Stochastic integrals are important in the study of stochastic differential equations
and ... define the Ito stochastic integral and some important properties.
Tampere University of Technology Post-Graduate Seminar on Applied Mathematics presentation by Tamás Bencsik
Stochastic Integration
Introduction In this chapter we will study two type of integrals:
Ÿa
t
f Hs, wL „ s and Ÿ gHs, wL „ W Hs, wL t
a
for a § t § b
where f , g stochastic process on HW, , PL. If f and g satisfy certain conditions and are stochastic process in Hilbert space HSP , then the integrals will also be stochastic process in HSP . Stochastic integrals are important in the study of stochastic differential equations and properties of stochastic integrals determine properties of stochastic differential equations. In this chapter we will - define the Ito stochastic integral and some important properties - describe a method to approximate Ito stochastic integrals - describe the stochastic differentials - derive the Ito's formula - define the Stratonovich stochastic integrals
2
Stochastic Integration.nb
Integrlas of the form Ÿa f Hs, wL „s t
Denotation:
J H f L = J H f L HwL :=Ÿ f Hs, wL „ s b
for a § t § b
a
J H f L HtL = J H f L Ht, wL :=Ÿ f Hs, wL „ s t
for a § t § b
a
In this and the next section there are 3 other conditions in addition to f œ HSP Hc1L Hc2L Hc3L
f HaL œ HRV . Hence, ∞ f HaL¥2RV = E † f HaL§2 § k1 , where k1 is a positive constant
∞ f Ht2 L - f Ht1 L¥2RV = E † f Ht2 L - f Ht1 L§2 § k2 †t2 - t1 § for any t1 , t2 e @a, bD for a positive constant k2 f is nonanticipating on @a, bD
Notice that if f œ HSP satisfies Hc1L and Hc2L, then ∞ f HtL¥RV §
k2 Hb - aL + ∞ f HaL¥RV
for any t e @a, bD
The third conditions means that f Ht, wL doesn't depend on time t ' for t ' > t. Example:
f1 HtL = 3 CosIW 2 HtLM + 4 W HtL - 5 t is nonanticipating f2 HtL = W I
Hence,
t+b M 2
is anticipating for a t b.
EH f1 HtL HW Ht 'L - W HtLLL = 0 for t ' > t EH f2 HtL HW Ht 'L - W HtLLL =
t ' - t, for t t ' § b-t , 2
for
t+b 2
t+b 2
t' § b
To motivate the definition of J H f L = J H f L HwL :=Ÿ f Hs, wL „ s for f œ HSP , b
a
the integral for a step function approximation to f is considered.
Definition: Let fm Ht, wL =⁄i=0 fiHmL HwL Ii HtL be an element of SSP. Then m-1
J H fm L = Ÿ fm HsL „ s = ⁄i=0 fiHmL Hti+1 - ti L, b
m-1
a
where a = t0 t1 ... tm = b and maxi †ti+1 - ti § Ø 0 as m Ø ¶, fiHmL HwL = f Hti , wL and let Ii HtL = for i = 0, ..., m - 1 Notice: J H fm L œ HRV proof!
1 for ti § t ti+1 0 otherwise
Stochastic Integration.nb
Notice: 8J H fm L
