Adding stochastic variables random variables where one has time component

Question

Based on GLS regression I have identified two random variables lets call them A & B

variable A is normal variable with  
E(A) = 15 
STDEV(A) = 18

variable B is composed as follows   
B = X * t
E(X) = 0.15
STDEV(X) = 0.02

Given the above, is it possible to make a forecast of STDEV(A+B) at time (t)?

I have also created time varying model for the standard deviation of B.

Dependent Variable: B               
Method: ML - ARCH (Marquardt) - Normal distribution             
Date: 17/06/14   Time: 13:02                
Sample: 641 778             
Included observations: 138              
Convergence achieved after 24 iterations                
Presample variance: backcast (parameter = 0.7)              
GARCH = C(3) + C(4)*GARCH(-1) + C(5)*DAY-639                    

Variable    Coefficient Std. Error  z-Statistic Prob.  

C          -0.332209    1.544698    -0.215064   0.8297
DAY-639     0.156594    0.022257    7.035607    0.0000

    Variance Equation           

C          -2.821977    0.647196    -4.360314   0.0000
GARCH(-1)   0.030162    0.393119     0.076726   0.9388
DAY-639     1.358799    0.544513     2.495440   0.0126

Answer 1

Since $A$ and $B$ are independent we have $$ \text{var}(A+B) = \text{var}(A) + \text{var}(B) = \text{var}(A) + \text{var}(tX) = 18^2 + t^2\text{var}(X) = 18^2 + t^2\cdot 0.02^2, $$ so that $\text{STDEV}(A+B) = \sqrt{18^2 + t^2\cdot 0.02^2}$.