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Moments computation
What's wrong in this procedure to compute the moments of a random variable X?I have a variable X which follows a lognormal GBM and it is defined asdX(t)=mu A(t)dt+sigma A(t) dW_tBy Ito's formula we haved ln A(t)=(mu -0.5 sigma^2)dt + sigma dW_tand by integrating we obtainA(t)=A(0) exp[(mu -0.5 sigma^2)t + sigma W_t]by using a Gaussian distribution the first and second moment of A areM(1)=exp(mu+sigma^2/2)M(2)=exp (2*mu + 2*sigma^2)thank youJL
Moments computation
I believe you meantdA(t)=mu A(t)dt+sigma A(t) dW_tSince A(t)=A(0) exp[(mu -0.5 sigma^2)t + sigma W_t].E[A(t)] = A(0) exp[(mu -0.5 sigma^2)t] E[exp(sigma W_t)].Now sigma W_t ~ N(0, sigma^2 t), soE[A(t)] = A(0) exp[(mu -0.5 sigma^2)t] exp[0.5 sigma^2 t].= A(0) exp[mu*t]Similarly,E[A(t)^2] = A(0)^2 exp[(2*mu - sigma^2)t] E[exp(2*sigma W_t)].= A(0)^2 exp[(2*mu - sigma^2)t] exp[0.5*(2*sigma)^2 * t]= A(0)^2 exp[(2*mu + sigma^2)t]I think you're leaving your "-0.5*sigma^2" term below.