Type of Expansions and Differential Equations

[courtrigrad]

Hello all

Could someone show me how we get: $$M(t+dt) - M(t) \doteq \frac{dM}{dt}dt + ...$$. I know that you use a Taylor Series expansion, but I need to see how it is done as I am new in this subject. How would you derive the formula $$e^{-r(T-t)}$$? Also could someone explain the concept of separation of Variables?

Thanks

 

[courtrigrad]

I think I use a log expansion. Is that right?

 

[dextercioby]

How about the definition of the derivative??

$$\frac{dv}{dt}=:\lim_{\Delta t \rightarrow 0} \frac{v(t+\Delta t)-v(t)}{\Delta t}$$

For the second part,where does that exponential come from??

Daniel.

 

[courtrigrad]

ok I get so we multiply by $$dt$$ to get the differential $$dM$$

 

[dextercioby]

Pretty much so.In this case "dt" is the one in the limit (Delta t,when it goes to zero).

Daniel.