{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# lecture-11 - Chapter 11 Markov Examples Section 11.1 nds...

This preview shows pages 1–3. Sign up to view the full content.

Chapter 11 Markov Examples Section 11.1 finds the transition kernels for the Wiener process, as an example of how to manipulate such things. Section 11.2 looks at the evolution of densities under the action of the logistic map; this shows how deterministic dynamical systems can be brought under the sway of the theory we’ve developed for Markov processes. 11.1 Transition Kernels for the Wiener Process We have previously defined the Wiener process (Examples 38 and 78) as the real-valued process on R + with the following properties: 1. W (0) = 0; 2. For any three times t 1 t 2 t 3 , W ( t 3 ) - W ( t 2 ) | = W ( t 2 ) - W ( t 1 ) (inde- pendent increments); 3. For any two times t 1 t 2 , W ( t 2 ) - W ( t 1 ) N (0 , t 2 - t 1 ) (Gaussian increments); 4. Continuous sample paths (in the sense of Definition 72). Here we will use the Gaussian increment property to construct a transition kernel, and then use the independent increment property to show that these keernels satisfy the Chapman-Kolmogorov equation, and hence that there exist Markov processes with the desired finite-dimensional distributions. 60

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
CHAPTER 11. MARKOV EXAMPLES 61 First, notice that the Gaussian increments property gives us the transition probabilities: P ( W ( t 2 ) B | W ( t 1 ) = w 1 ) = P ( W ( t 2 ) - W ( t 1 ) B - w 1 ) (11.1) = B - w 1 du 1 2 π ( t 2 - t 1 ) e - u 2 2( t 2 - t 1 ) (11.2) = B dw 2 1 2 π ( t 2 - t 1 ) e - ( w 2 - w 1 ) 2 2( t 2 - t 1 ) (11.3) μ t 1 ,t 2 ( w 1 , B ) (11.4) To show that W ( t ) is a Markov process, we must show that, for any three times t 1 t 2 t 3 , μ t 1 ,t 2 μ t 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

lecture-11 - Chapter 11 Markov Examples Section 11.1 nds...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online