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3 3. (stochastic differential equations) { dx(t) =f(t, X)dt + G(t, X)dW (t), t [,T], (3.) X( )=X X(t) : [,T] R d (d ) f(t, X) : [,T] R d R d (drift term) G(t, X) : [,T] R d R d m (diffusion term) W (t) : m Wiener (Wiener process) m Wiener W i (t) (i =,,,m) W (t) =(W (t),w (t),,w m (t)) T Wiener dx(t) =f(t, X)dt + g(t, X)dW (t), t [,T], X( )=X (3.) (3.) (stochastic initial-value problem) (3.) X(t, ω) (stochastic process, random process) t (probability space) (Ω, A,P) (random variable) Ω: (sample space) A: σ (σ algebra) P : (probability measure) 3

t X(t, ) ω X(,ω) (trajectory) E: Ω (mean) X(ω) E(X(ω)) ω X(t) E(X) 953. 978. 3.. Wiener Wiener Wiener Nobert Wiener Brown 88 R. Brown Wiener W (t) (t ) (i) P (W () = ) =, (ii) E(W (t)) = t [, ), (iii) E(W (t)w (s)) = min(t, s) Gauss (normal process) Wiener E(W (t) W (s)) =, E((W (t) W (s)) )=t s, s t E({W (t ) W (t )}{W (t 4 ) W (t 3 )}) =, t t t 3 t 4 (3.3) Wiener (3.3) W (t) W (s) ( s t), t s (normal distribution) N(; t s) Wiener t [,T] h = t k+ t k = T/N : t k (k =,,...,N) Wiener (increment) W k = W (t k+ ) W (t k ) 3

Wiener W (t n ) n W (t n )= W k, n =,,...,N (3.4) k= W k, h N(; h) Wiener, N(; ) (random numbers) ξ k W k = ξ k h Box Muller (Box-Muller method) Marsaglia (Polar Marsaglia method) Wiener 3. (i) (pseudo random numbers) C drand48() (ii) Box Muller (iii) drand48() (seed) 3. 3. Wiener Wiener Wiener W (N) (t, ω) =W (t k,ω)+(w(t k+,ω) W (t k,ω)) t t k, t k+ t k t k t t k+ (3.5) h (N ) W (N) (t, ω) Wiener W (t, ω) Wiener dx(t) =dw (t), X() = (3.4) 3.. (3.) X(t) dx(t) 3 3

.5 -.5 X - -.5 - -.5..4.6.8 t 3.: Wiener (stochastic integral equations) X(t) =X( )+ f(s, X(s)) ds + (stochastic integral) k= g(s, X(s)) dw (s) (3.6) n g(s, X(s)) dw (s) = lim g (t k,λx(t k+ )+( λ)x(t k )) W k (3.7) t h λ λ {t k } <t < <t k <t k+ < <t n = t W k = W (t k+ ) W (t k ),h =max(t k+ t k ) λ λ =, (3.7) (Itô-type) λ =/ Stratonovich (Stratonovich-type) λ = λ =/ (3.) Stratonovich S 3 4

S g(s, X(s)) dw (s) g(s, X(s)) dw (s) S S dx(t) =f(t, X)dt + g(t, X)dW (t), dx(t) = f(t, X)dt + g(t, X) dw (t) (diffusion process) martingale S martingale S b a g(s, X(s)) dw (s) = b a g(s, X(s)) dw (s)+ S b dx = f(t, X)dt + g(t, X) dw (t) [ dx = f + ] g x g (t, X)dt + g(t, X)dW (t) dx = f(t, X)dt + g(t, X)dW (t) S [ dx = f ] g x g (t, X)dt + g(t, X) dw (t) a [ ] g x g (s, X(s)) ds. (3.8) 3 5

3. 3.. f(t, x) g(t, x) (t, x) [,T] R. t [,T], x, y R K (a) f(t, x) f(t, y) + g(t, x) g(t, y) K x y. (b) f(t, x) + g(t, x) K ( + x ). 3. X W (t)(t >) E((X ) ). (3.6) X(t) [,T] sup [t,t ] E((X(t)) ) X = {X(t),t [,T]} Y = {Y (t),t [,T]} ( ) P sup X(t) Y (t) = [,T ] = (Itô formula) 3. φ(t, x) [,T] R X(t) φ t (t, x) =φ t, φ x (t, x) =φ x, dx(t) =f(t)dt + g(t)dw (t) φ x (t, x) =φ xx [,T] Y (t) =φ(t, X(t)) [,T] dy (t) =[φ t + fφ x + g φ xx ](t, X(t)) dt +[gφ x ](t, X(t)) dw (t) (3.9) L f = t + f x + g x, L g = g x 3 6

