Questions tagged [poisson-process]

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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 ...
7
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3answers
342 views

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 ...
3
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1answer
113 views

Quadratic covariation of two correlated Poisson processes

Let $N_t \sim \text{Poisson}(\lambda t)$ and $M_t \sim \text{Poisson}(\theta \lambda t)$. We know that if $N$ and $M$ were independent, $dNdM = 0$ using polarization identity. We also know that $(dN)^...
3
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1answer
138 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 $\mathcal{F_t}$-...
3
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1answer
159 views

Ito multiplication with jumps

Let $\{N_t|0<t\leqslant T \}$ and $\{M_t|0<t\leqslant T \}$ be two Poisson processes with intensities $\lambda_n, \lambda_m>0$, respectively. Based on the implicit results of Corollaries 1 ...
3
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2answers
102 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, ...
2
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0answers
72 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 $...
-1
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1answer
282 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 ...
4
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1answer
116 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=-...
3
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1answer
208 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. ...
5
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0answers
49 views

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". ...
6
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0answers
116 views

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 ...
3
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0answers
115 views

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 ...
3
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1answer
294 views

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 ...
1
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1answer
67 views

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