No caso determinístico, de uma equação diferencial
d x d t = f ( t , x ) ,
\frac{\mathrm{d}x}{\mathrm{d}t} = f(t, x),
d t d x = f ( t , x ) ,
com condição inicial x ( 0 ) = x 0 , x(0) = x_0, x ( 0 ) = x 0 , o método de Euler
x j n = x j − 1 n + Δ t f ( t j − 1 x j − 1 n ) , x j n ∣ j = 0 = x 0 ,
x_{j}^n = x_{j-1}^n + \Delta t f(t_{j-1} x_{j-1}^n), \qquad x_j^n|_{j = 0} = x_0,
x j n = x j − 1 n + Δ t f ( t j − 1 x j − 1 n ) , x j n ∣ j = 0 = x 0 ,
em uma malha temporal uniforme t j = j T / n , t_j = jT/n, t j = j T / n , j = 0 , … , n , j = 0, \ldots, n, j = 0 , … , n , com Δ t = T / n , \Delta t = T/n, Δ t = T / n , converge uniformemente, no intervalo [ 0 , T ] , [0, T], [ 0 , T ] , para a solução do problema de valor inicial. Além disso, essa convergência é de ordem um. Mais precisamente, existem C > 0 C > 0 C > 0 e δ > 0 \delta > 0 δ > 0 tais que
max j ∣ x ( t j ) − x j ∣ ≤ C Δ t , 0 < Δ t ≤ δ .
\max_{j}|x(t_j) - x_j| \leq C \Delta t, \qquad 0 < \Delta t \leq \delta.
j max ∣ x ( t j ) − x j ∣ ≤ C Δ t , 0 < Δ t ≤ δ .
Isso sob a hipótese de f f f ser localmente Lipschitz contínuas.
Por outro lado, no caso estocástico, com um ruído multiplicativo g = g ( t , X t ) , g = g(t, X_t), g = g ( t , X t ) ,
d X t = f ( t , X t ) d t + g ( t , X t ) d W t , t ≥ 0 ,
\mathrm{d}X_t = f(t, X_t)\mathrm{d}t + g(t, X_t)\mathrm{d}W_t, \qquad t \geq 0,
d X t = f ( t , X t ) d t + g ( t , X t ) d W t , t ≥ 0 ,
com uma condição inicial
X t ∣ t = 0 = X 0 ,
\left.X_t\right|_{t = 0} = X_0,
X t ∣ t = 0 = X 0 ,
a convergência forte é apenas de ordem 1 / 2 1/2 1/2 e isso sob a hipótese mais exigente de f f f e g g g serem globalmente Lipschitz contínuas. Mas é importante ressaltar que isso acontece no caso multiplicativo. Se o ruído for aditivo, g = g ( t , X t ) = g ( t ) , g = g(t, X_t) = g(t), g = g ( t , X t ) = g ( t ) , então ainda temos a convergência forte de ordem 1. 1. 1.
A diferença, no caso multiplicativo, vem, essencialmente, do fato de que, na equação estocástica, além dos termos de erro da ordem de Δ t , \Delta t, Δ t , há termos da ordem de Δ W . \Delta W. Δ W . Em um sentido apropriado, vale ( Δ W ) 2 ∼ Δ t , (\Delta W)^2 \sim \Delta t, ( Δ W ) 2 ∼ Δ t , o que nos dá um erro da ordem de ( Δ t ) 1 / 2 . (\Delta t)^{1/2}. ( Δ t ) 1/2 .
Outro ponto importante é que, no caso discreto, a constante C C C que aparece na ordem de convergência depende da condição inicial e explora o fato de que, com a condição inicial fixa, podemos limitar a solução exata e a aproximação. Por outro lado, no caso estocástico, considera-se, implicitamente, diversas condições iniciais X 0 ( ω ) X_0(\omega) X 0 ( ω ) e não temos esse controle, por isso a necessidade de se assumir que os termos f f f e g g g sejam globalmente Lipschitz contínuos. Esse problema aparece mesmo no caso de ruído aditivo e apenas f f f não globalmente Lipschitz.
Por último, um ponto um pouco mais técnico, é que, enquanto no caso discreto estimamos diretamente a diferença ∣ x ( t j ) − x j n ∣ , |x(t_j) - x_j^n|, ∣ x ( t j ) − x j n ∣ , no caso estocástico precisamos nos ancorar na isometria de Itô, de modo que o mais natural é olharmos para E [ ∣ X t j − X j n ∣ 2 ] . \mathbb{E}\left[|X_{t_j} - X_j^n|^2 \right]. E [ ∣ X t j − X j n ∣ 2 ] .
Em resumo, a hipótese de continuidade Lipschitz global é para garantir que o método convirja. E a presença de d W t ∼ d t \mathrm{d}W_t \sim \sqrt{\mathrm{d}t} d W t ∼ d t nos dá uma convergência forte apenas de ordem 1 / 2 , 1/2, 1/2 , no caso multiplicativo. Vejamos os detalhes.
Primeiramente, temos que
x ( t j ) = x ( t j − 1 ) + ∫ t j − 1 t j x ′ ( s ) d s = x ( t j − 1 ) + Δ t x ′ ( t j − 1 ) + ∫ t j − 1 t j ( x ′ ( s ) − x ′ ( t j − 1 ) ) d s .
x(t_j) = x(t_{j-1}) + \int_{t_{j-1}}^{t_j} x'(s)\;\mathrm{d}s = x(t_{j-1}) + \Delta t x'(t_{j-1}) + \int_{t_{j-1}}^{t_j} (x'(s) - x'(t_{j-1}))\;\mathrm{d}s.
x ( t j ) = x ( t j − 1 ) + ∫ t j − 1 t j x ′ ( s ) d s = x ( t j − 1 ) + Δ t x ′ ( t j − 1 ) + ∫ t j − 1 t j ( x ′ ( s ) − x ′ ( t j − 1 )) d s .
De acordo com a equação diferencial,
x ( t j ) = x ( t j − 1 ) + Δ t f ( t j − 1 , x ( t j − 1 ) ) + ∫ t j − 1 t j ( f ( s , x ( s ) ) − f ( t j − 1 , x ( t j − 1 ) ) ) d s .
x(t_j) = x(t_{j-1}) + \Delta t f(t_{j-1}, x(t_{j-1})) + \int_{t_{j-1}}^{t_j} (f(s, x(s)) - f(t_{j-1}, x(t_{j-1})))\;\mathrm{d}s.
x ( t j ) = x ( t j − 1 ) + Δ t f ( t j − 1 , x ( t j − 1 )) + ∫ t j − 1 t j ( f ( s , x ( s )) − f ( t j − 1 , x ( t j − 1 ))) d s .
Assim, nos pontos j = 1 , … , n j = 1, \ldots, n j = 1 , … , n da malha,
∣ x ( t j ) − x j ∣ ≤ ∣ x ( t j − 1 ) − x j − 1 ∣ + Δ t ∣ f ( t j − 1 , x ( t j − 1 ) ) − f ( t j − 1 , x j − 1 ) ∣ + ∫ t j − 1 t j ∣ f ( s , x ( s ) ) − f ( t j − 1 , x ( t j − 1 ) ) ∣ d s .
\begin{align*}
|x(t_j) - x_j| & \leq | x(t_{j-1}) - x_{j-1} | + \Delta t |f(t_{j-1}, x(t_{j-1})) - f(t_{j-1}, x_{j-1})| \\
& \quad + \int_{t_{j-1}}^{t_j} |f(s, x(s)) - f(t_{j-1}, x(t_{j-1}))|\;\mathrm{d}s.
\end{align*}
∣ x ( t j ) − x j ∣ ≤ ∣ x ( t j − 1 ) − x j − 1 ∣ + Δ t ∣ f ( t j − 1 , x ( t j − 1 )) − f ( t j − 1 , x j − 1 ) ∣ + ∫ t j − 1 t j ∣ f ( s , x ( s )) − f ( t j − 1 , x ( t j − 1 )) ∣ d s .
Como a solução é contínua, ela é limitada no intervalo [ 0 , T ] , [0, T], [ 0 , T ] , i.e.
