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Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it Markov. For instance, $(M_t, B_t)$ and $(M_t, M_t - B_t)$ are Markov processes.

This prompts the question: what else can we augment $M_t$ with to get a Markov process? In particular, consider $$ X_t := (M_t, t - \sup\{s \leq t \mid B_s = M_t\}) = (M_t, \text{"time since running maximum was last attained"}). $$$$ \begin{aligned} X_t &:= (M_t, t - \sup\{s \leq t \mid B_s = M_t\}) \\ & = (M_t, \text{"time since running maximum was last attained"}). \end{aligned} $$ I'm wondering:

  • Is $X_t$ a Markov process?
  • Is there a standard name and/or reference for $X_t$?
  • Does the conclusion change if instead of $B_t$, we use a different diffusion process?

Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it Markov. For instance, $(M_t, B_t)$ and $(M_t, M_t - B_t)$ are Markov processes.

This prompts the question: what else can we augment $M_t$ with to get a Markov process? In particular, consider $$ X_t := (M_t, t - \sup\{s \leq t \mid B_s = M_t\}) = (M_t, \text{"time since running maximum was last attained"}). $$ I'm wondering:

  • Is $X_t$ a Markov process?
  • Is there a standard name and/or reference for $X_t$?
  • Does the conclusion change if instead of $B_t$, we use a different diffusion process?

Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it Markov. For instance, $(M_t, B_t)$ and $(M_t, M_t - B_t)$ are Markov processes.

This prompts the question: what else can we augment $M_t$ with to get a Markov process? In particular, consider $$ \begin{aligned} X_t &:= (M_t, t - \sup\{s \leq t \mid B_s = M_t\}) \\ & = (M_t, \text{"time since running maximum was last attained"}). \end{aligned} $$ I'm wondering:

  • Is $X_t$ a Markov process?
  • Is there a standard name and/or reference for $X_t$?
  • Does the conclusion change if instead of $B_t$, we use a different diffusion process?
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Ziv
Ziv
  • 398
  • 1
  • 6

Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it Markov. For instance, $(M_t, B_t)$ and $(M_t, M_t - B_t)$ are Markov processes.

This prompts the question: what else can we augment $M_t$ with to get a Markov process? In particular, consider $$ X_t := (M_t, t - \sup\{s \leq t \mid B_s = M_t\}) = (M_t, ``\text{time since running maximum was last attained''}). $$$$ X_t := (M_t, t - \sup\{s \leq t \mid B_s = M_t\}) = (M_t, \text{"time since running maximum was last attained"}). $$ I'm wondering:

  • Is $X_t$ a Markov process?
  • Is there a standard name and/or reference for $X_t$?
  • Does the conclusion change if instead of $B_t$, we use a different diffusion process?

Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it Markov. For instance, $(M_t, B_t)$ and $(M_t, M_t - B_t)$ are Markov processes.

This prompts the question: what else can we augment $M_t$ with to get a Markov process? In particular, consider $$ X_t := (M_t, t - \sup\{s \leq t \mid B_s = M_t\}) = (M_t, ``\text{time since running maximum was last attained''}). $$ I'm wondering:

  • Is $X_t$ a Markov process?
  • Is there a standard name and/or reference for $X_t$?
  • Does the conclusion change if instead of $B_t$, we use a different diffusion process?

Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it Markov. For instance, $(M_t, B_t)$ and $(M_t, M_t - B_t)$ are Markov processes.

This prompts the question: what else can we augment $M_t$ with to get a Markov process? In particular, consider $$ X_t := (M_t, t - \sup\{s \leq t \mid B_s = M_t\}) = (M_t, \text{"time since running maximum was last attained"}). $$ I'm wondering:

  • Is $X_t$ a Markov process?
  • Is there a standard name and/or reference for $X_t$?
  • Does the conclusion change if instead of $B_t$, we use a different diffusion process?
Source Link
Ziv
Ziv
  • 398
  • 1
  • 6

Running maximum/supremum of Brownian motion: add information to make it a Markov process?

Let $B_t$ be standard Brownian motion, and let $M_t = \sup_{0 \leq s \leq t} B_s$ be its running maximum. $M_t$ is not a Markov process, but we can augment it with additional information to make it Markov. For instance, $(M_t, B_t)$ and $(M_t, M_t - B_t)$ are Markov processes.

This prompts the question: what else can we augment $M_t$ with to get a Markov process? In particular, consider $$ X_t := (M_t, t - \sup\{s \leq t \mid B_s = M_t\}) = (M_t, ``\text{time since running maximum was last attained''}). $$ I'm wondering:

  • Is $X_t$ a Markov process?
  • Is there a standard name and/or reference for $X_t$?
  • Does the conclusion change if instead of $B_t$, we use a different diffusion process?