Let X = X(t) be a stochastic process with SDE: dx = atX dt + BX² d X(0) = 1 (a and ß are constants) Let y(t) = X(t)² Using Ito's formula work out the SDE for Y(T) Should get dy =(some function of Y and t) dt+ (some function of Y and t) dw
Let X = X(t) be a stochastic process with SDE: dx = atX dt + BX² d X(0) = 1 (a and ß are constants) Let y(t) = X(t)² Using Ito's formula work out the SDE for Y(T) Should get dy =(some function of Y and t) dt+ (some function of Y and t) dw
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Let X = X(t) be a stochastic process with SDE:
dx = atX dt + ßX² d
X(0) = 1
(a and ß are constants)
Let y(t) = X(t)²
Using Ito's formula work out the SDE for Y(T)
Should get dy =(some function of Y and t) dt+ (some function of Y and t) dW
Expert Solution
Step 1
To use Ito's formula to derive the SDE for Y(t), we need to first find the differential of Y(t) using the chain rule:
dY = d(X^2) = 2X dX + (dX)^2
Since X satisfies the SDE dx = alpha tX dt + beta X^2 dW, we can substitute this expression for dX and simplify using Ito's lemma:
dY = 2X dX + (dX)^2
= 2X (alpha tX dt + beta X^2 dW) + (alpha tX dt + beta X^2 dW)^2
= 2alpha tX^2 dt + 2beta X^3 dW + alpha^2 t^2 X^2 dt^2 + 2alpha beta t X^3 dW dt + beta^2 X^4 dW^2
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