Oh crap! I was worried if I made an error in the expression and looks like I did

A simple sanity check is as follows

YOu have x/2 and x/2 invested ie 5K and 5K say.

And the index drops by 10%

So the total portfolio should be

5000*( 1-10%) +5000 = 5000 ( 2 - 10%)

algebraically - x/2*(2 - theta)

I messed up the sign on the expressions;

Here's the CORRECTED VERSION.

Subtractive side ie CASH LEFT on BTD at each step

C[i]= C[i-1]* ( Theta - Alpha + Theta* Alhpa]

Invested side

P[i]=P[i-1]* (1-theta) + P[i-1]*(1-theta)*(1+alpha)
which simplifies to

P[i]= P[i-1]* (1 - theta)*(2+alpha)

And you can check that if you put initial half and half or x/2 and add these 2
it will equal the x/2 (2 - theta) which is what you ought to have when portfolio drops by theta

On the STP side it goes the other way - ie you need to subtract that term from the invested side and add it back to the Cash side.

Hope this helps!