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Questions tagged [stochastic-processes]

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7
votes
2answers
223 views

Application of Poisson process in economic modelling

To understand the emergence of constitution, Myerson (2008) models a scernario that a political leader gathers supports from captains in order to defeat challengers whose arrival is modelled by a ...
1
vote
1answer
40 views

Purpose of Semidefinite Integral

I want to know the meaning of Semidefinite Integral. I am used to read definite and indefinite integral but I want to know the meaning of such equation : $\pi(e)\left(1-F\left[-\frac{a}{\pi(e)}\...
-1
votes
1answer
95 views

Stochastic process difference equation: stationary distribution

How can I find the stationary distribution (as t goes to infinity) of stochastic difference equations in the form: $x_{t+1} = a*x_t + b*N(0,1)$ where N(0,1) is a standard normal pdf I have ...
1
vote
0answers
80 views

Stochastic Dynamic Programming: Deriving the Steady-State for a Lottery

I am working through the basic examples of the stochastic RBC models in the book by McCandless (2008): The ABCs of RBCs, pp. 71 - 75 A Standard Stochastic Dynamic Programming Problem Here is a ...
3
votes
1answer
59 views

Profit maximization under uncertainity

I have a seller say S and I have a buyer say B. Buyer’s willing to pay is equal to x which is private information. But Seller believe that it falls in the range [0,x1]. Seller’s belief distribution is ...
5
votes
2answers
126 views

Decomposition of an additive functional into a Martingale part and other

This question relates to a theorem about the decomposition of additive functionals---a technique especially useful in macroeconomics and finance. This question has two objective. First, I don't have a ...
2
votes
1answer
62 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 \...
1
vote
0answers
29 views

Generalization of Tauchen 1986 approach to a case of time-varying volatility

My question is about generalization of Tauchen'86 approach to a case of time-varying volatility. Say, I have a process $$z_{t+1}=\rho z_t+\sigma_t \varepsilon_{t+1}$$ where $\varepsilon\sim \mathcal{...
1
vote
0answers
82 views

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

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 $\mathscr F_t ...
11
votes
3answers
797 views

Understanding the construction of stochastic processes

I've seen stochastic processes modeled/constructed in the following way. Consider the probability space $(\Omega, \mathcal F, Pr)$ and let $\mathbb S$ be the (measurable) transformation $\...
8
votes
2answers
1k views

Transition Matrix: Discrete -> Continuous Time

I have the code corresponding to Tauchen (1986) (Python equivalent of this), which generates a discrete approximation of a discrete time AR(1) process. For example, if you set up grid size as 3, it ...
4
votes
1answer
725 views

Proving there exists no arbitrage opportunities given 3 states and 2 assets

Assume there are 3 states of the world: w1, w2, and w3. Assume there are two assets: a risk-free asset returning Rf in each state, and a risky asset with Return R1 in state w1, R2 in state w2, and R3 ...
5
votes
2answers
98 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 $$dS_t=15dW_t$$ $$S_0=95$$ ...
6
votes
1answer
99 views

Showing that a transformation is measure preserving

Note: This question is related to this question about the construction of stochastic processes. Specifically, it relates to the transformation $\mathbb S: \Omega \rightarrow \Omega$ that is mentioned. ...