I have expression for implicit iteration (backward Euler Method):
$$X_{n+1} = X_{n}+\delta tX_{n+1}+\sqrt{\delta t}X_{n+1}Z_n,$$
where $n=0,...,N-1; Z_n iid\sim N(0,1)$
and explicit iteration (forward Euler Method):
$$X_{n+1} = X_{n}+\delta tX_{n}+\sqrt{\delta t}X_{n}Z_n.$$
I want to determine a solution formula for $X(T)$ for both expressions (implicit and explicit iteration) in the limit case $\delta t\to 0, T=N\delta t=const$.
How can i do this?
Any help is very appreciated.
You get to combine the factors of the infinite products in the resulting recursion formula via $$ \prod(1+a_i)=\exp(\sum\ln(1+a_i))=\exp(\sum a_i-\frac12\sum a_i^2+\frac13 \sum a_i^3+...) $$
Next you need to understand the chain of scales $δt\ll\sqrt{δt}\ll1$ to know which of the terms to keep, and then just expand the logarithms $$ \ln(1-\sqrt{δt}Z_n-δt) ~~ \text{ resp. } ~~ \ln(1+\sqrt{δt}Z_n+δt) $$ into power series. Then in the exponent the infinite products become infinite sums that can be seen as approximations of integrals.