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Computing the Scaling Limit of a Nonnegative Markov Chain

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1

$begingroup$

Fix $alpha >0$, and for $h > 0$, consider the Markov kernel $K_h$ derived by composing the following two `moves':

1. From $x^t$, move to $x^t+1/2 = x^t + y^t$, where $y^t sim textGamma(alpha h, 1)$ .
   
   
   • To fix notation, by this I mean $p(y) propto y^alpha h-1 exp(-y)$.
   
   


2. From $x^t+1/2$, move to $x^t+1 = x^t cdot z^t$, where $z^t sim textBeta(alpha, alpha h)$.
   
   
   • Again, for clarity, by this I mean $p(z) propto z^alpha-1 (1-z)^alpha h-1$.
   
   

I use $K_h$ to denote the composite Markov kernel which takes me from $x^t$ to $x^t+1$.

I know that this chain has $textGamma(alpha, 1)$ as an invariant measure, and moreover, that the chain is reversible with respect to this measure.

I want to compute the behaviour as $h to 0^+$ of this chain. Initially, I thought that the chain would admit a diffusion limit (something like the CIR process), but after carrying out some simulations, it appears more likely that it's something like a jump-diffusion, or even a pure jump process.

Anyhow, I'm not sure how I should go about i) identifying a limiting process, and ii) proving rigorously that it is the true limit of these Markov kernels. I can identify that if I define

$$B(x;h) triangleq mathbfE_K_h [x^t+1 - x^t | x^t = x]$$

$$V(x;h) triangleq textbfVar_K_h [x^t+1 - x^t | x^t = x], $$

then

beginalign
B(x;h) &= h (alpha - x ) + o(h) \
V(x;h) &= h left alpha^2 + fracalphaalpha + 1 x^2 right + o(h).
endalign

I initially though that this would mean the limiting process would be

$$dX_t = ( alpha - X_t) dt + sqrtalpha^2 + fracalphaalpha + 1 x^2 dW_t$$

but this both i) doesn't have $textGamma(alpha, 1)$ as a stationary measure, and ii) doesn't account for the jump behaviour I observe in the simulations.

Any advice would be well-received.

probability-theory stochastic-processes markov-chains markov-process weak-convergence

share|cite|improve this question

edited Mar 11 at 20:28

πr8

asked Feb 14 at 21:53

πr8πr8

9,69331025

$endgroup$

add a comment | 

1

$begingroup$

Fix $alpha >0$, and for $h > 0$, consider the Markov kernel $K_h$ derived by composing the following two `moves':

1. From $x^t$, move to $x^t+1/2 = x^t + y^t$, where $y^t sim textGamma(alpha h, 1)$ .
   
   
   • To fix notation, by this I mean $p(y) propto y^alpha h-1 exp(-y)$.
   
   


2. From $x^t+1/2$, move to $x^t+1 = x^t cdot z^t$, where $z^t sim textBeta(alpha, alpha h)$.
   
   
   • Again, for clarity, by this I mean $p(z) propto z^alpha-1 (1-z)^alpha h-1$.
   
   

I use $K_h$ to denote the composite Markov kernel which takes me from $x^t$ to $x^t+1$.

I know that this chain has $textGamma(alpha, 1)$ as an invariant measure, and moreover, that the chain is reversible with respect to this measure.

I want to compute the behaviour as $h to 0^+$ of this chain. Initially, I thought that the chain would admit a diffusion limit (something like the CIR process), but after carrying out some simulations, it appears more likely that it's something like a jump-diffusion, or even a pure jump process.

Anyhow, I'm not sure how I should go about i) identifying a limiting process, and ii) proving rigorously that it is the true limit of these Markov kernels. I can identify that if I define

$$B(x;h) triangleq mathbfE_K_h [x^t+1 - x^t | x^t = x]$$

$$V(x;h) triangleq textbfVar_K_h [x^t+1 - x^t | x^t = x], $$

then

beginalign
B(x;h) &= h (alpha - x ) + o(h) \
V(x;h) &= h left alpha^2 + fracalphaalpha + 1 x^2 right + o(h).
endalign

I initially though that this would mean the limiting process would be

$$dX_t = ( alpha - X_t) dt + sqrtalpha^2 + fracalphaalpha + 1 x^2 dW_t$$

but this both i) doesn't have $textGamma(alpha, 1)$ as a stationary measure, and ii) doesn't account for the jump behaviour I observe in the simulations.

Any advice would be well-received.

probability-theory stochastic-processes markov-chains markov-process weak-convergence

share|cite|improve this question

edited Mar 11 at 20:28

πr8

asked Feb 14 at 21:53

πr8πr8

9,69331025

$endgroup$

add a comment | 

$begingroup$

Fix $alpha >0$, and for $h > 0$, consider the Markov kernel $K_h$ derived by composing the following two `moves':

1. From $x^t$, move to $x^t+1/2 = x^t + y^t$, where $y^t sim textGamma(alpha h, 1)$ .
   
   
   • To fix notation, by this I mean $p(y) propto y^alpha h-1 exp(-y)$.
   
   


2. From $x^t+1/2$, move to $x^t+1 = x^t cdot z^t$, where $z^t sim textBeta(alpha, alpha h)$.
   
   
   • Again, for clarity, by this I mean $p(z) propto z^alpha-1 (1-z)^alpha h-1$.
   
   

I use $K_h$ to denote the composite Markov kernel which takes me from $x^t$ to $x^t+1$.

I know that this chain has $textGamma(alpha, 1)$ as an invariant measure, and moreover, that the chain is reversible with respect to this measure.

I want to compute the behaviour as $h to 0^+$ of this chain. Initially, I thought that the chain would admit a diffusion limit (something like the CIR process), but after carrying out some simulations, it appears more likely that it's something like a jump-diffusion, or even a pure jump process.

Anyhow, I'm not sure how I should go about i) identifying a limiting process, and ii) proving rigorously that it is the true limit of these Markov kernels. I can identify that if I define

$$B(x;h) triangleq mathbfE_K_h [x^t+1 - x^t | x^t = x]$$

$$V(x;h) triangleq textbfVar_K_h [x^t+1 - x^t | x^t = x], $$

then

beginalign
B(x;h) &= h (alpha - x ) + o(h) \
V(x;h) &= h left alpha^2 + fracalphaalpha + 1 x^2 right + o(h).
endalign

I initially though that this would mean the limiting process would be

$$dX_t = ( alpha - X_t) dt + sqrtalpha^2 + fracalphaalpha + 1 x^2 dW_t$$

but this both i) doesn't have $textGamma(alpha, 1)$ as a stationary measure, and ii) doesn't account for the jump behaviour I observe in the simulations.

Any advice would be well-received.

probability-theory stochastic-processes markov-chains markov-process weak-convergence

share|cite|improve this question

edited Mar 11 at 20:28

πr8

asked Feb 14 at 21:53

πr8πr8

9,69331025

$endgroup$

share|cite|improve this question

edited Mar 11 at 20:28

πr8

asked Feb 14 at 21:53

πr8πr8

9,69331025

share|cite|improve this question

edited Mar 11 at 20:28

πr8

asked Feb 14 at 21:53

πr8πr8

9,69331025

asked Feb 14 at 21:53

πr8πr8

9,69331025

asked Feb 14 at 21:53

πr8πr8

9,69331025