Multiresolution Hilbert Approach to Multidimensional Gauss-Markov Processes

Figure 3

(a) Basis for the construction of the Integrated Wiener process (𝑑=2). Plain red curve: 𝜓_1,1, dashed red: 𝜓_2,1, plain blue: 𝜓_1,2 and dashed blue: 𝜓_2,2. (b) 10-steps construction of the process. Moreover, we observe that each basis accounts separately for different aspects of sample path: 𝜓_2,1 fixes the value of the integrated process at the middle point 𝑚_𝑛,𝑘 and 𝜓_2,2 the value of the derivative of the process at the endpoints {𝑙_𝑛,𝑘,𝑟_𝑛,𝑘} in relationship with the value of the Wiener process at the middle point, whose contributions are split between functions 𝜓_1,1 and 𝜓_1,2 (see Figure 3).

(a)

(b)