Stochastic Numerical Analysis

Transcription

1 Stochastic Numerical Analysis Prof. Mike Giles Oxford University Mathematical Institute Stoch. NA, Lecture 3 p. 1

2 Multi-dimensional SDEs So far we have considered scalar SDEs with a single driving Wiener path. Now we consider the generalisation to a vector SDE ds t = a(s t,t)dt+b(s t,t) dw t where a(s,t) is a vector, b(s,t) is a matrix and W t is a vector of Wiener paths. In some cases, the different components of W t are independent, so over a time interval of length h we have E[ W W T ] = h I where I is the identity matrix. Stoch. NA, Lecture 3 p. 2

3 Multi-dimensional SDEs In other cases we may have a correlation between the components so that W N(,hΣ) meaning that the components of W have a joint Normal distribution with covariance E[ W W T ] = h Σ W can be simulated by letting W = h LZ where Z is a vector of independent N(,1) random variables and LL T = Σ, since E[LZZ T L T ] = L E[ZZ T ] L T = LL T = Σ Stoch. NA, Lecture 3 p. 3

4 Itô isometry The generalisation of the Itô isometry is 2 T E f j dw j = E j [ T E T f j f k Σ jk dt j,k f T Σf dt ] since E[dW j dw k ] = Σ jk dt. Stoch. NA, Lecture 3 p. 4

5 Itô lemma The generalisation of the Itô lemma for f(s,t) is df = f t dt+ j j,k 2 f S j S k f S j ds j l,m b jl b km Σ lm dt which comes from the fact that E[dS j ds k ] = l,m b jl b km E[dW l dw m ] = l,m b jl b km Σ lm dt Stoch. NA, Lecture 3 p. 5

6 Euler-Maruyama method For the vector SDE ds = a(s,t) dt+b(s,t) dw the Euler-Maruyama approximation is again Ŝ n+1 = Ŝ n +a(ŝ n,t n )h+b(ŝ n,t n ) W n Very simple this is one of the great attractions of the Euler-Maruyama method. The strong error remains O(h 1/2 ), and the weak error is still O(h) in most applications. Stoch. NA, Lecture 3 p. 6

7 Milstein Method In the vector case, the SDE ds i = a i (S,t) dt+ j b ij (S,t) dw j corresponds to the integral equation: t t S i (t) = S i ()+ a i (S(s),s)ds+ j b ij (S(s),s)dW j (s) Stoch. NA, Lecture 3 p. 7

8 Milstein Method An asymptotic expansion gives S i (t) S i () j b ij (S(),)W j (t) and hence b ij (S(t),t) b ij + l b ij + k,l b ij S l (S l (t) S l ()) b ij S l b lk W k (t) with b and its derivatives evaluated at (S(),). Stoch. NA, Lecture 3 p. 8

9 Milstein Method This then leads to S i (h) S i ()+a i h+ j b ij W j (h) + j,k,l b ij S l b lk h W k (t) dw j (t) where a,b and its derivatives all evaluated at (S(),). The problem now is to evaluate the iterated Itô integral I jk h W k (t) dw j (t) Stoch. NA, Lecture 3 p. 9

10 Lévy areas Itô calculus gives us d(w j W k ) = W j dw k (t)+w k dw j (t)+σ jk dt where Σ jk is the correlation between dw j and dw k. Hence, W j (h)w k (h) Σ jk h = I kj +I jk If we define the Lévy area to be A jk = I kj I jk = h W j (t) dw k (t) W k (t) dw j (t) then I jk = 1 2 ( Wj (h)w k (h) Σ jk h A jk ) Stoch. NA, Lecture 3 p. 1

11 Lévy areas The problem is that it is hard to simulate the Lévy areas: conditional distribution depends on W j (h) and W k (h) so can t simply invert a cumulative distribution function Lyons & Gaines have an efficient technique in 2-dimensions but for higher dimensions, need to simulate Brownian motion within each timestep to approximate the Lévy area Two notes for later use: E[A jk W] = E[A 2 jk ] = h2 if E[dW j dw k ] =. Stoch. NA, Lecture 3 p. 11

