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Bond Price expression

I've researching some mathematical finance and I've stumbled upon something I can't seems to find sources on. I'm probably overlooking something, but I hope someone can enlighten me and give me some ...
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Assumptions: Let $T > 0$, and let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathbb P = \tilde{\mathbb P}$ (risk-neutral measure) and $... 5 votes 1 answer 299 views What exactly is/How exactly do we interpret the binomial model's Radon-Nikodym derivative? Related: Lewis' triviality result? As I recall the one-step binomial model goes like this: The time periods are now$t=0$and later$t=1$. We have 2.1. a stock that pays off$u$for going up or$d$... 1 vote 2 answers 571 views Do real life economists and financial analysts actually use calculus in their jobs? From what I understand a lot of calculus is used at university level when studying economics and some finance courses (correct me if I'm wrong) but I was just wondering if economists and financial ... 6 votes 1 answer 2k views Apply Ito's Lemma to exponential martingale$\newcommand{\dd}{\, \mathrm{d}}$Consider the exponential martingale, $$\xi_t^\lambda = \exp \left\{ - \int_0^t \lambda_s \dd z_s - \frac 12 \int_0^T \lambda_s^2 \dd s \right\},$$ that is used in ... 5 votes 2 answers 2k views Intuition of the Kolmogorov Equations So I understand the derivation of the Kolmogorov Forward and Backward Equations, but I don't quite understand the intuition. Here is from Stokey, 2008: "The backward equation involves time$t$and ... 20 votes 0 answers 898 views How do I use the Malliavin calculus to solve for the optimal trading strategy in the classic Merton problem? How do I use the Malliavin calculus to solve for the optimal trading strategy in the classic Merton problem? In Duffie's book "Dynamic Asset Pricing," he outlines the "Martingale method" of solving ... 2 votes 1 answer 87 views Given$\mathbb Q$and$X_t$is$\mathbb Q$-Brownian, find$\frac{d\mathbb Q}{d\mathbb P}$/ Uniqueness of Brownian or Radon-Nikodym derivative The problem: Let$T >0$, and let$(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$be a filtered probability space where$\mathscr F_t = \mathscr F_t^W$where$W = \{W_t\}_{t \in ...
I'm getting into asset pricing and was looking at Ito's Lemma, but cannot understand a few steps that are given. Ito's Lemma states that given $$dx_t = \mu dt + \sigma dz_t \\ y_t = f(t, x_t)$$ ...