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$dX_t = aX_tdt + bX_tdW_t$, $X_0 = x$ is the question and the solution for this is $X_t = xexp{(a-\frac{b^2}{2})t+bW_t}$

a and b are positive constant numbers

randn('state',100)
lambda = 0.5; mu = 1; Xzero = 0.5; 
T = 1; N = 2^8; dt = T/N;
dW = sqrt(dt)*randn(1,N); 
W = cumsum(dW); 
R = 4; Dt = R*dt; L = N/R; 
Xem = zeros(1,L);
Xtemp = Xzero;

for j = 1:L
    Winc = sum(dW(R*(j-1)+1:R*j));
    Xtemp = Xtemp - Dt * lambda * Xtemp + mu*Winc;
    Xem(j) = Xtemp;
end

plot([0:Dt:T],[Xzero,Xem],'r--*')

this plots the graph, and because I had the lambda as 1, should i change that to 0.5 and that would mean strong convergence?

When i do it like this I get a strong convergence of 0.5

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