12 Milstein method The Milstein method is Ŝ i,n+1 = Ŝ i,n +a i,n h+ j b ij,n W j,n j,k,l b ij S l b lk,n ( Wj,n W k,n Σ jk h A jk,n ) with A jk,n = tn+1 t n (W j (t) W j (t n ))dw k (t) (W k (t) W k (t n ))dw j (t) The strong error is O(h), but the problem is the Lévy areas. Stoch. NA, Lecture 3 p. 12

13 Milstein method However, using A jk = A kj, j,k,l b ij S l b lk A jk,n = j,k,l b ij S l b lk A kj,n j,k,l b ik = b lj A jk,n S l j,k,l ( = 2 1 bij b lk b ) ik b lj S l S l A jk,n and so the Lévy areas are not needed if, for all i,j,k, l b ij S l b lk b ik S l b lj =. Stoch. NA, Lecture 3 p. 13

14 Milstein method If b is a non-singular diagonal matrix, so each component of S(t) is driven by a separate component of W(t), the commutativity condition reduces to b ij S k b kk b ik S j b jj = if either i=j=k, or i j and i k, this is satisfied if i=j and i k, it requires b ii S k = if i=k and i j, it requires b ii S j = hence, OK provided b ii depends only on S i Stoch. NA, Lecture 3 p. 14

15 Clark & Cameron paper We now come to an important result in the paper: The maximum rate of convergence of discrete approximations for stochastic differential equations, J.M.C. Clark & R.J. Cameron, pp in Lecture Notes in Control and Information Sciences, Volume 25, 198. Their analysis considers a very simple 2-dimensional SDE, and proves that no numerical approximation is capable of better than O(h 1/2 ) strong convergence if it is based solely on the discrete Brownian increments W n. Implication: in general you can t do better than Euler-Maruyama method. Stoch. NA, Lecture 3 p. 15

16 Clark & Cameron paper Their model SDE is dx 1 = dw 1 dx 2 = X 1 dw 2 with X 1 () = X 2 () = and independent dw 1,dW 2 could hardly be simpler! They don t consider a numerical approximation, but instead look at how much we know about the analytic solution given knowledge only of the Brownian increments. Stoch. NA, Lecture 3 p. 16

17 Clark & Cameron paper Over a time interval [,h], can integrate to obtain X 1 (h) = W 1 (h) X 2 = h X 1 dw 2 Now, Itô s lemma gives d(w 1 W 2 ) = W 1 dw 2 +W 2 dw 1 = W 1 (h) W 2 (h) = h (W 1 dw 2 +W 2 dw 1 ) = h W 1 dw 2 = 1 2 W 1(h) W 2 (h)+ 1 2 A where A is Lévy area for time interval [,h]. Stoch. NA, Lecture 3 p. 17

18 Clark & Cameron paper Repeating this for N timesteps of size h = T/N we obtain X 1 (T) = n X 2 (T) = n W 1,n ( ) X 1 (t n ) W 2,n W 1,n W 2,n A n Now recall that E[A n W n ] =, and E[A 2 n] = h 2, and also that V[Y] = E[Y 2 ] (E[Y]) 2 for any random variable Y. Hence, for any numerical approximation X we have... Stoch. NA, Lecture 3 p. 18

19 Clark & Cameron paper E [ (X 2 (T) X2 (T)) 2] [ = E E[(X 2 (T) X2 (T)) 2 W] ] E[ V[X 2 (T) W] ] [ N 1 = 1 4 E n= [ N 1 = 1 4 E = 1 4 [ N 1 n= n= V[A n W] E[A 2 n W] E[A 2 n] ] = 1 4 T h. Stoch. NA, Lecture 3 p. 19 ] ]

20 Clark & Cameron paper Hence, the minimum RMS error is O(h 1/2 ), and the best numerical approximation is the one which gives X 2 (T) = E[X 2 (T) W] which corresponds to neglecting the Lévy areas A n. As well as proving that in general you can t achieve a better order of strong convergence than the Euler-Maruyama method (when using only Brownian increments) it also shows the Milstein approximation without the Lévy areas gives the best asymptotic accuracy. Stoch. NA, Lecture 3 p. 2