Falling object problem

Question

A mass is falling down in the air (with drag force in $kv^n$ form )
Drag force is $kv^1$ here .When mass is falling ,we have $mg-f=ma $ there is $ma=m\frac {dv}{dt}$ so : $$mg-kv=m\frac{dv}{dt}$$ Suppose velocity is in form of $v+\alpha W_t$ (be cause of temperature ,humidity ,dust , ...) so finally we have $$dv = (g - \frac{k}{m}v)dt - \frac{k}{m}\alpha {W_t}.dt\\ dv = (g - \frac{k}{m}v)dt - \frac{k}{m}\alpha d{B_t}$$ ($W_t$ is a wiener process ) To solve SDE ,I did like this $${e^{\frac{k}{m}t}}dv + {e^{\frac{k}{m}t}}\frac{k}{m}vdt = {e^{\frac{k}{m}t}}gdt - {e^{\frac{k}{m}t}}\frac{k}{m}\alpha d{B_t}\\d({e^{\frac{k}{m}t}}v) = {e^{\frac{k}{m}t}}gdt - {e^{\frac{k}{m}t}}\frac{k}{m}\alpha d{B_t}\\\int\limits_0^t {d({e^{\frac{k}{m}s}}v)} = \int\limits_0^t {{e^{\frac{k}{m}s}}gds} - \int\limits_0^t {{e^{\frac{k}{m}s}}\frac{k}{m}\alpha d{B_s}} \\ {e^{\frac{k}{m}t}}v - {e^{\frac{k}{m}}v_0} = g\int\limits_0^t {{e^{\frac{k}{m}s}}ds} - \frac{k}{m}\alpha \int\limits_0^t {{e^{\frac{k}{m}s}}d{B_s}} $$we know $v_0=0$ so $${v_t} = g\frac{m}{k}(1 - {e^{ - \mathop {}\limits_{} \frac{k}{m}t}}) - \frac{k}{m}\alpha {e^{ - \mathop {}\limits_{} \frac{k}{m}t}}\int\limits_0^t {{e^{\frac{k}{m}s}}d{B_s}} $$ This was my try to write an analytic solution ,then I did numerical solution ,by euler maruyama scheme ,and obtain some simulation . I need to establish an Error bound for the problem . But I don't know how to start !? (the path was clear for ODE's Error analysis but here ...)
I can calculate Root Mean Square Error for numerical result ,But I can't write somehow analytical Error calculation .
I am thankful ,If you look at my work and check if I am in a right way (?) ,then Hint me for a analytical Error analyzing . $$\color{green} {E[v] = g\frac{m}{k}E[(1 - {e^{ - \mathop {}\limits_{} \frac{k}{m}t}})]\\{\mathop{\rm var}} [v] = {\mathop{\rm var}} [{v_0}] + \frac{k}{{2m}}{\alpha ^2}(1 - {e^{\frac{{ - 2k}}{m}t}})}$$