Question 1: Incorrect approach

Aproximating b(t, X(t)) by the trapezodial rule:

dX(t) = a(t, X(t))dt + b(t, X(t))dW_{t}  (0.0.1)

X_{n+1}(t) = X_{n} + a(t_{n}, X(t_{n}))Δ_{t} + b(\frac{1}{2}(t_{n+1} + t_{n})), \frac{1}{2}(X_{t_{n+1}}, X_{t_{n}}))ΔW_{n}  (0.0.2)

What is wrong?

Let’s consider the process: dX(t) = bdW (t) for constant b. Using above approximation we obtain X_{n+1}(t) = X_{n} + b(\frac{1}{2}(X_{t_{n+1}}, X_{t_{n}}))ΔW_{n}

X_{n+1}(t)(1 − \frac{1}{2}bΔW_{n}) = X_{n}(1 + \frac{1}{2}bΔW_{n})

X_{n+1}(t) = fracX_{n}(1 + \frac{1}{2}bΔW_{n})(1 − \frac{1}{2}bΔW_{n}) Computing E(X_{1}^{2}) we have

E(X_{1}^{2}) = (X_{0})^{2} ∫_{∞}^{∞} \frac{((1 + \frac{1}{2}bz))^{2}}{((1 − \frac{1}{2}b)^{2}z)}e^{−z^{2}/(2Δt)}dz  (0.0.3)

where we have used the fact that E(f(x)) = ∫ (f(x)dp(x)) where p(x) is the probability density of x.

Above integral is of the form ∫_{∞}^{∞} \frac{(1+x)^{2}}{(1−x)^{2}} e^{(} − x^{2})dx which does not converge.