R 0 = max 0 ≤ t ≤ T ∣ x ( t ) ∣ < ∞ .
R_0 = \max_{0\leq t \leq T}|x(t)| < \infty.
R 0 = 0 ≤ t ≤ T max ∣ x ( t ) ∣ < ∞.
Seja R > R 0 R > R_0 R > R 0 e considere as constantes de Lipschitz L 1 = L 1 ( R ) L_1 = L_1(R) L 1 = L 1 ( R ) e L 2 = L 2 ( R ) L_2 = L_2(R) L 2 = L 2 ( R ) tais que
∣ f ( t , x ) − f ( s , y ) ∣ ≤ L 1 ( R ) ∣ t − s ∣ + L 2 ( R ) ∣ x − y ∣ , ∀ 0 ≤ t , s ≤ T , ∀ x , y , ∣ x ∣ , ∣ y ∣ ≤ R .
|f(t, x) - f(s, y)| \leq L_1(R)|t - s| + L_2(R)|x - y|, \quad \forall 0 \leq t, s \leq T, \;\forall x, y, \; |x|, |y| \leq R.
∣ f ( t , x ) − f ( s , y ) ∣ ≤ L 1 ( R ) ∣ t − s ∣ + L 2 ( R ) ∣ x − y ∣ , ∀0 ≤ t , s ≤ T , ∀ x , y , ∣ x ∣ , ∣ y ∣ ≤ R .
Assuma, por indução, que ∣ x j − 1 ∣ ≤ R . |x_{j-1}| \leq R. ∣ x j − 1 ∣ ≤ R . Com isso,
∣ x ( t j ) − x j ∣ ≤ ∣ x ( t j − 1 ) − x j − 1 ∣ + L 2 Δ t ∣ x ( t j − 1 ) − x j − 1 ∣ + ∫ t j − 1 t j ( L 1 ∣ s − t j − 1 ∣ + L 2 ∣ x ( s ) − x ( t j − 1 ) ∣ ) d s ≤ ∣ x ( t j − 1 ) − x j − 1 ∣ + L 2 Δ t ∣ x ( t j − 1 ) − x j − 1 ∣ + L 1 2 ∣ t j − t j − 1 ∣ 2 + L 2 ∫ t j − 1 t j ∣ x ( s ) − x ( t j − 1 ) ∣ d s .
\begin{align*}
|x(t_j) - x_j| & \leq |x(t_{j-1}) - x_{j-1}| + L_2 \Delta t |x(t_{j-1}) - x_{j-1}| \\
& \quad +\int_{t_{j-1}}^{t_j} \left( L_1 |s - t_{j-1}| + L_2 |x(s) - x(t_{j-1})|\right)\;\mathrm{d}s \\
& \leq |x(t_{j-1}) - x_{j-1}| + L_2 \Delta t |x(t_{j-1}) - x_{j-1}| \\
& \quad + \frac{L_1}{2}|t_j - t_{j-1}|^2 + L_2 \int_{t_{j-1}}^{t_j}|x(s) - x(t_{j-1})|\;\mathrm{d}s.
\end{align*}
∣ x ( t j ) − x j ∣ ≤ ∣ x ( t j − 1 ) − x j − 1 ∣ + L 2 Δ t ∣ x ( t j − 1 ) − x j − 1 ∣ + ∫ t j − 1 t j ( L 1 ∣ s − t j − 1 ∣ + L 2 ∣ x ( s ) − x ( t j − 1 ) ∣ ) d s ≤ ∣ x ( t j − 1 ) − x j − 1 ∣ + L 2 Δ t ∣ x ( t j − 1 ) − x j − 1 ∣ + 2 L 1 ∣ t j − t j − 1 ∣ 2 + L 2 ∫ t j − 1 t j ∣ x ( s ) − x ( t j − 1 ) ∣ d s .
O integrando do último termo pode ser estimado por
∫ t j − 1 t j ∣ x ( s ) − x ( t j − 1 ) ∣ d s ≤ ∫ t j − 1 t j ∫ t j − 1 s ∣ x ′ ( τ ) ∣ d τ d s ≤ ∫ t j − 1 t j ∫ t j − 1 s ∣ f ( τ , x ( τ ) ) ∣ d τ d s + ∫ t j − 1 t j ∫ t j − 1 s ( ∣ f ( τ , x ( τ ) ) − f ( τ , 0 ) ∣ + ∣ f ( τ , 0 ) ∣ ) d τ d s + ∫ t j − 1 t j ∫ t j − 1 s ( L 2 ∣ x ( τ ) ∣ + ∣ f ( τ , 0 ) ∣ ) d τ d s ≤ ( L 1 R 0 + C 0 ) Δ t 2 ,
\begin{align*}
\int_{t_{j-1}}^{t_j} |x(s) - x(t_{j-1})|\;\mathrm{d}s & \leq \int_{t_{j-1}}^{t_j} \int_{t_{j-1}}^s |x'(\tau)|\;\mathrm{d}\tau\;\mathrm{d}s \\
& \leq \int_{t_{j-1}}^{t_j} \int_{t_{j-1}}^s |f(\tau, x(\tau))|\;\mathrm{d}\tau\;\mathrm{d}s \\
& \quad + \int_{t_{j-1}}^{t_j} \int_{t_{j-1}}^s \left(|f(\tau, x(\tau)) - f(\tau, 0)| + |f(\tau, 0)|\right)\;\mathrm{d}\tau\;\mathrm{d}s \\
& \quad + \int_{t_{j-1}}^{t_j} \int_{t_{j-1}}^s \left(L_2|x(\tau)| + |f(\tau, 0)|\right) \;\mathrm{d}\tau\;\mathrm{d}s \\
& \leq (L_1 R_0 + C_0) \Delta t^2,
\end{align*}
∫ t j − 1 t j ∣ x ( s ) − x ( t j − 1 ) ∣ d s ≤ ∫ t j − 1 t j ∫ t j − 1 s ∣ x ′ ( τ ) ∣ d τ d s ≤ ∫ t j − 1 t j ∫ t j − 1 s ∣ f ( τ , x ( τ )) ∣ d τ d s + ∫ t j − 1 t j ∫ t j − 1 s ( ∣ f ( τ , x ( τ )) − f ( τ , 0 ) ∣ + ∣ f ( τ , 0 ) ∣ ) d τ d s + ∫ t j − 1 t j ∫ t j − 1 s ( L 2 ∣ x ( τ ) ∣ + ∣ f ( τ , 0 ) ∣ ) d τ d s ≤ ( L 1 R 0 + C 0 ) Δ t 2 ,
onde
C 0 = max 0 ≤ t ≤ T ∣ f ( τ , 0 ) ∣ .
C_0 = \max_{0 \leq t \leq T} |f(\tau, 0)|.
C 0 = 0 ≤ t ≤ T max ∣ f ( τ , 0 ) ∣.
Assim,
∣ x ( t j ) − x j ∣ ≤ ( 1 + L 2 Δ t ) ∣ x ( t j − 1 ) − x j − 1 ∣ + M Δ t 2 ,
|x(t_j) - x_j| \leq (1 + L_2\Delta t)|x(t_{j-1}) - x_{j-1}| + M \Delta t^2,
∣ x ( t j ) − x j ∣ ≤ ( 1 + L 2 Δ t ) ∣ x ( t j − 1 ) − x j − 1 ∣ + M Δ t 2 ,
onde
M = L 1 2 + L 1 R 0 + C 0 .
M = \frac{L_1}{2} + L_1 R_0 + C_0.
M = 2 L 1 + L 1 R 0 + C 0 .
Iterando essa estimativa, chegamos a
∣ x ( t j ) − x j ∣ ≤ ( 1 + L 2 Δ t ) 2 ∣ x ( t j − 2 ) − x j − 2 ∣ + M Δ t 2 ( 1 + ( 1 + L Δ t ) ) ≤ … ≤ ( 1 + L 2 Δ t ) j ∣ x ( t 0 ) − x 0 ∣ + M Δ t 2 ( 1 + ( 1 + L 2 Δ t ) + … + ( 1 + L 2 Δ t ) j − 1 ) .
\begin{align*}
|x(t_j) - x_j| & \leq (1 + L_2\Delta t)^2|x(t_{j-2}) - x_{j-2}| + M \Delta t^2(1 + (1 + L\Delta t)) \\
& \leq \ldots \\
& \leq (1 + L_2\Delta t)^j|x(t_0) - x_0| + M \Delta t^2(1 + (1 + L_2\Delta t) + \ldots + (1 + L_2\Delta t)^{j-1}).