(3.9) dy (t) =[L f φ](t, X(t)) dt +[L g φ](t, X(t)) dw (t) (3.) t [,T] Y (t) =Y ( )+ [L f φ](s, X(s)) ds + [L g φ](s, X(s)) dw (s) (3.) 3..3 3. ( logistic ) logistic dx dt =[α βx(t)]x(t) (3.) x x /x x α βx α, β α dx(t) =X(α βx)dt + σx dw (t) (3.3) W (t) Wiener σ σ 3. 3.6 α =,β= σ =.,.5,.,.5,. X() = h = 4 Euler σ (3.) g(t, x) (i) x g(t) (additive noise) (ii) (multiplicative noise) Langevin 3. dx = axdt + bxdw (t) (3.4) a, b (3.4) Brown (geometric Brownian motion) 3 7

4 3 X - 5 5 t 3.: logistic (σ =.,.) 4 3 X - 5 5 t 3.3: logistic (σ =.5) 4 3 X - 5 5 t 3.4: logistic (σ =.) 3 8

4 3 X - 5 5 t 3.5: logistic (σ =.5) 4 3 X - 5 5 t 3.6: logistic (σ =.) X( )=X (3.4) ( X(t) =X exp (a ) b )(t )+b(w(t) W ( )) (3.5) Runge-Kutta 3. Euler (Euler-Maruyama scheme, EM scheme) Gisiro Maruyama, Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo (), 4 (955), 48 9 3 9

3.. EM (3.) 3.. Riemann (3.7) λ = EM {t n } (t n = + nh, T = t N ): h [t n,t n+ ] X n X n+ X n+ = X n + f(t n, X n )h + g(t n, X n ) W n (3.6) h = t n+ t n : g(t, x) EM (3.6) Euler Euler Wiener Wiener 3.. EM 3.3 (EM ) Brown 3. a = b = { dx(t) =X dt + X dw (t), t [,.], (3.7) X() =. EM (3.5) ( ) X(t) =exp t + W (t) (3.8) h = 3 EM T = 3.7 3.. (strong approximation) (pathwise approximation): (weak approximation): 3. X n α h h K e(h) =E( X(T ) X N ) Kh α (3.9) 3

.8.6.4. X.8.6.4...4.6.8 t 3.7: EM e(h): 3. X n β h h ϕ K E(ϕ(X(T ))) E(ϕ(X N )) Kh β (3.) 3..3 EM (3.) 3.. -Taylor (3.) X(t) =X( )+ f(s, X(s))ds + f(s, X),g(s, X) 3 g(s, X(s))dW (s) (3.)

(3.) f(s, X(s)) g(s, X(s)) (3.) f(s, X(s)) = f(,x( )) + g(s, X(s)) = g(,x( )) + (3.) X( )+f(,x( )) [L f f](r, X(r)) dr + [L g f](r, X(r)) dw (r), [L f g](r, X(r)) dr + [L g g](r, X(r)) dw (r) ds + g(,x( )) dw (s) + [L f f](r, X(r)) dr ds + [L g f](r, X(r)) dw (r)ds (3.) + [L f g](r, X(r)) dr dw (s)+ [L g g](r, X(r)) dw (r)dw (s) R X(t) X(t) X(t) =X( )+f(x( )) t t = + h ds + g(x( )) dw (s) X(t )=X( )+f(x( ))h + g(x( )){W (t ) W ( )} {W (t ) W ( )} W EM R E( R )=O(h ) + h, +h,..., + Nh E( X(T ) X N ) Kh (3.3) EM / -Taylor EM (3.) L g g L g g(s, X(s)) = L g g(,x( )) + (3.) X( )+f(,x( )) [L f L g g](r, X(r)) dr + ds + g(,x( )) +[L g g](,x( )) dw (r)dw(s)+r 3 [L g L g g](r, X(r)) dw (r) dw (s) (3.4)

