3. Consider the stochastic differential

Question

3. Consider the stochastic differential equation (SDE)
dX= tX+ dt + X{ dBť, Xo = 1.
(a) Let Yt = ln Xt, compute the stochastic differential dYt.
(b) Find a formula for Yt, hence solve the SDE for Xt.

Kunduz · Accepted Answer

For
OLTLos let to [OT]
d x = act] Xy dt + x4 dBz
Xo=1
linear growth ES TIX J² dt <
dxt
w
emprtxJubs- = 7x p - - 7Xp
alt, sel+ 1 t, x/xb< clitxl
son (x+) dwt
& quore root process
- St sgn (Ws) dwg
(
-syn(
Son (Y+Jd we
dx = (*- xt) dz tv rxz dwt
but Yt = ln xt
d x, - dyt son rx ) dwtte,
) Wzt a
dY+- Son (X4) dwz + ln xt
CXz
Smail.corly formula
dY+ = san (xt) d wet enxt.
Y = 0 + (xo- oj e kty od wzten
Hence Anawes
2
t
=
6
SDE

3. Consider the stochastic differential equation (SDE) dX= tX+ dt + X{ dBť, Xo = 1. (a) Let Yt = ln Xt, compute the stochastic d