\end{align*}
∣ x ( t j ) − x j ∣ ≤ ( 1 + L 2 Δ t ) 2 ∣ x ( t j − 2 ) − x j − 2 ∣ + M Δ t 2 ( 1 + ( 1 + L Δ t )) ≤ … ≤ ( 1 + L 2 Δ t ) j ∣ x ( t 0 ) − x 0 ∣ + M Δ t 2 ( 1 + ( 1 + L 2 Δ t ) + … + ( 1 + L 2 Δ t ) j − 1 ) .
Usando que 1 + a ≤ e a , 1 + a \leq e^a, 1 + a ≤ e a , para todo a ≥ 0 , a \geq 0, a ≥ 0 , temos
( 1 + L 2 Δ t ) j ≤ e L 2 j Δ t = e L 2 t j .
(1 + L_2\Delta t)^j \leq e^{L_2j\Delta t} = e^{L_2 t_j}.
( 1 + L 2 Δ t ) j ≤ e L 2 j Δ t = e L 2 t j .
Além disso,
1 + ( 1 + L 2 Δ t ) + … + ( 1 + L 2 Δ t ) j − 1 = ( 1 + L 2 Δ t ) j − 1 ( 1 + L 2 Δ t ) − 1 = 1 L 2 Δ t ( 1 + L 2 Δ t ) j ≤ 1 L 2 Δ t e L 2 t j .
1 + (1 + L_2\Delta t) + \ldots + (1 + L_2\Delta t)^{j-1} = \frac{(1 + L_2\Delta t)^j - 1}{(1 + L_2\Delta t) - 1} = \frac{1}{L_2\Delta t}(1 + L_2\Delta t)^j \leq \frac{1}{L_2\Delta t}e^{L_2 t_j}.
1 + ( 1 + L 2 Δ t ) + … + ( 1 + L 2 Δ t ) j − 1 = ( 1 + L 2 Δ t ) − 1 ( 1 + L 2 Δ t ) j − 1 = L 2 Δ t 1 ( 1 + L 2 Δ t ) j ≤ L 2 Δ t 1 e L 2 t j .
Portanto,
∣ x ( t j ) − x j ∣ ≤ e L 2 T ∣ x ( t 0 ) − x 0 ∣ + M L 2 e L 2 T Δ t .
|x(t_j) - x_j| \leq e^{L_2T}|x(t_0) - x_0| + \frac{M}{L_2}e^{L_2T}\Delta t.
∣ x ( t j ) − x j ∣ ≤ e L 2 T ∣ x ( t 0 ) − x 0 ∣ + L 2 M e L 2 T Δ t .
Considerando que x 0 = x ( t 0 ) , x_0 = x(t_0), x 0 = x ( t 0 ) , obtemos
∣ x ( t j ) − x j ∣ ≤ M L 2 e L 2 T Δ t .
|x(t_j) - x_j| \leq \frac{M}{L_2}e^{L_2T}\Delta t.
∣ x ( t j ) − x j ∣ ≤ L 2 M e L 2 T Δ t .
Lembrando que L 2 = L 2 ( R ) , L_2=L_2(R), L 2 = L 2 ( R ) , para Δ t \Delta t Δ t suficientemente pequeno tal que
M L 2 ( R ) e L 2 ( R ) T Δ t ≤ R − R 0 ,
\frac{M}{L_2(R)}e^{L_2(R)T}\Delta t \leq R - R_0,
L 2 ( R ) M e L 2 ( R ) T Δ t ≤ R − R 0 ,
podemos garantir que
∣ x j ∣ ≤ R ,
|x_j| \leq R,
∣ x j ∣ ≤ R ,
obtendo, por indução, que
max j = 0 , … , n ∣ x j n ∣ ≤ R , max j = 0 , … , n ∣ x ( t j ) − x j ∣ ≤ M L 2 e L 2 T Δ t ,
\max_{j=0, \ldots, n}|x_j^n| \leq R, \qquad \max_{j=0, \ldots, n} |x(t_j) - x_j| \leq \frac{M}{L_2}e^{L_2T}\Delta t,
j = 0 , … , n max ∣ x j n ∣ ≤ R , j = 0 , … , n max ∣ x ( t j ) − x j ∣ ≤ L 2 M e L 2 T Δ t ,
mostrando que o método de Euler é de primeira ordem.
Considere, agora, a equação estocástica
d X t = f ( t , X t ) d t + g ( t , X t ) d W t , t ≥ 0 ,
\mathrm{d}X_t = f(t, X_t)\mathrm{d}t + g(t, X_t)\mathrm{d}W_t, \qquad t \geq 0,
d X t = f ( t , X t ) d t + g ( t , X t ) d W t , t ≥ 0 ,
com uma condição inicial
X t ∣ t = 0 = X 0 .
\left.X_t\right|_{t = 0} = X_0.
X t ∣ t = 0 = X 0 .
Nesse caso, temos
X t = X 0 + ∫ 0 t f ( s , X s ) d s + ∫ 0 t g ( s , X s ) d W s .
X_t = X_0 + \int_0^t f(s, X_s)\;\mathrm{d}s + \int_0^t g(s, X_s)\;\mathrm{d}W_s.
X t = X 0 + ∫ 0 t f ( s , X s ) d s + ∫ 0 t g ( s , X s ) d W s .
Já a aproximação pelo método de Euler-Maruyama é dada por
X j n = X j − 1 n + f ( t j − 1 , X j − 1 n ) Δ t + g ( t j − 1 , X j − 1 n ) Δ W j ,
X_j^n = X_{j-1}^n + f(t_{j-1}, X_{j-1}^n) \Delta t + g(t_{j-1}, X_{j-1}^n) \Delta W_j,
X j n = X j − 1 n + f ( t j − 1 , X j − 1 n ) Δ t + g ( t j − 1 , X j − 1 n ) Δ W j ,
onde X 0 n = X 0 X_0^n = X_0 X 0 n = X 0 e Δ W j . \Delta W_j. Δ W j .
Assumimos f f f e g g g globalmente Lipschitz contínuas em x x x e globalmente Hölder contínuas em t . t. t . Mais precisamente, assumimos que
∣ f ( t , x ) − f ( t , y ) ∣ ≤ L f ∣ x − y ∣
|f(t, x) - f(t, y)| \leq L_f|x - y|
∣ f ( t , x ) − f ( t , y ) ∣ ≤ L f ∣ x − y ∣
e
∣ f ( t , x ) − f ( s , x ) ∣ ≤ H f ( 1 + ∣ x ∣ ) ∣ t − s ∣ 1 / 2 , ∣ g ( t ) − g ( s ) ∣ ≤ H g ( 1 + ∣ x ∣ ) ∣ t − s ∣ 1 / 2 ,
|f(t, x) - f(s, x)| \leq H_f(1 + |x|)|t - s|^{1/2}, \quad |g(t) - g(s)| \leq H_g(1 + |x|)|t - s|^{1/2},
∣ f ( t , x ) − f ( s , x ) ∣ ≤ H f ( 1 + ∣ x ∣ ) ∣ t − s ∣ 1/2 , ∣ g ( t ) − g ( s ) ∣ ≤ H g ( 1 + ∣ x ∣ ) ∣ t − s ∣ 1/2 ,
para x , y ∈ R x, y\in\mathbb{R} x , y ∈ R e 0 ≤ t , s ≤ T , 0\leq t, s \leq T, 0 ≤ t , s ≤ T , onde H f , H_f, H f , L f , L_f, L f , H g , H_g, H g , L g > 0 L_g > 0 L g > 0 são constantes apropriadas.
