in a way to guarantee 1., 2., 3. and 5. holds?

5. Does the definition of the product space given above imply independence of $M$ and $X$, or does one need to consider additional conditions to assure independence of $M$ and $X$, perhaps completion of the product space is necessary?

• Answers to all 1-5 are positive. No completion is necessary. Aug 18, 2017 at 14:13
• @zhoraster Would you mind eloborate this as an answer so i could accept it? Aug 27, 2017 at 17:34