Questions tagged [poisson-process]
The poisson-process tag has no usage guidance.
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American put option. Exercise time is a random variable, calculation of expected payoff
I got an American put option, where the payoff is $V_\tau = \max(K - X_{\tau}, 0)$ and $X_{\tau}$ is the price of an underlying at the stopping time $\tau < T$. The underlying follows a standard ...
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Estimation of right truncated poisson process
I have following problem:
Imagine I generate large number of homogenous poisson process sample paths (by sample path I mean a sequence of arrival times $\tau_i$ all with the same intensity. However ...
5
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1
answer
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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
$$d\mu_t=-...
5
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1
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Marked poisson process vs compounded
I am a bit fuzzy about difference between compounded poisson process defined as
$$\sum_{i=1}^{N_t} D_i $$
where $N_t$ is poisson process and $ D_i $ are iid random variables
and marked poisson ...
5
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0
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Martingale property of inhomogenous poisson process
I have found this martingale property for an inhomogenous poisson process with intensity $\lambda(s)$ which I don't know how to prove. The text itself advises: "proceed using Monotone class theorem".
...
4
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1
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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.
...
4
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2
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249
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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, ...
3
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1
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How do I learn the stochastic calculus of Poisson processes?
I'm looking for references on the stochastic calculus of Poisson processes. My books tend to focus on derivative pricing, where Brownian motion reigns supreme. Maybe some jump-diffusion models thrown ...
3
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1
answer
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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}$-...
3
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1
answer
411
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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. ...
2
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0
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130
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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 $...
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1
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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 ...
1
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1
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198
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Prove the jump times of a poisson process in a given interval are uniformly distributed
Can someone provide a reference for this fact on the internet?
While I know this fact is proved in many text books, but I cannot find a proof of this fact very easily on the internet
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1
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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 ...
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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 ...
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Diffusive Limits for compound poisson process
I was reading about compounded Hawkes process and came across diffusive limit theorems.
Where can I find diffusive limit theorems for Poisson processes.
I am new to this area, is there a nice ...
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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 ...
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What’s the Ito’s lemma of compound Poisson process with two-sided jump and mean-reverting jump size?
In the book Cartea and Jaimungal (Algorithmic and High Frequency Trading, page 249.), the midprice dynamics follows
$ dS_t=\sigma dW_t+\epsilon^+ dM^+_t-\epsilon^- dM^-_t$ (10.22)
where $M^+_t$ and $M^...
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(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 ...
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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 ...