Para uma estimativa adequada do termo estocástico, precisamos da isometria de Itô, e para isso precisamos trabalhar com a média quadrática. Mais precisamente, vamos estimar
max i = 0 , … , n E [ ∣ X t i − X i n ∣ 2 ] .
\max_{i = 0, \ldots, n} \mathbb{E}\left[ |X_{t_i} - X_i^n|^2\right].
i = 0 , … , n max E [ ∣ X t i − X i n ∣ 2 ] .
Em relação à média quadrática, lembremos das estimativas
E [ X t 2 ] ≤ M T ,
\mathbb{E}\left[X_t^2\right] \leq M_T,
E [ X t 2 ] ≤ M T ,
e
E [ ∣ X t + τ − X t ∣ 2 ] ≤ C T 2 τ ,
\mathbb{E}\left[ |X_{t+\tau} - X_t|^2\right] \leq C_T^2\tau,
E [ ∣ X t + τ − X t ∣ 2 ] ≤ C T 2 τ ,
para 0 ≤ t ≤ t + τ ≤ T , 0\leq t \leq t + \tau \leq T, 0 ≤ t ≤ t + τ ≤ T , para constantes apropriadas C T , M T > 0. C_T, M_T > 0. C T , M T > 0.
Agora, por conta também da necessidade de trabalharmos com a média quadrática, devemos considerar uma expressão global para o erro, escrevendo
X t j = X 0 + ∫ 0 t j f ( s , X s ) d s + ∫ 0 t j g ( s , X s ) d W s
X_{t_j} = X_0 + \int_0^{t_j} f(s, X_s)\;\mathrm{d}s + \int_0^{t_j} g(s, X_s)\;\mathrm{d}W_s
X t j = X 0 + ∫ 0 t j f ( s , X s ) d s + ∫ 0 t j g ( s , X s ) d W s
e
X j n = X 0 + ∑ i = 1 j f ( t i − 1 , X i − 1 n ) Δ t i − 1 + ∑ i = 1 j g ( t i − 1 , X i − 1 n ) Δ W i − 1 .
X_j^n = X_0 + \sum_{i=1}^j f(t_{i-1}, X_{i-1}^n)\Delta t_{i-1} + \sum_{i=1}^j g(t_{i-1}, X_{i-1}^n)\Delta W_{i-1}.
X j n = X 0 + i = 1 ∑ j f ( t i − 1 , X i − 1 n ) Δ t i − 1 + i = 1 ∑ j g ( t i − 1 , X i − 1 n ) Δ W i − 1 .
Não funciona estimarmos de maneira recursiva, pois, por conta da desigualdade ( a 1 + ⋯ + a k ) 2 ≤ k ( a 1 2 + ⋯ + a k 2 ) , (a_1 + \cdots + a_k)^2 \leq k(a_1^2 + \cdots + a_k^2), ( a 1 + ⋯ + a k ) 2 ≤ k ( a 1 2 + ⋯ + a k 2 ) , teríamos algo do tipo d j ≤ C 1 d j − 1 + C 0 , d_j \leq C_1d_{j-1} + C_0, d j ≤ C 1 d j − 1 + C 0 , com C > 1 , C>1, C > 1 , de forma que as iterações nos dariam um termo acumulado C j , C^j, C j , que explode à medida que a malha é refinada, pois não está acompanhado do passo de tempo Δ t . \Delta t. Δ t .
Assim, escrevendo o erro de t = 0 t=0 t = 0 a t = t j , t=t_j, t = t j , temos
X t j − X j n = ∫ 0 t j f ( s , X s ) d s + ∫ 0 t j g ( s , X s ) d W s − ∑ i = 1 j f ( t i − 1 , X i − 1 n ) Δ t i − 1 − ∑ i = 1 j g ( t i − 1 , X i − 1 n ) Δ W i − 1 .
X_{t_j} - X_j^n = \int_0^{t_j} f(s, X_s)\;\mathrm{d}s + \int_0^{t_j} g(s, X_s)\;\mathrm{d}W_s - \sum_{i=1}^j f(t_{i-1}, X_{i-1}^n) \Delta t_{i-1} - \sum_{i=1}^j g(t_{i-1}, X_{i-1}^n) \Delta W_{i-1}.
X t j − X j n = ∫ 0 t j f ( s , X s ) d s + ∫ 0 t j g ( s , X s ) d W s − i = 1 ∑ j f ( t i − 1 , X i − 1 n ) Δ t i − 1 − i = 1 ∑ j g ( t i − 1 , X i − 1 n ) Δ W i − 1 .
Podemos escrever isso na forma
X t j − X j n = ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) ) d s + ∫ 0 t j ( g ( s , X s ) − g ( t i n ( s ) , X t n ( s ) ) ) d W s + ∑ i = 1 j ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n ) ) Δ t i − 1 + ∑ i = 1 j ( g ( t i − 1 , X t i − 1 ) − g ( t i − 1 , X i − 1 n ) ) Δ W i − 1 ,
\begin{align*}
X_{t_j} - X_j^n & = \int_0^{t_j} (f(s, X_s) - f(t^n(s), X_{t^n(s)}))\;\mathrm{d}s + \int_0^{t_j} (g(s, X_s) - g(t_{i^n(s)}, X_{t^n(s)}))\;\mathrm{d}W_s \\
& \quad + \sum_{i=1}^j (f(t_{i-1}, X_{t_{i-1}}) - f(t_{i-1}, X_{i-1}^n)) \Delta t_{i-1} + \sum_{i=1}^j (g(t_{i-1}, X_{t_{i-1}}) - g(t_{i-1}, X_{i-1}^n)) \Delta W_{i-1},
\end{align*}
X t j − X j n = ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) )) d s + ∫ 0 t j ( g ( s , X s ) − g ( t i n ( s ) , X t n ( s ) )) d W s + i = 1 ∑ j ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n )) Δ t i − 1 + i = 1 ∑ j ( g ( t i − 1 , X t i − 1 ) − g ( t i − 1 , X i − 1 n )) Δ W i − 1 ,
onde t n t^n t n e i n i^n i n são as funções de malha
i n ( t ) = max j = 0 , … , n { j ; t j ≤ t } ,
i^n(t) = \max_{j=0, \ldots, n}\{j; \;t_j \leq t\},
i n ( t ) = j = 0 , … , n max { j ; t j ≤ t } ,
e
t n ( t ) = t i n ( t ) = max { t i ≤ t ; i = 0 , … , n } ,
t^n(t) = t_{i^n(t)} = \max\{t_i \leq t; \; i = 0, \ldots, n\},
t n ( t ) = t i n ( t ) = max { t i ≤ t ; i = 0 , … , n } ,
que nos dão o índice i n ( t ) i^n(t) i n ( t ) do ponto da malha que está mais próximo e à esquerda de um instante t t t e o ponto correspondente t n ( t ) = t i n ( t ) t^n(t) = t_{i^n(t)} t n ( t ) = t i n ( t ) da malha.
Elevando ao quadrado e usando que ( a 1 + … + a 4 ) 2 ≤ 4 ( a 1 2 + … + a 4 2 ) , (a_1 + \ldots + a_4)^2 \leq 4(a_1^2 + \ldots + a_4^2), ( a 1 + … + a 4 ) 2 ≤ 4 ( a 1 2 + … + a 4 2 ) ,
( X t j − X j n ) 2 = 4 ( ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) ) d s ) 2 + 4 ( ∫ 0 t j ( g ( s , X s ) − g ( t n ( s ) , X t n ( s ) ) ) d W s ) 2 + 4 ( ∑ i = 1 j ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n ) ) Δ t i − 1 ) + 4 ( ∑ i = 1 j ( g ( t i − 1 , X t i − 1 ) − g ( t i − 1 , X i − 1 n ) ) Δ W i − 1 ) 2 .
\begin{align*}
\left(X_{t_j} - X_j^n\right)^2 & = 4\left(\int_0^{t_j} (f(s, X_s) - f(t^n(s), X_{t^n(s)}))\;\mathrm{d}s\right)^2 + 4\left(\int_0^{t_j} (g(s, X_s) - g(t^n(s), X_{t^n(s)}))\;\mathrm{d}W_s\right)^2 \\
& \quad + 4\left(\sum_{i=1}^j (f(t_{i-1}, X_{t_{i-1}}) - f(t_{i-1}, X_{i-1}^n)) \Delta t_{i-1}\right) + 4\left(\sum_{i=1}^j (g(t_{i-1}, X_{t_{i-1}}) - g(t_{i-1}, X_{i-1}^n)) \Delta W_{i-1}\right)^2.