R R = [L f f](r, X(r)) dr ds + + [L f g](r, X(r)) dr dw (s)+ [L g f](r, X(r)) dw (r)ds r [L f L g g](z, X(z)) dz dw (r)dw (s) R r + [L g L g g](z, X(z)) dw (z)dw(r)dw(s) E( R )=O(h 3 ) (3.5) (3.4) EM EM [L g g](,x( )) dw (r)dw (s) (3.6) (3.6) [ ] g x g (,X( )) (W (t) W ()) (t ) (3.7) G.N. Milstein Milstein [ ] g X n+ = X n + f(t n, X n )h + g(t n, X n ) W n + x g (t n, X n ) ( W n) h (3.8) E( X(T ) X N ) Kh (3.9) Milstein e(h) (3.7) EM Milstein (3.7) ( ) X(t) =exp t + W (t) (3.3) Wiener W (t) 3 3

h 3 4 5 EM.7 5.8 3.67 Milstein 4.77.44 6.86 3.: 3.. (3.3) W (t) Wiener W (N) (t) ( ) X (N) (t) =exp t + W (N)(t) (3.3) (realized exact solution) E( X(T ) Y N ) E((X(T ) Y N ) ) error = M M (X(N) i (t) Xi N ) (3.3) i= X(N) i (t), Xi N : i 3.: h = 3, 4, 5 T = EM Milstein Milstein EM 3.8: log h log (error) EM.4, Milstein.7 3..4 -Taylor 3. EM X n+ = X n + f(t n, X n )h + g(t n, X n ) W n 3 4

.5 Euler Milstein -.5 log error - -.5 - -.5-5 -4-3 log h 3.8: -Taylor Taylor [ ] g X n+ = X n + f(t n, X n )h + g(t n, X n ) W n + x g (t n, X n ) ( W n) h [ ] [ f f + x g (t n, X n ) Z n + x f + ] f x g (t n, X n ) h [ g + x f + ] g x g (t n, X n ){ W n h Z n } (3.33) Z n Z n = n+ t n t n dw (r)ds (3.34) Gauss W n E( W n )=E( Z n )=, E(( W n ) )=h, E(( Z n ) )= h3 3, E( W n Z n )= h (3.35) Wiener W n ξ n W n = ξ n h (3.35) Z n ξ n ξ n Z n = h 3 ( ξ n + 3 ξn ) (3.36) 3 5

t = t n µ N µ = N k= X n,k N: X n,k : t = t n k µ M M (batch) j µ j ˆµ ˆσ µ j = N N X n,k,j, k= j =,,...,M ˆµ = M M j= µ j, ˆσ = M M (µ j ˆµ) j= M Student t ( α)% µ ( ) ˆσ ˆσ ˆµ t α,m M, ˆµ + t α,m M t α,m 3.4 ( ) { dx(t) =X dt + X dw (t), t [,.], (3.37) X() =. E(X(t)) = e t h = 3 T = M = N = 9%.9,9 =.73 EM 3.9 Taylor 3. EM Taylor 3 6

3.5 E(X).5.5..4.6.8 t 3.9: EM 3.5 E(X).5.5..4.6.8 t 3.: 3 7

EM Wiener W n Ŵn E( Ŵn) =E(( Ŵn) 3 )=, E(( Ŵn) )=h. (3.38) (two-point distributed random variable) P ( Ŵn = ± h)= (3.39) Ŵn = h(u n /) (3.4) U n [, ) (simplified scheme) X n+ = X n + f(t n, X n )h + g(t n, X n ) Ŵn (3.4) Taylor (3.33) X n+ = X n + f(t n, X n )h + g(t n, X n ) Ŵn + [ f + ] (t n, X n ) h Ŵn x g [ g + x f + g x g ] + [ ] g x g (t n, X n ) ( Ŵn) h [ f x f + ] f x g (t n, X n ) h (t n, X n ) h Ŵn Ŵn (3.4) E( Ŵn) =E(( Ŵn) 3 )=E(( Ŵn) 5 )=, E(( Ŵn) )=h, E(( Ŵn) 4 )=3h. (3.43) Ŵn (three-point distributed random variable) P ( Ŵn = ± 3h) = 6, P( Ŵn =)= 3 (3.44) 3.3.. SDEs 3. 4. 3 8