Asaf Cohen Department of Mathematics University of Michigan Financial Mathematics Seminar University of Michigan September 10,

Contents Motivation: Brownian Motion with Unknown Drift Introduction: Bayesian Parameter Estimation, Examples Current & Future Work Scaled Renewal Processes Brownian Motion (with unknown drift) The Posterior Processes and their Limits 2

Contents Motivation: Brownian Motion with Unknown Drift Introduction: Bayesian Parameter Estimation, Examples Current & Future Work The Posterior Processes and their Limits 2 Scaled Renewal Processes Brownian Motion (with unknown drift)

A DM (Decision Maker) wants to estimate a parameter. His i.i.d. (given ) observations are Introduction: Bayesian Parameter Estimation, Examples Bayesian Parameter Estimation has a prior distribution The DM chooses an estimator that minimizes the expected loss function Bayesian A fundamental tool is the posterior distribution Parameter Estimation 3 e.g.,

Sampling is stopped accordance with a stopping rule The sample size is not fixed in advance. 4 Introduction: Bayesian Parameter Estimation, Examples Bayesian Sequential Parameter Estimation A fundamental tool is the posterior process The DM chooses an estimator that minimizes the expected loss function has a prior distribution Bayesian Parameter Estimation Sequential e.g., A DM wants to estimate a parameter

The samples are observed continuously 5 Introduction: Bayesian Parameter Estimation, Examples Bayesian Sequential Parameter Estimation with Renewal Process has a prior distribution t 2 3 Time 1 The posterior process A DM wants to estimate a parameter

The DM continuously observes 6 Introduction: Bayesian Parameter Estimation, Examples Bayesian Sequential Parameter Estimation with Brownian Noise has a prior distribution The posterior process A DM wants to estimate a parameter Standard Brownian Motion noise coefficient

The DM continuously observes 6 Introduction: Bayesian Parameter Estimation, Examples Brownian Motion with Unknown Drift has a prior distribution The posterior process A DM wants to estimate a parameter Standard Brownian Motion noise coefficient

Contents Motivation: Brownian Motion with Unknown Drift Introduction: Bayesian Parameter Estimation, Examples Current & Future Work The Posterior Processes and their Limits 7 Scaled Renewal Processes Brownian Motion (with unknown drift)

Observation: 8 The Model has a prior distribution The posterior process A DM wants to estimate a parameter Motivation: Brownian Motion with Unknown Drift

9 The Model has a prior distribution e.g., A common solution: A First Exit Time of A First Exit Time of the posterior process A DM wants to estimate a parameter. Observation:

10 Literature Motivation: Brownian Motion with Unknown Drift Kalman and Bucy (1961) Zakai (1969) - Bayesian Posterior Process Filtering Theory Shiryaev (1978) - formulated the problem (discrete and continuous time) Gapeev and Peskir (2004) - Finite Horizon Gapeev and Shiryaev (2011) - General Diffusion Zhitlukhin and Shiryaev (2011) - Three hypotheses Buonaguidi and Muliere (2013) - Levy Processes Sequential 2-Hypothesis Testing Berry and Friestedt (1985) - formulated the problem C. and Solan (2013) - Levy Processes Bayesian Brownian Bandit Economics Bolton and Harris (1999), Felli and Harris (1996), Bergemann and Valimaki (1997), Keller and Rady (1999), Moscarini (2005), Jovanovic (1979)...

11 Motivation: Brownian Motion with Unknown Drift

11 Motivation: Brownian Motion with Unknown Drift

12 Is This Model Useful? has a prior distribution e.g., A common solution: A First Exit Time of Observation: A First Exit Time of the posterior process A DM wants to estimate a parameter. a “discrete process” What if ? Advantages of the Brownian model: easier for investigation Motivation: Bayesian Parameter Estimation with Brownian Noise Slide from before

13 Is This Model Useful? Motivation: Bayesian Parameter Estimation with Brownian Noise Kalman and Bucy (1961) Zakai (1969) - Bayesian Posterior Process Filtering Theory Shiryaev (1978) - formulated the problem (discrete and continuous time) Gapeev and Peskir (2004) - Finite Horizon Gapeev and Shiryaev (2011) - General Diffusion Zhitlukhin and Shiryaev (2011) - Three hypotheses Buonaguidi and Muliere (2013) - Levy Processes Sequential 2-Hypothesis Testing Berry and Friestedt (1985) - formulated the problem C. and Solan (2013) - Levy Processes Bayesian Brownian Bandit Economics Bolton and Harris (1999), Felli and Harris (1996), Bergemann and Valimaki (1997), Keller and Rady (1999), Moscarini (2005), Jovanovic (1979)... No Justification Slide from before