\end{align*}
( X t j − X j n ) 2 = 4 ( ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) )) d s ) 2 + 4 ( ∫ 0 t j ( g ( s , X s ) − g ( t n ( s ) , X t n ( s ) )) d W s ) 2 + 4 ( i = 1 ∑ j ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n )) Δ t i − 1 ) + 4 ( i = 1 ∑ j ( g ( t i − 1 , X t i − 1 ) − g ( t i − 1 , X i − 1 n )) Δ W i − 1 ) 2 .
Usando a desigualdade de Cauchy-Schwartz na primeira integral e no primeiro somatório, obtemos
( X t j − X j n ) 2 ≤ 4 t j ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) ) 2 d s + 4 ( ∫ 0 t j ( g ( s , X s ) − g ( t n ( s ) , X t n ( s ) ) ) d W s ) 2 + 4 t j ∑ i = 1 j ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n ) ) 2 Δ t i − 1 + 4 ( ∑ i = 1 j ( g ( t i − 1 , X t i − 1 ) − g ( t i − 1 , X i − 1 n ) ) Δ W i − 1 ) 2 .
\begin{align*}
\left(X_{t_j} - X_j^n\right)^2 & \leq 4t_j\int_0^{t_j} (f(s, X_s) - f(t^n(s), X_{t^n(s)}))^2\;\mathrm{d}s + 4\left(\int_0^{t_j} (g(s, X_s) - g(t^n(s), X_{t^n(s)}))\;\mathrm{d}W_s\right)^2 \\
& \quad + 4t_j\sum_{i=1}^j (f(t_{i-1}, X_{t_{i-1}}) - f(t_{i-1}, X_{i-1}^n))^2 \Delta t_{i-1} + 4\left(\sum_{i=1}^j (g(t_{i-1}, X_{t_{i-1}}) - g(t_{i-1}, X_{i-1}^n)) \Delta W_{i-1}\right)^2.
\end{align*}
( X t j − X j n ) 2 ≤ 4 t j ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) ) 2 d s + 4 ( ∫ 0 t j ( g ( s , X s ) − g ( t n ( s ) , X t n ( s ) )) d W s ) 2 + 4 t j i = 1 ∑ j ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n ) ) 2 Δ t i − 1 + 4 ( i = 1 ∑ j ( g ( t i − 1 , X t i − 1 ) − g ( t i − 1 , X i − 1 n )) Δ W i − 1 ) 2 .
Tomando o valor esperado e usando a isometria de Itô na integral e no somatório (que é a isometria de Itô numa função escada e que também pode ser deduzido diretamente pelas independências dos saltos em intervalos distintos e pale condição de não antecipação),
E [ ( X t j − X j n ) 2 ] ≤ 4 t j ∫ 0 t j E [ ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) ) 2 ] d s + 4 ∫ 0 t j E [ ( g ( s , X s ) − g ( t n ( s ) , X t n ( s ) ) ) 2 ] d s + 4 t j ∑ i = 1 j E [ ( f ( t n ( s ) , X t i − 1 ) − f ( t i − 1 , X i − 1 n ) ) 2 ] Δ t i − 1 + 4 ∑ i = 1 j E [ ( g ( t i − 1 , X t i − 1 ) − g ( t i − 1 , X i − 1 n ) ) 2 ] Δ t i − 1 .
\begin{align*}
\mathbb{E}\left[\left(X_{t_j} - X_j^n\right)^2\right] & \leq 4t_j\int_0^{t_j} \mathbb{E}\left[(f(s, X_s) - f(t^n(s), X_{t^n(s)}))^2\right]\;\mathrm{d}s + 4\int_0^{t_j} \mathbb{E}\left[(g(s, X_s) - g(t^n(s), X_{t^n(s)}))^2\right]\;\mathrm{d}s \\
& \quad + 4t_j\sum_{i=1}^j \mathbb{E}\left[(f(t^n(s), X_{t_{i-1}}) - f(t_{i-1}, X_{i-1}^n))^2\right] \Delta t_{i-1} + 4\sum_{i=1}^j \mathbb{E}\left[(g(t_{i-1}, X_{t_{i-1}}) - g(t_{i-1}, X_{i-1}^n))^2\right] \Delta t_{i-1}.
\end{align*}
E [ ( X t j − X j n ) 2 ] ≤ 4 t j ∫ 0 t j E [ ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) ) 2 ] d s + 4 ∫ 0 t j E [ ( g ( s , X s ) − g ( t n ( s ) , X t n ( s ) ) ) 2 ] d s + 4 t j i = 1 ∑ j E [ ( f ( t n ( s ) , X t i − 1 ) − f ( t i − 1 , X i − 1 n ) ) 2 ] Δ t i − 1 + 4 i = 1 ∑ j E [ ( g ( t i − 1 , X t i − 1 ) − g ( t i − 1 , X i − 1 n ) ) 2 ] Δ t i − 1 .
Os dois primeiros termos integrais podem ser estimados por
∫ 0 t j E [ ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) ) 2 ] d s ≤ ∑ i = 1 j ∫ t i − 1 t i ( 2 H f 2 ∣ s − t i − 1 ∣ ( 1 + E [ X s 2 ] ) + 2 L f 2 E [ ∣ X s − X t i − 1 ∣ 2 ] ) d s ≤ ( 2 H f 2 ( 1 + M T ) + 2 L f 2 C T ) ∑ i = 1 j ∫ t i − 1 t i ( s − t i − 1 ) d s ≤ ( H f 2 ( 1 + M T ) + L f 2 C T ) ∑ i = 1 j ( t i − t i − 1 ) ≤ t j ( H f 2 ( 1 + M T ) + L f 2 C T ) max i ( t i − t i − 1 ) ≤ T ( H f 2 ( 1 + M T ) + L f 2 C T ) Δ t .
\begin{align*}
\int_0^{t_j} \mathbb{E}\left[(f(s, X_s) - f(t^n(s), X_{t^n(s)}))^2\right]\;\mathrm{d}s & \leq \sum_{i=1}^j\int_{t_{i-1}}^{t_i} \left(2H_f^2|s - t_{i-1}|(1 + \mathbb{E}\left[X_s^2\right]) + 2L_f^2\mathbb{E}\left[|X_s - X_{t_{i-1}}|^2\right]\right)\;\mathrm{d}s \\
& \leq \left(2H_f^2(1 + M_T) + 2L_f^2C_T\right)\sum_{i=1}^j\int_{t_{i-1}}^{t_i} (s - t_{i-1}) \;\mathrm{d}s \\
& \leq \left(H_f^2(1 + M_T) + L_f^2C_T\right)\sum_{i=1}^j(t_i - t_{i-1}) \\
& \leq t_j\left(H_f^2(1 + M_T) + L_f^2C_T\right)\max_i (t_i - t_{i-1}) \\
& \leq T\left(H_f^2(1 + M_T) + L_f^2C_T\right)\Delta t.
\end{align*}
∫ 0 t j E [ ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) ) 2 ] d s ≤ i = 1 ∑ j ∫ t i − 1 t i ( 2 H f 2 ∣ s − t i − 1 ∣ ( 1 + E [ X s 2 ] ) + 2 L f 2 E [ ∣ X s − X t i − 1 ∣ 2 ] ) d s ≤ ( 2 H f 2 ( 1 + M T ) + 2 L f 2 C T ) i = 1 ∑ j ∫ t i − 1 t i ( s − t i − 1 ) d s ≤ ( H f 2 ( 1 + M T ) + L f 2 C T ) i = 1 ∑ j ( t i − t i − 1 ) ≤ t j ( H f 2 ( 1 + M T ) + L f 2 C T ) i max ( t i − t i − 1 ) ≤ T ( H f 2 ( 1 + M T ) + L f 2 C T ) Δ t .
e, analogamente,
∫ 0 t j E [ ( g ( s , X s ) − g ( t n ( s ) , X t n ( s ) ) ) 2 ] d s ≤ T ( H g 2 ( 1 + M T ) + L g 2 C T ) Δ t .
