Chapter 11: Stochastic integration Flashcards
area under f, a random continuous funct
is a random quantity
∫{a,b} f(t) dt
= limit δ→0 [Σ{i=1,n} [f(t_{i−1})[t_i −t_{i−1}]
δ=max_i {|t_{i+1}-t_i}
stochastic process, (Adapted stoch process, adapted to the same filtration as brownian motion)
ito calculus
∫{a,b} f(t) dB_T
= limit δ→0 [Σ{i=1,n} [f(t_{i−1})[B_t_i −B_t_{i−1}]
this models decision and effect
* choose value of f(t_i) at time t_i
*decision at time t_i is f(t_i) and brownian motion term is indep of what happened before
- world changes between t_i and t_{i+1}
- =sum gives cumulative effect of decision at t+0,t_1,..t_n
At each time ti−1 a decision is taken for the value of f(ti), based only on previously available information, then the world changes randomly during ti−1 7→ ti, and at time ti we receive and add the random effect of our decision:
f(t_{i−1{)[B_{ti} −B_{ti−1}].
chapter 11
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REMARK martingale transform
martingale transform (f ◦B)_n = Σ_{i=1,m} f_i (B_{I+1} -B_i) modelled roulette using the martingale transform if set t_i=I, C_n f(t_n) and M_n=B_{t_n} = B_n, then M_n is a discrete time martingale
(Take F_n = σ(B_i ; i ≤ n) and
∫{a,b} f(t) dB_T
= limit δ→0 [Σ{i=1,n} [f(t_{i−1})[B_t_i −B_t_{i−1}]
is this martingale transform
(C ◦M)n = Σ{i=1,n} [C_{i-1}(M_i - M_{i-1})
chapter 11
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REMARK stochastic=capital
Notation
∫_{a,b} F_t dB_t
stochastic=capital
chapter 11
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REMARK writing derivatives
We don’t write
∫{a,b} F_t dB_t
=
∫{a,b} F_t dB_t /dt . dt as deriv doesn’t exist
B_t isn’t differentiable so dBt/dt doesn’t exist
f_t
f_t is the generated filtration of BM B_t
A stochastic process (F_t) os LOCALLY SQUARE INTEGRABLE
A stochastic process (F_t) os LOCALLY SQUARE INTEGRABLE is for all t bigger than 0
∫_{0,t} [E[F_t ^2].dt less than infinity
Set H^2 (curly H)
to be the set of such (F_t)
THM 11.2.1
∫_{0,t} Ft .dBt
For any (F_t) ∈ H^2, and any t ∈ [0,∞) the Ito integral
∫_{0,t} Ft .dBt exists, and is a continuous martingale with mean and variance given by
E[∫{0,t} F_t dB_t]= 0,
E[ (∫{0,t} F_t dB_t)^2]
= ∫_{0,t}E[F^2_t ]dt.
THM 11.2.1
Ito integral
exists, and is a continuous martingale with mean and variance given by
For any (F_t) ∈ H^2, and any t ∈ [0,∞) the Ito integral
∫_{0,t} Ft .dBt exists, and is a continuous martingale
(is an L^2 limit and (I_t) is a continuous time martingale)
with mean and variance given by
E[∫{0,t} F_t dB_t]= 0,
E[( ∫{0,t} Ft dBt)^2]=
∫_{0,t} E[F^2_t ]dt.
E(I_t)=0
Var(I_t^2)=E(I_t^2) =∫_{0,t} E[F^2_t ]dt.
EXTENDING ITO INTEGRAL TO OVER [a,b] properties:
self consistency
linearity
∫{a,c} F_t dB_t =
∫{a,b} F_t dB_t + ∫_{b,c} F_t dB_t
for a ≤ b ≤ c
for α, β in R
∫{a,b} (αF_t + βG_t) dB_t =
α∫{a,b} F_t dB_t + β∫_{a,b} G_t dB_t
however many ways ito integral does not behave like classical integral
Integrate 0
classical vs ito integral
∫{0,t} 0 du= 0
∫{0,t} 0 dBu = 0
set f ≡ 0
integrate 1
classical vs ito integral
∫{0,t} 1 du= t
∫{0,t} 1 dBu = B_t
(set f≡ 1, sum from i=1 to n of [ f(t_{i-1}[B_{t_i} - B_{t_i-1}] = B_t - B_0 = B_t as B_0 is 0 for standard brownian motion)
Ito integration behaves differently to classical integration
classical vs ito integral
∫{0,t} u du= t^2/2
∫{0,t} B_u dBu = B_t^2/2 - t/2
set f(x) ≡ x
sum from i-1 to n of x(B_{t_{i+1}}- B_{t_i}
DEF 11.4.1
When is a stochastic process and ito process?
A stochastic process X, (X_t) is known as an Ito process if X0 is F0 measurable and X can be written in the form
X_t = X_0 + ∫{0,t} F_u du +
∫{0,t} G_u dB_u
Here, G ∈ H^2 and F is a continuous adapted process.
(curly H only)
- classical integral + ito integral
- F is continuous and adapted stoch process
- G is adapted and in H^2
*to specify X_t we need both F_u and G_u and initial X_0
LEMMA 11.4.2
Expectations and integrals
2 For a continuous stochastic process F, (F_t) if one (which ⇒ both) of the two sides
is finite, then we have
E[ ∫_{0,t} F_u du]
=
∫_{0,t} E[F_u] du
ONLY WORKS FOR CLASSICAL INTEGRALS
DOES NOT WORK FOR ITO INTEGRALS
e.g
E(X_t)
where X_t = X_0 + ∫{0,t} F_u du +
∫{0,t} G_u dB_u
E(X_t) =
(linearity of expectation)
E[X_0] + E[ ∫{0,t} F_u du] + E[∫{0,t} G_u dB_u]
(ito integrals are martingales with zero-mean THM 11.2.1)
(lemma 11.4.2 swap E and int for classical)
= E[X_0] + ∫_{0,t} E [F_u] du.
Example 11.4.3 Let Xt be the Ito process satisfying
Xt = 1 + ∫_{0,t} [2B^2u du] + ∫{0,t} [3u dB_u]
Then
E[X_t] =
E[X_t] = E[1] + E[∫_{0,t} [2B^2u du] ] + ∫{0,t} [3u dB_u]
=
1 + 2 ∫_{0,t} E [B^2u ].du
= 1 + 2 ∫{0,t} u.du
= 1+t^2
BROWNIAN MOTION IS AN ITO PROCESS
B_t = B_0 + ∫{0,t} 0 du + ∫{0,t} 1 dB_u = B_0 + B_t
LEMMA 11.4.5
EQUATING COEFFS IN AN ITO PROCESS
Suppose that
X_t = X_0 +∫{0,t} F^X_u du +
∫{0,t} G^X_u dB_u
Y_t = Y_0 + ∫{0,t} F^Y_u du + ∫{0,t} G^Y_u dB_u
are Ito processes and that
P[for all t, Xt = Yt] = 1.
Then,
P[for all t, F^X_t = F^Y_t and G^X = G^Y_t] = 1.
key to the argument that will allow to hedge financial derivatives in continuous time.
*ie different initial condition, different values however probabilities of being equal are 1
we can compare the F and G coeffs of two ito processes
