이토 곱셈 테이블
📂확률미분방정식 이토 곱셈 테이블 빌드업 s < t < t + u s< t < t+u s < t < t + u 확률과정 { W t } \left\{ W_{t} \right\} { W t } 위너 프로세스 라 한다.
(i): W 0 = 0 W_{0} = 0 W 0 = 0 (ii): ( W t + u − W t ) ⊥ W s \left( W_{t+u} - W_{t} \right) \perp W_{s} ( W t + u − W t ) ⊥ W s (iii): ( W t + u − W t ) ∼ N ( 0 , u ) \left( W_{t+u} - W_{t} \right) \sim N ( 0, u ) ( W t + u − W t ) ∼ N ( 0 , u ) (iv): W t W_{t} W t 거의 어디서나 연속이다. 위너 프로세스는 다음과 같은 성질들을 가진다.
[1]: W t ∼ N ( 0 , t ) \displaystyle W_{t} \sim N ( 0 , t ) W t ∼ N ( 0 , t ) [2]: E ( W t ) = 0 \displaystyle E ( W_{t} ) = 0 E ( W t ) = 0 [3]: Var ( W t ) = t \displaystyle \operatorname{Var} ( W_{t} ) = t Var ( W t ) = t [4]: cov ( W t , W s ) = 1 2 ( ∣ t ∣ + ∣ s ∣ − ∣ t − s ∣ ) = min { t , s } \displaystyle \text{cov} ( W_{t} , W_{s} ) = {{1} \over {2}} (|t| + |s| - |t-s|) = \min \left\{ t , s \right\} cov ( W t , W s ) = 2 1 ( ∣ t ∣ + ∣ s ∣ − ∣ t − s ∣ ) = min { t , s } 위너 프로세스 { W t } t ≥ 0 \left\{ W_{t} \right\}_{t \ge 0} { W t } t ≥ 0 [ t , t + d t ] \left[ t , t + d t \right] [ t , t + d t ] d t > 0 dt > 0 d t > 0 ( d t ) 1 / 2 > 0 \left( dt \right)^{1/2} > 0 ( d t ) 1/2 > 0 k = 2 , 3 , ⋯ k = 2 , 3, \cdots k = 2 , 3 , ⋯ ( d t ) k = 0 \left( dt \right)^{k} = 0 ( d t ) k = 0 α + β d t \alpha + \beta dt α + β d t 이원수 로 다루는 것이다.
이제 d W t : = W t + d t − W t dW_{t} := W_{t + dt} - W_{t} d W t := W t + d t − W t d t dt d t d W t d W_{t} d W t
Part 1. ( d t ) 2 = 0 \left( dt \right)^{2} = 0 ( d t ) 2 = 0
물론 d t > 0 dt > 0 d t > 0 d t dt d t ( d t ) 2 = 0 \left( dt \right)^{2} = 0 ( d t ) 2 = 0
Part 2. d t d W t = 0 dt d W_{t} = 0 d t d W t = 0
W t W_{t} W t 정규분포 를 따라서 d W t ∼ N ( 0 , d t 2 ) d W_{t} \sim N \left( 0, \sqrt{dt}^{2} \right) d W t ∼ N ( 0 , d t 2 )
평균과 분산의 성질들 :
[2]: E ( a X + b ) = a E ( X ) + b E(aX + b) = a E(X) + b E ( a X + b ) = a E ( X ) + b [5]: Var ( a X + b ) = a 2 Var ( X ) \operatorname{Var} (aX + b) = a^2 \operatorname{Var} (X) Var ( a X + b ) = a 2 Var ( X ) d t d W t dt d W_{t} d t d W t d t dt d t E ( d t d W t ) = d t E ( d W t ) = d t ⋅ 0 = 0
E \left( dt d W_{t} \right) = dt E \left( d W_{t} \right) = dt \cdot 0 = 0
E ( d t d W t ) = d tE ( d W t ) = d t ⋅ 0 = 0 d t d W t dt d W_{t} d t d W t d t dt d t Var ( d t d W t ) = ( d t ) 2 Var ( d W t ) = 0 ⋅ Var ( d W t ) = 0
\operatorname{Var} \left( dt d W_{t} \right) = (dt)^{2} \operatorname{Var} \left( d W_{t} \right) = 0 \cdot \operatorname{Var} \left( d W_{t} \right) = 0
Var ( d t d W t ) = ( d t ) 2 Var ( d W t ) = 0 ⋅ Var ( d W t ) = 0 d t d W t dt d W_{t} d t d W t 0 0 0 0 0 0 d t d W t = d W t d t = 0
dt d W_{t} = d W_{t} dt = 0
d t d W t = d W t d t = 0
Part 3. ( d W t ) 2 = d t \left( d W_{t} \right)^{2} = dt ( d W t ) 2 = d t
Var ( d W t ) = d t \operatorname{Var} \left( d W_{t} \right) = dt Var ( d W t ) = d t d W t ⋅ d W t d W_{t} \cdot d W_{t} d W t ⋅ d W t d t = Var ( d W t ) = E ( ( d W t ) 2 ) − [ E ( d W t ) ] 2 = E ( ( d W t ) 2 ) − 0 2
\begin{align*}
dt =& \operatorname{Var} \left( d W_{t} \right)
\\ =& E \left( \left( d W_{t} \right)^{2} \right) - \left[ E \left( d W_{t} \right) \right]^{2}
\\ =& E \left( \left( d W_{t} \right)^{2} \right) - 0^{2}
\end{align*}
d t = = = Var ( d W t ) E ( ( d W t ) 2 ) − [ E ( d W t ) ] 2 E ( ( d W t ) 2 ) − 0 2 E ( ( d W t ) 2 ) = d t E \left( \left( d W_{t} \right)^{2} \right) = dt E ( ( d W t ) 2 ) = d t
평균이 0인 정규분포를 따르는 확률변수 거듭제곱의 기대값 : 확률변수 X X X 정규분포 N ( 0 , σ 2 ) N \left( 0 , \sigma^{2} \right) N ( 0 , σ 2 ) X n X^{n} X n 기대값 은 다음과 같이 재귀적인 공식으로 나타난다.