\int_0^{t_j} \mathbb{E}\left[(g(s, X_s) - g(t^n(s), X_{t^n(s)}))^2\right]\;\mathrm{d}s \leq T\left(H_g^2(1 + M_T) + L_g^2C_T\right)\Delta t.
∫ 0 t j E [ ( g ( s , X s ) − g ( t n ( s ) , X t n ( s ) ) ) 2 ] d s ≤ T ( H g 2 ( 1 + M T ) + L g 2 C T ) Δ t .
Já os somatórios nos dão
∑ i = 1 j E [ ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n ) ) 2 ] Δ t i − 1 ≤ L f , 2 2 ∑ i = 1 j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1
\sum_{i=1}^j \mathbb{E}\left[(f(t_{i-1}, X_{t_{i-1}}) - f(t_{i-1}, X_{i-1}^n))^2\right] \Delta t_{i-1} \leq L_{f, 2}^2\sum_{i=1}^j \mathbb{E}\left[(X_{t_{i-1}} - X_{i-1}^n)^2\right] \Delta t_{i-1}
i = 1 ∑ j E [ ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n ) ) 2 ] Δ t i − 1 ≤ L f , 2 2 i = 1 ∑ j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1
e
∑ i = 1 j E [ ( g ( t i − 1 , X t i − 1 ) − g ( t i − 1 , X i − 1 n ) ) 2 ] Δ t i − 1 ≤ L g , 2 2 ∑ i = 1 j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1 .
\sum_{i=1}^j \mathbb{E}\left[(g(t_{i-1}, X_{t_{i-1}}) - g(t_{i-1}, X_{i-1}^n))^2\right] \Delta t_{i-1} \leq L_{g, 2}^2\sum_{i=1}^j \mathbb{E}\left[(X_{t_{i-1}} - X_{i-1}^n)^2\right] \Delta t_{i-1}.
i = 1 ∑ j E [ ( g ( t i − 1 , X t i − 1 ) − g ( t i − 1 , X i − 1 n ) ) 2 ] Δ t i − 1 ≤ L g , 2 2 i = 1 ∑ j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1 .
Juntando as estimativas,
E [ ( X t j − X j n ) 2 ] ≤ 4 t j ( L f , 1 2 + L f , 2 2 C T ) Δ t + 4 ( L g , 1 2 + L g , 2 2 C T ) Δ t + 4 t j L f , 2 2 ∑ i = 1 j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1 + 4 L g , 2 2 ∑ i = 1 j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1 .
\begin{align*}
\mathbb{E}\left[\left(X_{t_j} - X_j^n\right)^2\right] & \leq 4t_j\left(L_{f,1}^2 + L_{f, 2}^2C_T\right)\Delta t + 4\left(L_{g,1}^2 + L_{g, 2}^2C_T\right)\Delta t \\
& \quad + 4t_jL_{f, 2}^2\sum_{i=1}^j \mathbb{E}\left[(X_{t_{i-1}} - X_{i-1}^n)^2\right] \Delta t_{i-1} + 4L_{g, 2}^2\sum_{i=1}^j \mathbb{E}\left[(X_{t_{i-1}} - X_{i-1}^n)^2\right] \Delta t_{i-1}.
\end{align*}
E [ ( X t j − X j n ) 2 ] ≤ 4 t j ( L f , 1 2 + L f , 2 2 C T ) Δ t + 4 ( L g , 1 2 + L g , 2 2 C T ) Δ t + 4 t j L f , 2 2 i = 1 ∑ j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1 + 4 L g , 2 2 i = 1 ∑ j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1 .
Ou seja,
E [ ( X t j − X j n ) 2 ] ≤ C 2 Δ t + 2 L ∑ i = 1 j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1 ,
\mathbb{E}\left[\left(X_{t_j} - X_j^n\right)^2\right] \leq C^2 \Delta t + 2L \sum_{i=1}^j \mathbb{E}\left[(X_{t_{i-1}} - X_{i-1}^n)^2\right] \Delta t_{i-1},
E [ ( X t j − X j n ) 2 ] ≤ C 2 Δ t + 2 L i = 1 ∑ j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1 ,
para C , L > 0 C, L > 0 C , L > 0 apropriadas. Pela desigualdade de Gronwall discreta, isso nos dá
E [ ( X t j − X j n ) 2 ] ≤ C 2 e 2 L t j Δ t .
\mathbb{E}\left[\left(X_{t_j} - X_j^n\right)^2\right] \leq C^2e^{2Lt_j}\Delta t.
E [ ( X t j − X j n ) 2 ] ≤ C 2 e 2 L t j Δ t .
Considerando a norma forte, obtemos
E [ ∣ X t j − X j n ∣ ] ≤ E [ ( X t j − X j n ) 2 ] 1 / 2 ≤ C e L t j Δ t 1 / 2 .
\mathbb{E}\left[\left|X_{t_j} - X_j^n\right|\right] \leq \mathbb{E}\left[\left(X_{t_j} - X_j^n\right)^2\right]^{1/2} \leq Ce^{Lt_j}\Delta t^{1/2}.
E [ ∣ ∣ X t j − X j n ∣ ∣ ] ≤ E [ ( X t j − X j n ) 2 ] 1/2 ≤ C e L t j Δ t 1/2 .
mostrando que o método de Euler-Maruyama é de ordem forte 1 / 2. 1/2. 1/2.
Quando g = g ( t ) g=g(t) g = g ( t ) não depende de x x x e quando f = f ( t , x ) f=f(t, x) f = f ( t , x ) e g = g ( t ) g=g(t) g = g ( t ) são mais suaves, podemos mostrar que a convergência forte é, na verdade, de order 1. Mais precisamente, pedimos que f f f e g g g sejam continuamente diferenciáveis em t t t e que f f f seja duas vezes continuamente diferenciáveis em x , x, x , com limitações uniformes,
∣ ( ∂ t f ) ( t , x ) ∣ ≤ H f , ∣ ( ∂ x f ) ( t , x ) ∣ ≤ L f , ∣ ( ∂ x x f ) ( t , x ) ∣ ≤ L f f .
|(\partial_t f)(t, x)| \leq H_f, \quad |(\partial_x f)(t, x)| \leq L_f, \quad |(\partial_{xx} f)(t, x)| \leq L_{ff}.
∣ ( ∂ t f ) ( t , x ) ∣ ≤ H f , ∣ ( ∂ x f ) ( t , x ) ∣ ≤ L f , ∣ ( ∂ xx f ) ( t , x ) ∣ ≤ L ff .
Isso tudo em 0 ≤ t ≤ T , 0\leq t \leq T, 0 ≤ t ≤ T , x ∈ R . x\in \mathbb{R}. x ∈ R . Como g = g ( t ) g=g(t) g = g ( t ) só depende de t t t e o intevalo [ 0 , T ] [0, T] [ 0 , T ] é limitado, temos, pela suavidade de g , g, g , que
∣ g ( t ) ∣ ≤ M g , ∣ ( ∂ t g ) ( t ) ∣ ≤ H g .
|g(t)| \leq M_g, \quad |(\partial_t g)(t)| \leq H_g.
∣ g ( t ) ∣ ≤ M g , ∣ ( ∂ t g ) ( t ) ∣ ≤ H g .
em 0 ≤ t ≤ T . 0\leq t \leq T. 0 ≤ t ≤ T .
