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Given some random process that increments by independant gausian process Y, ie in time 2, X2=Y1+Y2: If i look at the process starting at some point not zero say at 4 ie X4 up to 7 then can:

 P(X₄X₇>a)  be written as:  P(X₇>a/X₄)=P(Y₅+Y₆+Y₇>a/X₄)  ie at the new starting point can X4 be taken as non random  and just some number? 

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No, but close. You are missing a term: $P\left(X_7 \gt \frac{a}{X_4}\right) = P\left(X_4 + Y_5+Y_6+Y_7 \gt \frac{a}{X_4}\right)=P\left( Y_5+Y_6+Y_7 \gt \frac{a}{X_4} - X_4 \right).$

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    @user39325: I have included it because I thought the definition of $X_7$ was $X_4 + Y_5+Y_6+Y_7$ since you said "some random process that increments by independent Gaussian process $Y$"2012-10-12