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Unformatted text preview: Numerical Methods of SDEsDr. Ra´ul Tempone Homework #3 Exercise 1 Formulate and motivate a forward Euler method for approximation of the Stratonovich SDE dX t = a ( t,X t ) dt + b ( t,X t ) ◦ dW t (1) First approach: (Incorrect approach!) We approximated b ( t,X t ) by the midpoint rule: X t n +1 = X n + a ( t n ,X t n )Δ t n + b 1 2 ( t n +1 + t n ) , 1 2 ( X t n +1 + X t n ) Δ W n (2) What is wrong with approach (2)? Consider the following process as an example to illustrate why formulation (2) was incorrect. dX ( t ) = bX t dW ( t ) with a constant b, (3) Using (2) to approximate (3) we obtain: X t n +1 = X t n + b 1 2 ( X t n +1 + X t n ) Δ W n X t n +1 (1 1 2 b Δ W n ) = X t n (1 + 1 2 b Δ W n ) X t n +1 = X t n 1 + 1 2 b Δ W n 1 1 2 b Δ W n (4) Consider the computation of E ( X 2 1 ) E ( X 2 1 ) = ( X ) 2 √ 2 π Δ t Z ∞∞ (1 + 1 2 bz ) 2 (1 1 2 bz ) 2 e z 2 2Δ t dz (5) where we have used the fact that E ( f ( x )) = R ( f ( x ) dp ( x )) where p ( x ) is the probability density of x . From (5), consider a similiar integral: Z ∞∞ (1 + x ) 2 (1 x ) 2 e x 2 dx (6) However, the integral in (6) does not converge. Formulation: Rewrite (1), so that may discretize using the forward Euler method. X t = X + Z t a ( s,X ( s )) ds + 1 2 Z t b ( s,X s ) b x ( s,X s ) ds + Z t b ( s,X s ) dW s (7) From (7), considering one interval, [ t n ,t n +1 ] X n +1 = X n + Z t n +1 t n a ( x,X ( s )) ds + 1 2 Z t n +1 t n b ( s,X s ) b x ( s,X s ) ds + Z t n +1 t n b ( s,X s ) dW s (8) Discretization of (8) to obtain the following: X t n +1 = X t n + a ( t n ,X t n )Δ t n + 1 2 b ( t n ,X t n ) ∂b ( t n ,X t n ) ∂x Δ t n + b Δ t n Δ W t n (9) Exercise 2 Consider the deterministic differential equation dZ t = a ( Z t ) dt (10) Z (0) = x ≤ t ≤ T, and consider a peturbation of (10), the It ˆ o stochastic differential equation dX t = a ( X t ) dt + bdW t (11) X (0) = x...
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 Fall '06
 gallivan

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