Escrevemos a diferença entre a solução e a aproximação na forma
X t j − X j n = ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) ) d s + ∫ 0 t j ( g ( s , X s ) − g ( t i n ( s ) , X t n ( s ) ) ) d W s + ∑ i = 1 j ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n ) ) Δ t i − 1 + ∑ i = 1 j ( g ( t i − 1 , X t i − 1 ) − g ( t i − 1 , X i − 1 n ) ) Δ W i − 1 ,
\begin{align*}
X_{t_j} - X_j^n & = \int_0^{t_j} (f(s, X_s) - f(t^n(s), X_{t^n(s)}))\;\mathrm{d}s + \int_0^{t_j} (g(s, X_s) - g(t_{i^n(s)}, X_{t^n(s)}))\;\mathrm{d}W_s \\
& \quad + \sum_{i=1}^j (f(t_{i-1}, X_{t_{i-1}}) - f(t_{i-1}, X_{i-1}^n)) \Delta t_{i-1} + \sum_{i=1}^j (g(t_{i-1}, X_{t_{i-1}}) - g(t_{i-1}, X_{i-1}^n)) \Delta W_{i-1},
\end{align*}
X t j − X j n = ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) )) d s + ∫ 0 t j ( g ( s , X s ) − g ( t i n ( s ) , X t n ( s ) )) d W s + i = 1 ∑ j ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n )) Δ t i − 1 + i = 1 ∑ j ( g ( t i − 1 , X t i − 1 ) − g ( t i − 1 , X i − 1 n )) Δ W i − 1 ,
No caso em que g = g ( t ) , g=g(t), g = g ( t ) , o último termo desaparece (mas que não é um termo problemático) e o segundo termo simplifica (esse sim é problemático),
X t j − X j n = ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) ) d s + ∫ 0 t j ( g ( s ) − g ( t i n ( s ) ) ) d W s + ∑ i = 1 j ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n ) ) Δ t i − 1 .
\begin{align*}
X_{t_j} - X_j^n & = \int_0^{t_j} (f(s, X_s) - f(t^n(s), X_{t^n(s)}))\;\mathrm{d}s + \int_0^{t_j} (g(s) - g(t_{i^n(s)}))\;\mathrm{d}W_s \\
& \quad + \sum_{i=1}^j (f(t_{i-1}, X_{t_{i-1}}) - f(t_{i-1}, X_{i-1}^n)) \Delta t_{i-1}.
\end{align*}
X t j − X j n = ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) )) d s + ∫ 0 t j ( g ( s ) − g ( t i n ( s ) )) d W s + i = 1 ∑ j ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n )) Δ t i − 1 .
O último termo é como antes e nos dá
∑ i = 1 j E [ ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n ) ) 2 ] Δ t i − 1 ≤ L f , 2 2 ∑ i = 1 j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1 .
\sum_{i=1}^j \mathbb{E}\left[(f(t_{i-1}, X_{t_{i-1}}) - f(t_{i-1}, X_{i-1}^n))^2\right] \Delta t_{i-1} \leq L_{f, 2}^2\sum_{i=1}^j \mathbb{E}\left[(X_{t_{i-1}} - X_{i-1}^n)^2\right] \Delta t_{i-1}.
i = 1 ∑ j E [ ( f ( t i − 1 , X t i − 1 ) − f ( t i − 1 , X i − 1 n ) ) 2 ] Δ t i − 1 ≤ L f , 2 2 i = 1 ∑ j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1 .
O segundo termo, agora sem a dependência em x x x e com continuidade Lipschitz em t , t, t , nos dá
E [ ( ∫ 0 t j ( g ( s ) − g ( t i n ( s ) ) ) d W s ) 2 ] = ∫ 0 t j E [ ( g ( s ) − g ( t i n ( s ) ) ) 2 ] d s ≤ H g 2 ∫ 0 t j ( s − t i n ( s ) ) 2 d s ≤ H g 2 t j Δ t 2 .
\begin{align*}
\mathbb{E}\left[\left(\int_0^{t_j} (g(s) - g(t_{i^n(s)}))\;\mathrm{d}W_s\right)^2\right] & = \int_0^{t_j} \mathbb{E}\left[\left(g(s) - g(t_{i^n(s)})\right)^2\right]\;\mathrm{d}s \\
& \leq H_g^2 \int_0^{t_j} \left(s - t_{i^n(s)}\right)^2\;\mathrm{d}s \\
& \leq H_g^2 t_j \Delta t^2.
\end{align*}
E [ ( ∫ 0 t j ( g ( s ) − g ( t i n ( s ) )) d W s ) 2 ] = ∫ 0 t j E [ ( g ( s ) − g ( t i n ( s ) ) ) 2 ] d s ≤ H g 2 ∫ 0 t j ( s − t i n ( s ) ) 2 d s ≤ H g 2 t j Δ t 2 .
O primeiro termo é o mais delicado e requer o uso da fórmula de Itô. Com ela, temos
f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) = ∫ t n ( s ) s ( ( ∂ t f ) ( ξ , X ξ ) f ( ξ , X ξ ) + 1 2 ( ∂ x x f ) ( ξ , X ξ ) g ( ξ ) 2 ) d ξ + ∫ t n ( s ) s ( ∂ x f ) ( ξ , X ξ ) g ( ξ ) d W ξ .
\begin{align*}
f(s, X_s) - f(t^n(s), X_{t^n(s)}) & = \int_{t^n(s)}^s \left((\partial_t f)(\xi, X_{\xi})f(\xi, X_{\xi}) + \frac{1}{2}(\partial_{xx} f)(\xi, X_{\xi})g(\xi)^2 \right)\;\mathrm{d}\xi \\
& \quad + \int_{t^n(s)}^s (\partial_x f)(\xi, X_{\xi})g(\xi)\;\mathrm{d}W_\xi.
\end{align*}
f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) = ∫ t n ( s ) s ( ( ∂ t f ) ( ξ , X ξ ) f ( ξ , X ξ ) + 2 1 ( ∂ xx f ) ( ξ , X ξ ) g ( ξ ) 2 ) d ξ + ∫ t n ( s ) s ( ∂ x f ) ( ξ , X ξ ) g ( ξ ) d W ξ .
O ponto chave é trocar a ordem de integração, usando uma versão estocástica do Teorema de Fubini na segunda integral. Assim,
∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) ) d s = ∫ 0 t j ∫ t n ( s ) s ( ( ∂ t f ) ( ξ , X ξ ) f ( ξ , X ξ ) + 1 2 ( ∂ x x f ) ( ξ , X ξ ) g ( ξ ) 2 ) d ξ d s + ∫ 0 t j ∫ t n ( s ) s ( ∂ x f ) ( ξ , X ξ ) g ( ξ ) d W ξ d s = ∫ 0 t j ∫ ξ t ~ n ( ξ ) ( ( ∂ t f ) ( ξ , X ξ ) f ( ξ , X ξ ) + 1 2 ( ∂ x x f ) ( ξ , X ξ ) g ( ξ ) 2 ) d s d ξ + ∫ 0 t j ∫ ξ t ~ n ( ξ ) ( ∂ x f ) ( ξ , X ξ ) g ( ξ ) d s d W ξ ,
\begin{align*}
\int_0^{t_j} (f(s, X_s) - f(t^n(s), X_{t^n(s)}))\;\mathrm{d}s & = \int_0^{t_j} \int_{t^n(s)}^s \left((\partial_t f)(\xi, X_{\xi})f(\xi, X_{\xi}) + \frac{1}{2}(\partial_{xx} f)(\xi, X_{\xi})g(\xi)^2 \right)\;\mathrm{d}\xi\;\mathrm{d}s \\
& \quad + \int_0^{t_j} \int_{t^n(s)}^s (\partial_x f)(\xi, X_{\xi})g(\xi)\;\mathrm{d}W_\xi\;\mathrm{d}s \\
& = \int_0^{t_j} \int_{\xi}^{\tilde t^{n}(\xi)} \left((\partial_t f)(\xi, X_{\xi})f(\xi, X_{\xi}) + \frac{1}{2}(\partial_{xx} f)(\xi, X_{\xi})g(\xi)^2 \right)\;\mathrm{d}s\;\mathrm{d}\xi \\
& \quad + \int_0^{t_j} \int_{\xi}^{\tilde t^{n}(\xi)} (\partial_x f)(\xi, X_{\xi})g(\xi)\;\mathrm{d}s\;\mathrm{d}W_\xi,
\end{align*}
∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) )) d s = ∫ 0 t j ∫ t n ( s ) s ( ( ∂ t f ) ( ξ , X ξ ) f ( ξ , X ξ ) + 2 1 ( ∂ xx f ) ( ξ , X ξ ) g ( ξ ) 2 ) d ξ d s + ∫ 0 t j ∫ t n ( s ) s ( ∂ x f ) ( ξ , X ξ ) g ( ξ ) d W ξ d s = ∫ 0 t j ∫ ξ t ~ n ( ξ ) ( ( ∂ t f ) ( ξ , X ξ ) f ( ξ , X ξ ) + 2 1 ( ∂ xx f ) ( ξ , X ξ ) g ( ξ ) 2 ) d s d ξ + ∫ 0 t j ∫ ξ t ~ n ( ξ ) ( ∂ x f ) ( ξ , X ξ ) g ( ξ ) d s d W ξ ,
onde
t ~ n ( t ) = min { t i ≥ t ; i = 0 , … , n }
\tilde t^{n}(t) = \min\{t_i \geq t; \; i = 0, \ldots, n\}
t ~ n ( t ) = min { t i ≥ t ; i = 0 , … , n }
é o ponto da malha que está mais próximo e à direita do instante t . t. t . Observe que o integrando não depende de s , s, s , de modo que o fato da integral em s s s ser no intervalo [ ξ , t ~ n ( ξ ) ] , [\xi, \tilde t^n(\xi)], [ ξ , t ~ n ( ξ )] , ou seja, posterior a ξ , \xi, ξ , viola nenhuma condição de não antecipação do integrando.
