Markov with epsilon memory and Quantitative Strong Markov property

Question

We have a process $\{X_{t}\}_{t\geq 0}$ ,with fixed parameter $\epsilon>0$, starting from zero that satisfies

• The process is strictly monotone $X_{t+r}-X_{t}>0$ with moments existing $p\in(-\infty, \beta)$ for some $\beta>0$. (In the interval [0,1] we also have the lower bound $X_{t+r}-X_{t}>cr^{b}$ where $b>1$ and c is a random constant).
• $X_{t+r}-X_{r}\stackrel{d}{=}X_{t}$ for $r\geq 0$ (stationary increments).
• For $a<b<c<d$ the increments $X_{b}-X_{a}$ and $X_{d}-X_{c}$ are independent when $b+\epsilon<c$ (independence when epsilon away).

So we have Markov up to the recent past: $$P(X_{t+r}-X_{t}|\sigma(X_{s}),s\leq t)=P(X_{t+r}-X_{t}|\sigma(X_{t}-X_{s}),s\in [t-\epsilon,t]).$$

Given the hitting times $T_{a_{k}}:=\inf\{s: X_{s}\geq a_{k}\}$ for points $a_{k}\geq 0$, we want to test if they satisfy some weak stationarity. The stationarity $$T_{c+b}-T_{c}\stackrel{d}{=}T_{b}$$

is not true because contrary to Brownian motion we can have $c>b$ but the increments are dependent when $b+\epsilon>c >b$.

Q1: Have you seen an analogous process to $X_{t}$ anywhere else that has been studied?

Q2:Can we make some deterministic choice of $a_{k}\geq 0$ to give $$T_{a_{k}+t}-T_{a_{k}}\stackrel{d}{=}T_{t}$$ or having comparability for the moments $$c_{1} E[(T_{t})^{p}] \leq E[(T_{a_{k}+t}-T_{a_{k}})^{p}]\leq c_{2} E[(T_{t})^{p}]$$ Q3: Is there a quantitative Strong Markov property $$ |P[x>X_{T_{a}+t}-X_{T_{a}}]-P[x>X_{t}]|=g(\epsilon,a,t,x),$$ where $g(\epsilon,a,t,x)\to 0$ as $\epsilon\to 0$.

Attempts

1) Can we estimate the difference $$ |P[T_{a_{k}+x}- T_{a_{k}}>t]-P[T_{x}>t]|=g(\epsilon,a_{k},t),$$ where $g(\epsilon,a)\to 0$ as $\epsilon\to 0$. By increasing monotonicity, we have that $X_{y}$ can be inverted as the inverse to $T_{x}$ i.e. $X_{T_{x}}=x$ and so we study the difference: $$ |P[x>X_{T_{a}+t}-X_{T_{a}}]-P[x>X_{t}]|=g(\epsilon,a),$$ where $g(\epsilon,a)\to 0$ as $\epsilon\to 0$.

2)Using a proof similar to the one of the strong Markov property for Brownian motion we obtain: $$ X_{T_{a}+\epsilon+t}-X_{T_{a}+\epsilon}\stackrel{d}{=}X_{t}$$ or equivalently $$T_{a_{\epsilon}+x}-T_{a_{\epsilon}}\stackrel{d}{=}T_{x},$$ where $a_{\epsilon}:=X_{T_{a}+\epsilon}$. So then the question becomes whether we have $$c_{1} E[(T_{a_{\epsilon}+x}-T_{a_{\epsilon}})^{p}] \leq E[(T_{a+x}-T_{a})^{p}]\leq c_{2} E[(T_{a_{\epsilon}+x}-T_{a_{\epsilon}})^{p}]$$ since $$ E[(T_{a_{\epsilon}+x}-T_{a_{\epsilon}})^{p}]= E[(T_{x})^{p}].$$

Answer 1

Q1: I have not seen such a process, but I can easily imagine one: Let $Z_t$ be any non-negative Lévy process with positive drift, and let $X_t = \int_{t-\epsilon}^t Z_s ds$.

Q2: No, unless we know something more about $X_t$. Imagine the process $X_t$ as above, with $Z_t$ having frequent extremely large jumps. As $a \to \infty$, the random variable $T_{a+b} - T_a$ will converge to zero (for typically $\tfrac{d}{dt} X_t$ at $t = T_a$ will grow indefinitely as $a \to \infty)$, so there is no hope that $T_{a+b} - T_a$ will ever has equal distribution as $T_b$.

(This is related to the overshoot of $Z_t$ when it crosses the level $a$: as $a$ goes to infinite, $Z_t$ will typically pass through $a$ with a very large jump, so that $T^Z_{a+b} - T^Z_a$ will be equal to zero with an increasing probability. In this case $T^X_{a+b} - T^X_a$ will be very small, too. Here of course $T^Z_a$ denotes the first passage time for $Z_t$.)

Answer 2

Q1: Have you seen an analogous process to $X _t$ anywhere else? $$$$ I have not. Are you sure they exist ? If $\epsilon = 0$ the increments must be from positive, infinitely divisible distribution, none of which has moments of all orders. The strictly monotone condition also bothers me. It seems to only work with b = 1. If b > 1 then $X_t > t^b$ which is too big, while if b < 1 $X_t > (\frac 1 {\Delta t}) (\Delta t)^\beta$ which is too big.