Linear Filtering of general Gaussian processes

Transcription

1 Linear Filtering of general Gaussian processes Vít Kubelka Advisor: prof. RNDr. Bohdan Maslowski, DrSc. Robust 2018 Department of Probability and Statistics Faculty of Mathematics and Physics Charles University in Prague 25. January 2018

2 Stochastic differential equation example - Vasicek model {X(t), t 0} 2 Linear Filtering of general Gaussian processes

3 Stochastic differential equation example - Vasicek model {X(t), t 0} dx t = λ(µ X t ) dt + σ dw t, X 0 = x, λ, σ > 0 3 Linear Filtering of general Gaussian processes

4 Stochastic differential equation example - Vasicek model {X(t), t 0} dx t = λ(µ X t ) dt + σ dw t, X 0 = x, λ, σ > 0 X t = e λt x + µ(1 e λt ) + σe λt t 0 e λs dw s 4 Linear Filtering of general Gaussian processes

5 Stochastic differential equation example - Vasicek model {X(t), t 0} dx t = λ(µ X t ) dt + σ dw t, X 0 = x, λ, σ > 0 X t = e λt x + µ(1 e λt ) + σe λt t 0 e λs dw s Vasicek model with parameters: µ = 5, λ = 0.5, σ = 1 and x = 20 5 Linear Filtering of general Gaussian processes

6 One - dimensional linear filtering example - simulation dy t = hx t dt + σ 1 dw 1 t, Y 0 = y 6 Linear Filtering of general Gaussian processes

7 One - dimensional linear filtering example - simulation dy t = hx t dt + σ 1 dw 1 t, Y 0 = y dx t = ax t dt + σ dw t, X 0 = x 7 Linear Filtering of general Gaussian processes

8 One - dimensional linear filtering example - simulation dy t = hx t dt + σ 1 dw 1 t, Y 0 = y dx t = ax t dt + σ dw t, X 0 = x W 1 and W are independent, h, σ 1, a, σ > 0 8 Linear Filtering of general Gaussian processes

9 One - dimensional linear filtering example - simulation dy t = hx t dt + σ 1 dw 1 t, Y 0 = y dx t = ax t dt + σ dw t, X 0 = x W 1 and W are independent, h, σ 1, a, σ > 0 Blue is the observation process Y, red is its drift process X (the signal) and green is the estimate of the drift process (the filter). Parameters are h = 3, σ 1 = 5, a = 2, σ = 10, y = 30 and x = 0. 9 Linear Filtering of general Gaussian processes

10 One - dimensional linear filtering example - stock prediction Stock evolution in time. Source: Kaggle - Two Sigma Financial Modelling Challenge, stock id Linear Filtering of general Gaussian processes

11 One - dimensional linear filtering example - stock prediction Stock evolution in time. Source: Kaggle - Two Sigma Financial Modelling Challenge, stock id 816. Slope of the stock modelled by Kalman bucy filter. Parameters of the modell: proces: h = 0.01 and σ 1 = 5, slope: a = 2 and σ = Linear Filtering of general Gaussian processes

12 Filtering of general Gaussian signal in separable Hilbert space Theorem Let us assume V is a general separable Hilbert space, signal {X t, t 0} is a general centered Gaussian process in V, the real - valued observation process {Y t, t 0} is given as Y t = t 0 A(s)Xs ds + Wt, Y 0 = y, where A is a bounded linear operator from V to IR and {W t, t 0} is a real - valued Wiener process independent of the signal X. Let {Ft Y, t 0} be the sigma algebra generated by observation process Y. Then the filter X t = IE[X t Ft Y ] is given by stochastic integral equation: X t = ( ) t Φ(t, 0 s)a (s) dy s (s)a(s) X s ds, where for all 0 s t and Φ(t, s) = IE[X tx s ] s 0 Φ(t, r)a (r)a(r)φ (s, r) dr Φ(t, t) = IE[(X t X t)(x t X t) ]. 12 Linear Filtering of general Gaussian processes

13 Applications Gaussian signal with values in space of continuous functions 13 Linear Filtering of general Gaussian processes

14 Applications Gaussian signal with values in space of continuous functions Heat equation with gaussian noise: X = {X(t, x), t 0, x D} dx(t, x) = X(t, x)dt + dw t, X(t, x) = n i=1 d 2 X(t,x) d 2 x i 14 Linear Filtering of general Gaussian processes

15 Applications Gaussian signal with values in space of continuous functions Heat equation with gaussian noise: X = {X(t, x), t 0, x D} dx(t, x) = X(t, x)dt + dw t, X(t, x) = n i=1 d 2 X(t,x) d 2 x i Musiela equation - forward rate curve evolution: dx(t, x) = ( dx(t,x) dx X = {X(t, x), t 0, x 0} + σ(t, x) x 0 σ(t, y) dy ) dt + σ(t, x) dw t 15 Linear Filtering of general Gaussian processes

16 Thank you for your attention References. [1] Kalman, R. E., & Bucy, R. S. (1961). New results in linear filtering and prediction theory. Journal of basic engineering, 83(1), [2] Kleptsyna, M. L., Kloeden, P. E., & Anh, V. V. (1998). Linear filtering with fractional Brownian motion. Stochastic Analysis and Applications, 16(5), [3] Kleptsyna, M. L., & Le Breton A. (2001). Optimal linear filtering of general multidimensional Gaussian processes and its application to Laplace transforms of quadratic functionals. Journal of Applied Mathematics and Stochastic Analysis, 14(3), Linear Filtering of general Gaussian processes