CODEWITHSUNDEEP
maple
itc
itc

Reputation: 175

Assigning products to constants or variables in Maple

I am doing Ito calculus workings in Maple. With B__t as a standard Brownian motion, the following results are true:

dB__t*dt = 0;
dB__t^2 = dt;
dt^2 = 0;

Now when I want to do the assignment:

dB__t*dt := 0;
dB__t^2 := dt;
dt^2 := 0;

I get the error message Error, illegal use of an object as a name. Why the need of the above assignments? Suppose I have:

Y__t := ln(1/X__t - 1);

dX__t := 0.5*X__t(-X__t + 1)*(-2*X__t + 1)*dt - X__t*(-X__t + 1)*dB__t;

Applying Ito's lemma on Y__t:

dY__t := diff(Y__t, t)*dt + diff(Y__t, X__t)*dX__t + 0.5*diff(Y__t, X__t, X__t)*dX__t^2

and expanding and simplying using:

dY__t := simplify(expand(dY__t))

yields: enter image description here

Now that I have an expanded dY__t expression, I want to then replace dt*dB__t, dB__t*dt and dt^2 with zero and to replace dB__t^2 with dt. How do I do that?

Upvotes: 1

Views: 105

Answers (1)

acer
acer

Reputation: 7271

Here are two ways to make such substitutions (with simplifications as one goes),

restart;
Y__t:=ln(1/X__t-1):
dX__t:=1/2*X__t*(-X__t+1)*(-2*X__t+1)*dt-X__t*(-X__t+1)*dB__t:
dY__t:= diff(Y__t,t)*dt + diff(Y__t,X__t)*dX__t+diff(Y__t,X__t,X__t)*dX__t^2:

First way,

simplify(dY__t,{dB__t*dt=0});
simplify(%,{dB__t^2=dt});
simplify(%,{dt^2=0});

Second way,

simplify(algsubs(dB__t*dt=0,expand(dY__t)));
simplify(algsubs(dB__t^2=dt,%));
simplify(algsubs(dt^2=0,%));

Upvotes: 1

Related Questions