E ( X n ) = ( n − 1 ) σ 2 E ( X n − 2 )
E \left( X^{n} \right) = (n - 1) \sigma^{2} E \left( X^{n-2} \right)
E ( X n ) = ( n − 1 ) σ 2 E ( X n − 2 ) E ( X n ) E \left( X^{n} \right) E ( X n ) n n n 0 0 0 짝수 일 때 다음과 같다.
E ( X 2 n ) = ( 2 n − 1 ) ! ! σ 2 n
E \left( X^{2n} \right) = \left( 2n - 1 \right)!! \sigma^{2n}
E ( X 2 n ) = ( 2 n − 1 ) !! σ 2 n k ! ! = k ⋅ ( k − 2 ) ⋯ k!! = k \cdot \left( k - 2 \right) \cdots k !! = k ⋅ ( k − 2 ) ⋯ 더블 팩토리얼 을 나타낸다.
d W t ∼ N ( 0 , d t 2 ) d W_{t} \sim N \left( 0, \sqrt{dt}^{2} \right) d W t ∼ N ( 0 , d t 2 ) d W t d W_{t} d W t 0 0 0 ( d W t ) 2 \left( d W_{t} \right)^{2} ( d W t ) 2 E ( X 2 n ) = ( 2 n − 1 ) ! ! σ 2 n E \left( X^{2n} \right) = \left( 2n - 1 \right)!! \sigma^{2n} E ( X 2 n ) = ( 2 n − 1 ) !! σ 2 n Var ( ( d W t ) 2 ) = E ( [ ( d W t ) 2 ] 2 ) − [ E ( ( d W t ) 2 ) ] 2 = E ( ( d W t ) 2 ⋅ 2 ) − [ d t ] 2 = ( 2 ⋅ 2 − 1 ) d t 2 ⋅ 2 − d t 2 = 3 d t 2 − d t 2 = 2 d t 2 = 0
\begin{align*}
\operatorname{Var} \left( \left( d W_{t} \right)^{2} \right) =& E \left( \left[ \left( d W_{t} \right)^{2} \right]^{2} \right) - \left[ E \left( \left( d W_{t} \right)^{2} \right) \right]^{2}
\\ =& E \left( \left( d W_{t} \right)^{2 \cdot 2} \right) - \left[ dt \right]^{2}
\\ =& \left( 2 \cdot 2 - 1 \right) \sqrt{dt}^{2 \cdot 2} - dt^{2}
\\ =& 3 dt^{2} - dt^{2}
\\ =& 2 dt^{2}
\\ =& 0
\end{align*}
Var ( ( d W t ) 2 ) = = = = = = E ( [ ( d W t ) 2 ] 2 ) − [ E ( ( d W t ) 2 ) ] 2 E ( ( d W t ) 2 ⋅ 2 ) − [ d t ] 2 ( 2 ⋅ 2 − 1 ) d t 2 ⋅ 2 − d t 2 3 d t 2 − d t 2 2 d t 2 0 ( d W t ) 2 \left( d W_{t} \right)^{2} ( d W t ) 2 0 0 0 d t dt d t ( d W t ) 2 = d t
\left( d W_{t} \right)^{2} = dt
( d W t ) 2 = d t
요약 α + β d t \alpha + \beta dt α + β d t 이원수 라 가정하자. d t dt d t d W t d W_{t} d W t ( d t ) 2 = 0 d t d W t = 0 d W t d t = 0 ( d W t ) 2 = d t
\begin{align*}
\left( dt \right)^{2} =& 0
\\ dt d W_{t} =& 0
\\ d W_{t} dt =& 0
\\ \left( d W_{t} \right)^{2} =& dt
\end{align*}
( d t ) 2 = d t d W t = d W t d t = ( d W t ) 2 = 0 0 0 d t
