How To Detect A Salami Slicer : A Stochastic Controller-And-Stopper Game With Unknown Competition

• 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.

Subject headings

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)

Keyword

fraud detection

optimal stopping

stochastic game theory

stochastic optimal control

Publication and Content Type

ref (subject category)

art (subject category)