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How To Detect A Sal...
How To Detect A Salami Slicer : A Stochastic Controller-And-Stopper Game With Unknown Competition
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- Ekström, Erik, 1977- (författare)
- Uppsala universitet,Sannolikhetsteori och kombinatorik
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- Lindensjö, Kristoffer, 1979- (författare)
- Stockholms universitet,Matematiska institutionen,Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden.
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- Olofsson, Marcus (författare)
- Umeå universitet,Institutionen för matematik och matematisk statistik,Umeå Univ, Dept Math & Math Stat, S-90187 Umeå, Sweden.
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(creator_code:org_t)
- Society for Industrial and Applied Mathematics Publications, 2022
- 2022
- Engelska.
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Ingår i: SIAM Journal of Control and Optimization. - : Society for Industrial and Applied Mathematics Publications. - 0363-0129 .- 1095-7138. ; 60:1, s. 545-574
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Abstract
Ämnesord
Stäng
- We consider a stochastic game of control and stopping specified in terms of a process $X_t=-\theta \Lambda_t+W_t$, representing the holdings of Player 1, where $W$ is a Brownian motion, $\theta$ is a Bernoulli random variable indicating whether Player 2 is active or not, and $\Lambda$ is a nondecreasing continuous process representing the accumulated "theft" or "fraud" performed by Player 2 (if active) against Player 1. Player 1 cannot observe $\theta$ or $\Lambda$ directly but can merely observe the path of the process $X$ and may choose a stopping rule $\tau$ to deactivate Player 2 at a cost $M$. Player 1 thus does not know if she is the victim of fraud or not and operates in this sense under unknown competition. Player 2 can observe both $\theta$ and $W$ and seeks to choose a fraud strategy $\Lambda$ that maximizes the expected discounted amount ${\mathbb E} \left [ \left. \int _0^{\tau} e^{-rs} d\Lambda_s \right \vert \theta=1\right ],$ whereas Player 1 seeks to choose the stopping strategy $\tau$ so as to minimize the expected discounted cost ${\mathbb E} \left [\theta \int _0^{\tau} e^{-rs} d\Lambda_s + e^{-r\tau}M\I{\tau<\infty} \right ].$ This non-zero-sum game belongs to a class of stochastic dynamic games with unknown competition and continuous controls and is motivated by applications in fraud detection; it combines filtering (detection), stochastic control, optimal stopping, strategic features (games), and asymmetric information. We derive Nash equilibria for this game; for some parameter values we find an equilibrium in pure strategies, and for other parameter values we find an equilibrium by allowing for randomized stopping strategies.
Ämnesord
- NATURVETENSKAP -- Matematik -- Sannolikhetsteori och statistik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Probability Theory and Statistics (hsv//eng)
- NATURVETENSKAP -- Matematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics (hsv//eng)
Nyckelord
- fraud detection
- optimal stopping
- stochastic game theory
- stochastic optimal control
Publikations- och innehållstyp
- ref (ämneskategori)
- art (ämneskategori)
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