Usando Cauchy-Schwartz e a isometria de Itô, obtemos a seguinte estimativa para a média quadrática desse termo.
E [ ( ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) ) d s ) 2 ] ≤ t j ∫ 0 t j ( t n ( ξ ) − ξ ) ∫ ξ t ~ n ( ξ ) E [ ( ( ∂ t f ) ( ξ , X ξ ) f ( ξ , X ξ ) + 1 2 ( ∂ x x f ) ( ξ , X ξ ) g ( ξ ) 2 ) 2 ] d s d ξ + ∫ 0 t j ( t ~ n ( ξ ) − ξ ) ∫ ξ t ~ n ( ξ ) E [ ( ( ∂ x f ) ( ξ , X ξ ) g ( ξ ) ) 2 ] d s d ξ .
\begin{align*}
\mathbb{E}&\left[\left(\int_0^{t_j} (f(s, X_s) - f(t^n(s), X_{t^n(s)}))\;\mathrm{d}s\right)^2\right] \\
& \leq t_j\int_0^{t_j} (t^{n}(\xi) - \xi) \int_{\xi}^{\tilde t^{n}(\xi)} \mathbb{E}\left[\left((\partial_t f)(\xi, X_{\xi})f(\xi, X_{\xi}) + \frac{1}{2}(\partial_{xx} f)(\xi, X_{\xi})g(\xi)^2 \right)^2\right]\;\mathrm{d}s\;\mathrm{d}\xi \\
& \quad + \int_0^{t_j} (\tilde t^{n}(\xi) - \xi) \int_{\xi}^{\tilde t^{n}(\xi)} \mathbb{E}\left[\left((\partial_x f)(\xi, X_{\xi})g(\xi)\right)^2\right]\;\mathrm{d}s\;\mathrm{d}\xi.
\end{align*}
E [ ( ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) )) d s ) 2 ] ≤ t j ∫ 0 t j ( t n ( ξ ) − ξ ) ∫ ξ t ~ n ( ξ ) E [ ( ( ∂ t f ) ( ξ , X ξ ) f ( ξ , X ξ ) + 2 1 ( ∂ xx f ) ( ξ , X ξ ) g ( ξ ) 2 ) 2 ] d s d ξ + ∫ 0 t j ( t ~ n ( ξ ) − ξ ) ∫ ξ t ~ n ( ξ ) E [ ( ( ∂ x f ) ( ξ , X ξ ) g ( ξ ) ) 2 ] d s d ξ .
Usando as estimativas para f , f, f , g g g e suas derivadas, obtemos
E [ ( ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) ) ) d s ) 2 ] ≤ t j ∫ 0 t j ( t n ( ξ ) − ξ ) ∫ ξ t ~ n ( ξ ) C 1 d s d ξ + ∫ 0 t j ( t ~ n ( ξ ) − ξ ) ∫ ξ t ~ n ( ξ ) C 2 d s d ξ = ( t j C 1 + C 2 ) ∫ 0 t j ( t n ( ξ ) − ξ ) 2 d ξ ≤ ( T C 1 + C 2 ) Δ t 2 .
\begin{align*}
\mathbb{E}\left[\left(\int_0^{t_j} (f(s, X_s) - f(t^n(s), X_{t^n(s)}))\;\mathrm{d}s\right)^2\right] & \leq t_j\int_0^{t_j} (t^{n}(\xi) - \xi) \int_{\xi}^{\tilde t^{n}(\xi)} C_1\;\mathrm{d}s\;\mathrm{d}\xi \\
& \quad + \int_0^{t_j} (\tilde t^{n}(\xi) - \xi) \int_{\xi}^{\tilde t^{n}(\xi)} C_2\;\mathrm{d}s\;\mathrm{d}\xi \\
& = (t_jC_1 + C_2)\int_0^{t_j} (t^{n}(\xi) - \xi)^2 \;\mathrm{d}\xi \\
& \leq (TC_1 + C_2)\Delta t^2.
\end{align*}
E [ ( ∫ 0 t j ( f ( s , X s ) − f ( t n ( s ) , X t n ( s ) )) d s ) 2 ] ≤ t j ∫ 0 t j ( t n ( ξ ) − ξ ) ∫ ξ t ~ n ( ξ ) C 1 d s d ξ + ∫ 0 t j ( t ~ n ( ξ ) − ξ ) ∫ ξ t ~ n ( ξ ) C 2 d s d ξ = ( t j C 1 + C 2 ) ∫ 0 t j ( t n ( ξ ) − ξ ) 2 d ξ ≤ ( T C 1 + C 2 ) Δ t 2 .
para constantes apropriadas C 1 , C 2 > 0. C_1, C_2 > 0. C 1 , C 2 > 0. Juntando as estimativas, obtemos
E [ ( X t j − X j n ) 2 ] ≤ C 2 Δ t 2 + 2 L ∑ i = 1 j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1 ,
\mathbb{E}\left[\left(X_{t_j} - X_j^n\right)^2\right] \leq C^2 \Delta t^2 + 2L \sum_{i=1}^j \mathbb{E}\left[(X_{t_{i-1}} - X_{i-1}^n)^2\right] \Delta t_{i-1},
E [ ( X t j − X j n ) 2 ] ≤ C 2 Δ t 2 + 2 L i = 1 ∑ j E [ ( X t i − 1 − X i − 1 n ) 2 ] Δ t i − 1 ,
para constante C , L > 0 C, L > 0 C , L > 0 apropriadas. Pela desigualdade de Gronwall discreta, isso nos dá
E [ ( X t j − X j n ) 2 ] ≤ C 2 e 2 L t j Δ t 2 .
\mathbb{E}\left[\left(X_{t_j} - X_j^n\right)^2\right] \leq C^2e^{2Lt_j}\Delta t^2.
E [ ( X t j − X j n ) 2 ] ≤ C 2 e 2 L t j Δ t 2 .
Considerando a norma forte, obtemos
E [ ∣ X t j − X j n ∣ ] ≤ E [ ( X t j − X j n ) 2 ] 1 / 2 ≤ C e L t j Δ t .
\mathbb{E}\left[\left|X_{t_j} - X_j^n\right|\right] \leq \mathbb{E}\left[\left(X_{t_j} - X_j^n\right)^2\right]^{1/2} \leq Ce^{Lt_j}\Delta t.
E [ ∣ ∣ X t j − X j n ∣ ∣ ] ≤ E [ ( X t j − X j n ) 2 ] 1/2 ≤ C e L t j Δ t .
mostrando que o método de Euler-Maruyama é de ordem forte 1. 1. 1.