Questions tagged [stochastic-calculus]
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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)...
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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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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 ...
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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$...
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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},\...
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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 $...
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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 ...
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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)$$
...
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Apply Ito's Lemma to exponential martingale
$\newcommand{\dd}{\, \mathrm{d}}$
Consider the exponential martingale,
$$
\xi_t^\lambda = \exp \left\{ - \int_0^t \lambda_s \dd z_s - \frac 12 \int_0^T \lambda_s^2 \dd s \right\},
$$
that is used in ...
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Intuition of the Kolmogorov Equations
So I understand the derivation of the Kolmogorov Forward and Backward Equations, but I don't quite understand the intuition. Here is from Stokey, 2008:
"The backward equation involves time $t$ and ...
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Urn balls and probabilities
Think of the following balls as individuals of populations.
Say I have $U$ urns, and some balls. Both numbers are really large. So large, that authors like Blanchard and Diamond have approximated ...
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Find probability that payoff function is in $[10,20]$
In moment $t=0$ we bought option with expiration date $T=2$. The payoff function of this option is given by:
$$f=(\max_{t\in[0,T]} S_t -110)^{+}$$
where $S_t$ satisfies
$$dS_t=15dW_t$$
$$S_0=95$$
...
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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 ...
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