14 Is This Model Useful? Suppose that 1. For a given rule of strategy does the losses satisfy 2.Are the optimal rules of strategies for the observed processes and relatively close? 3.Are the optimal losses relatively close? Usually not! So… Why to study Bayesian parameter estimation with Brownian noise? Motivation: Brownian Motion with Unknown Drift Questions:

14 Is This Model Useful? Suppose that 1. For a given rule of strategy does the losses satisfy 2.Are the optimal rules of strategies for the observed processes and relatively close? 3.Are the optimal losses relatively close? Usually not! So… Why to study Bayesian parameter estimation with Brownian noise? Motivation: Brownian Motion with Unknown Drift Questions: Weird phenomena: a different approximation works…. How to calculate? Fundamental tool: posterior processes Why different?

Contents Motivation: Brownian Motion with Unknown Drift Introduction: Bayesian Parameter Estimation, Examples Current & Future Work The Posterior Processes and their Limits 15 Scaled Renewal Processes Brownian Motion (with unknown drift)

are i.i.d. distributed as the RV 16 Renewal Process with Rate 1 Scaled Renewal Processes Brownian Motion (unknown drift) 2 3 Time 1 Renewal Process with Rate 2 3 Time 1 t

17 High Rate Take. How behaves? By proper scaling: Similar to Brownian Motion FCLT Scaled Renewal Processes Brownian Motion (unknown drift)

18 High and Unknown Rate - The Proper Scaling is chosen at time The DM continuously observes FCLT Why ? Scaled Renewal Processes Brownian Motion (unknown drift)

Contents Motivation: Brownian Motion with Unknown Drift Introduction: Bayesian Parameter Estimation, Examples Current & Future Work The Posterior Processes and their Limits 19 Scaled Renewal Processes Brownian Motion (with unknown drift)

20 The Posterior Processes and their Limits The Posterior Processes

21 The Posterior Processes and their Limits Diffusion Limits has a density with the support 2 3 Time 1

22 The Posterior Processes and their Limits Diffusion Limits has a density with the support Why ?

23 The Posterior Processes and their Limits ? Using the Taylor’s Series Origins of

24 The Posterior Processes and their Limits Heuristics for Why ? Recall that The smaller : “easier” to estimate larger When is a “large” R.V.? The interarrival time for the discussion, or

The Posterior Processes and their Limits Why ? recall that the smaller : “easier” to estimate larger When is a “large” R.V.? The interarrival time for the discussion, or “easier” to estimate ? ? ? x 25 Heuristics for

26 The Posterior Processes and their Limits The New, and equality holds iff. What is the relation between and ? In fact can be arbitrary large So, can be very different from

27 The Posterior Processes and their Limits Intuition (Theorem 2) is a sufficient statistic for : Usually is not a sufficient statistic for : However, if, it is sufficient! and since, then.

28 The Posterior Processes and their Limits Proof (Theorem 2) Cauchy– Schwarz Equality iff and are linearly dependent iff

29 The Posterior Processes and their Limits Back to the Motivation Since can be arbitrary large can be very different from

Contents Motivation: Brownian Motion with Unknown Drift Introduction: Bayesian Parameter Estimation, Examples The Posterior Processes and their Limits Current & Future Work 30 Scaled Renewal Processes Brownian Motion (with unknown drift)

2. - replace FCLT with stable FCLT. 31 Current & Future Work A) Behavior of posteriors in more general frameworks: B) The Disorder Problem 1. General diffusions (e.g., Zakai Equation type). Assume that at some unobservable random time the drift of a Brownian motion (rate of an arrival process) changes. similar structure. Find the posterior belief that the change has already occurred?

32 Current & Future Work C) Apply the results in context of queues with uncertainty about the service/arrival rates, asymptotic solution using Bayesian bandits. D) Apply the results in context of risk processes. Arrival process General CDF customers: Bandit (index solution) customers: non-classical Bandit (empty periods) Asymptotic solution under heavy traffic using BM with unknown drift router

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