The Mathematics of Financial Modelingand Investment Management

8-Stochastic Integrals Page 236 Wednesday, February 4, 2004 12:50 PM

236 The Mathematics of Financial Modeling and Investment Management

T T

If[]()w = ∫ ft( ω , ) dBt() ω = lim φ ( ntω , ) dt

n→ ∞∫

S S

where the functions φ ( n tω , ) ∈ Φ are a sequence of elementary functions

such that

T

E∫( f– φ ) n^2 dt → 0

S

The multistep procedure outlined above ensures that the sequence

φ ( n t ω , ) ∈ Φ exists. In addition, it can be demonstrated that the Itô

isometry holds in general for every ft( ω , ) ∈ Φ



2 T



T

E∫ ft( ω , ) dBt ()ω = Eft( ω , )

2

∫ dt

S  S

SOME PROPERTIES OF ITÔ STOCHASTIC INTEGRALS

Suppose that fg, ∈ Φ( ST, ) and let 0 < S< U< T. It can be demon-

strated that the following properties of Itô stochastic integrals hold:

T U T

∫ fBd t = ∫ fBd t+∫ fBd t for a.a. ω

S S U

T

Ef B∫ d t = 0

S

T T T

( cf+ dg) dB = cf Bd t+ dg B∫ d (^) t , for a.a. ω,,cdconstants
S S S

∫ t ∫

If we let the time interval vary, say (0,t), then the stochastic integral

becomes a stochastic process: