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Questions tagged [stochastic-calculus]

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1 vote
0 answers
28 views

About part of Romer's model. Is my math process correct?

$$\mathcal{L}=\int_{i=0}^Ap(i)L(i)di-\lambda([\int_{i=0}^AL(i)^\phi di]^{\frac{1}{\phi}}-1)\\ s.t.\int_{i=0}^{A}L(i)^{\phi}di=1 $$ $$ \begin{aligned} \frac{d\mathcal{L}}{dL}&=\int_{i=0}^{A}p(i)di-\...
2 votes
1 answer
55 views

Stochastic optimal control problem (calculus)

I am reading a paper with a stochastic optimal control problem. At one point the author faces the following Hamilton-Jacobi-Bellman (HJB) equation: $$\rho V(A)=\max_{c,w}\left\{ u(c)+V'(A)\left[(r(1-w)...
2 votes
1 answer
52 views

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 ...
2 votes
0 answers
109 views

How to use Girsanov theorem to prove $\hat{W_t}$ is $\hat{\mathbb P}$-Brownian motion?

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 ...
3 votes
2 answers
878 views

Ito's Lemma derivation

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 $$dx_t = \mu dt + \sigma dz_t \\ y_t = f(t, x_t)$$ ...
1 vote
0 answers
157 views

How to solve a variation of Merton's optimal portfolio problem?

Does anyone know how to solve the following problem? I have tried to solve this but I'm lost since I have never dealt with a Stochastic Dynamic Programming problem with many variables. $max_{c_{t},\...
1 vote
3 answers
303 views

Urn balls and probabilities

Think of the following balls as individuals of populations. Say I have $U$ urns, and some balls. Both numbers are really large. So large, that authors like Blanchard and Diamond have approximated ...
5 votes
2 answers
177 views

Find probability that payoff function is in $[10,20]$

In moment $t=0$ we bought option with expiration date $T=2$. The payoff function of this option is given by: $$f=(\max_{t\in[0,T]} S_t -110)^{+}$$ where $S_t$ satisfies $$dS_t=15dW_t$$ $$S_0=95$$ ...