Statement Let $B(t)$ be Brownian motion and let $f \in C^2(\mathbb{R})$. Then for $a < b$, [f(B(b)) - f(B(a)) = \int_a^b f’(B(t))\, dB(t) + \frac{1}{2}\int_a^b f’‘(B(t))\, dt \quad \text{a.s}....

Statement Let $\lambda$ and $\mu$ be positive finite measures on $\mathfrak{M}$. There exists a unique decomposition of positive finite measures \(\lambda = \lambda_a + \lambda_s\) where ...

Statement [\frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - rV = 0.] Assumptions Stock price dynamics The underly...

Statement [n! \sim \sqrt{2 \pi n} \left( \frac{n}{e} \right)^{n}.] Proof Asymptotic behavior of $n!$ Using monotonicity of \(\log x\), we have \[\sum_{k=1}^{n-1} \log k \leq \int_1^n \...