For $f_\epsilon(x)=(\epsilon^2+x^2)^{1/2}$, direct <differentiation> gives
$$
f_\epsilon'(x)=\frac{x}{\sqrt{\epsilon^2+x^2}},
\qquad
f_\epsilon''(x)=\frac{\epsilon^2}{(\epsilon^2+x^2)^{3/2}}.
$$
Since the <quadratic variation> of standard <Brownian motion> is $[B]_t=t$, the <Itô formula> gives
$$
f_\epsilon(B_t)=\epsilon+
\int_0^t\frac{B_s}{\sqrt{\epsilon^2+B_s^2}}\,dB_s
+\frac12\int_0^t\frac{\epsilon^2}{(\epsilon^2+B_s^2)^{3/2}}\,ds.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-202/1/b/solution