Questions tagged [poisson-process]
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21 questions
1 vote
1 answer
115 views
Variance of Poisson integral
I am looking for the solution to the expression $$\text{Var}(\int_t^Te^{-a(T-s)}\ln(J)\text{d}N_s)$$ where $\ln(J)\sim N(\mu, \sigma^2)$ and $N$ has intensity $\lambda$. The term comes from a mean ...
- 21
2 votes
1 answer
200 views
Continuous-time two-state Markov process
$\lambda_t$ is binary with $\lambda_H$ and $\lambda_L$, with instantaneous transition probailities of $\mu_H$ and $\mu_L$. What is $\mathbb{E}_t[\lambda_T]$, assuming $\lambda_t=\lambda_H$ or $\...
- 53
1 vote
0 answers
62 views
How to derive the the following integral in continuous time? [closed]
Here is a standard exogenous system in continuous time, where $\pi_t$ is the pricing kernel \begin{align*} \frac{dx_t}{x_t} &= \mu_x dt + \sigma_x dB_{xt}\\ \frac{dz_t}{z_t} &= \mu_z dt + \...
- 53
0 votes
0 answers
85 views
Ruin Probability Question
In an insurance company the number of claims are modelled as a Poisson process with rate $\lambda>0$. Assume that the size of all claims is a fixed amount $\alpha>0$, the initial surplus is ...
0 votes
1 answer
176 views
Pure jump process in Duffie, Pan and Singleton's paper
In page 1349 or Section 2.1 of "Duffie, D., Pan, J., & Singleton, K. (2000). Transform Analysis and Asset Pricing for Affine Jump-Diffusions. Econometrica, 68(6), 1343-1376" the pure ...
4 votes
1 answer
268 views
Quadratic Variation Of Mixed Brownian Motion and Poisson Process
I am trying to solve this problem where we're asked to compute the quadratic variation of a process. I assume that it is necessary to apply Ito's formula but not sure how to get the right solution. ...
- 43
0 votes
1 answer
558 views
Separating jumps and diffusion
I want to model energy prices. I have two markets, lets say market 1 and 2. Market 1 is continuously traded, and I will assume it follows brownian motion. So the value of the asset could be defined ...
- 115
0 votes
0 answers
152 views
(Non-)Homogeneous Poisson process and their corresponding inter-arrival time distribution:
When it is said that the number of claims follows a homogenous Poisson process (where the intensity is assumed to be a fix value), it means that we have the stationary and independent assumptions for ...
- 419
1 vote
1 answer
185 views
Second variation of a Brownian motion under jump-diffusion process
I am trying to solve exercise 15.3 from the book The concepts and practice of mathematical finance where it is asked Suppose the $\log S_t$ follows a Brownian motion over the period $[0, 1]$ except ...
- 386
3 votes
1 answer
2k views
Ito Lemma for Poisson Process
I'm new to stochastic calculus on jump processes and encountered a difficulty. I would appreciate some clarification from the community on the following question. Let $g_t$ be a $\mathcal{F_t}$-...
- 184
4 votes
2 answers
354 views
Poisson process under equivalent martingale measure
I have a stochastic process $N(t)$ which is equal to $n$ with probability $P\{N(t) = n\}=\frac{\left(\lambda t \right)^{n}}{n!}e^{-\lambda t }$ where $t$ represents the time period. In other words, ...
- 213
2 votes
0 answers
179 views
The distribution of the jump diffusion process
In the Merton jump diffusion model the process of the share price can be expressed as $$S_{t}=S_{0}\cdot\exp\left\{ X_{t}\right\} ,$$ where $$X_{t}=\mu t+\sigma W_{t}+\sum_{i=1}^{N_{t}}Y_{i}.$$ Here $...
- 279
0 votes
1 answer
2k views
How can I learn stochastic process & stochastic calculus in two weeks? [closed]
I am going for an interview for a quant job. The interview will focus on my mathematical knowledge about stochastic process & stochastic calculus, and I believe I will definitely be asked to solve ...
- 21
5 votes
1 answer
438 views
Expected Value of Mean-Reverting Jump Process
I cant see the link between my method of calculation and the method done in the book Cartea and Jaimungal (Algorithmic and High Frequency Trading, page 220.) We have a mean-reverting process $$d\mu_t=-...
- 165
3 votes
1 answer
490 views
Arbitrage free in a Black-Scholes/Poisson model
I am trying to solve the following exercise from Bjork's Arbitrage Theory in Continuous Time: Consider a model for the stock market where the short rate of interest $r$ is a deterministic constant. ...
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