close

Вход

Забыли?

вход по аккаунту

?

Фильтрация сигналов в стохастических системах диффузионно-скачкообразного типа на основе метода статистических испытаний.

код для вставкиСкачать
2015
ɇȺɍɑɇɕɃ ȼȿɋɌɇɂɄ ɆȽɌɍ ȽȺ
ʋ 220
ɍȾɄ 519.676
ɎɂɅɖɌɊȺɐɂə ɋɂȽɇȺɅɈȼ ȼ ɋɌɈɏȺɋɌɂɑȿɋɄɂɏ ɋɂɋɌȿɆȺɏ
ȾɂɎɎɍɁɂɈɇɇɈ-ɋɄȺɑɄɈɈȻɊȺɁɇɈȽɈ ɌɂɉȺ
ɇȺ ɈɋɇɈȼȿ ɆȿɌɈȾȺ ɋɌȺɌɂɋɌɂɑȿɋɄɂɏ ɂɋɉɕɌȺɇɂɃ1
Ʉ.Ⱥ. ɊɕȻȺɄɈȼ
ȼ ɫɬɚɬɶɟ ɪɚɡɜɢɜɚɸɬɫɹ ɦɟɬɨɞɵ ɫɜɟɞɟɧɢɹ ɡɚɞɚɱɢ ɨɩɬɢɦɚɥɶɧɨɣ ɧɟɥɢɧɟɣɧɨɣ ɮɢɥɶɬɪɚɰɢɢ ɜ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ
ɫɢɫɬɟɦɚɯ ɞɢɮɮɭɡɢɨɧɧɨ-ɫɤɚɱɤɨɨɛɪɚɡɧɨɝɨ ɬɢɩɚ ɤ ɡɚɞɚɱɟ ɚɧɚɥɢɡɚ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ,
ɬɪɚɟɤɬɨɪɢɢ ɤɨɬɨɪɨɣ ɦɨɝɭɬ ɩɨɥɭɱɚɬɶ ɫɥɭɱɚɣɧɵɟ ɩɪɢɪɚɳɟɧɢɹ, ɪɚɡɜɟɬɜɥɹɬɶɫɹ ɢ ɨɛɪɵɜɚɬɶɫɹ ɜ ɫɥɭɱɚɣɧɵɟ ɦɨɦɟɧɬɵ
ɜɪɟɦɟɧɢ. Ɋɚɧɟɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɦɟɬɨɞɵ ɢ ɚɥɝɨɪɢɬɦɵ ɛɵɥɢ ɩɪɟɞɥɨɠɟɧɵ ɢ ɚɩɪɨɛɢɪɨɜɚɧɵ ɞɥɹ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ
ɫɢɫɬɟɦ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɬɢɩɚ.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɚɩɨɫɬɟɪɢɨɪɧɚɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ, ɜɟɬɜɹɳɢɟɫɹ ɩɪɨɰɟɫɫɵ, ɦɟɬɨɞ Ɇɨɧɬɟ-Ʉɚɪɥɨ,
ɦɟɬɨɞ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɢɫɩɵɬɚɧɢɣ, ɨɩɬɢɦɚɥɶɧɚɹ ɮɢɥɶɬɪɚɰɢɹ, ɫɤɚɱɤɨɨɛɪɚɡɧɵɣ ɩɪɨɰɟɫɫ, ɫɬɨɯɚɫɬɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ,
ɭɪɚɜɧɟɧɢɟ Ⱦɭɧɤɚɧɚ–Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ.
ȼȼȿȾȿɇɂȿ
Ɋɚɧɟɟ ɜ ɪɚɛɨɬɚɯ [5], [6] ɛɵɥɢ ɩɪɟɞɥɨɠɟɧɵ ɧɨɜɵɟ ɦɟɬɨɞɵ ɢ ɚɥɝɨɪɢɬɦɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ
ɨɩɬɢɦɚɥɶɧɨɣ ɮɢɥɶɬɪɚɰɢɢ ɫɢɝɧɚɥɨɜ ɜ ɧɟɥɢɧɟɣɧɵɯ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɫɢɫɬɟɦɚɯ ɞɢɮɮɭɡɢɨɧɧɨɝɨ
ɬɢɩɚ, ɬ.ɟ. ɜ ɫɢɫɬɟɦɚɯ, ɦɨɞɟɥɢ ɨɛɴɟɤɬɚ ɧɚɛɥɸɞɟɧɢɹ ɢ ɢɡɦɟɪɢɬɟɥɹ ɞɥɹ ɤɨɬɨɪɵɯ ɨɩɢɫɵɜɚɸɬɫɹ ɫ
ɩɨɦɨɳɶɸ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ. ɉɪɟɞɥɨɠɟɧɧɵɟ ɦɟɬɨɞɵ ɢ ɚɥɝɨɪɢɬɦɵ
ɛɚɡɢɪɭɸɬɫɹ ɧɚ ɨɛɳɧɨɫɬɢ ɫɬɪɭɤɬɭɪɵ ɭɪɚɜɧɟɧɢɣ ɨɩɬɢɦɚɥɶɧɨɣ ɧɟɥɢɧɟɣɧɨɣ ɮɢɥɶɬɪɚɰɢɢ ɞɥɹ
ɧɟɧɨɪɦɢɪɨɜɚɧɧɨɣ ɚɩɨɫɬɟɪɢɨɪɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ ɨɛɴɟɤɬɚ
ɧɚɛɥɸɞɟɧɢɹ, ɚ ɢɦɟɧɧɨ ɭɪɚɜɧɟɧɢɣ ɬɢɩɚ Ⱦɭɧɤɚɧɚ–Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɵɯ
ɢɡɦɟɪɟɧɢɹɯ, ɢ ɨɛɨɛɳɟɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ Ɏɨɤɤɟɪɚ–ɉɥɚɧɤɚ–Ʉɨɥɦɨɝɨɪɨɜɚ, ɜɤɥɸɱɚɸɳɟɝɨ
ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɫɥɚɝɚɟɦɵɟ – ɮɭɧɤɰɢɢ ɩɨɝɥɨɳɟɧɢɹ ɢ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ [1], [3], [4].
Ɉɞɢɧ ɢɡ ɨɫɧɨɜɧɵɯ ɪɟɡɭɥɶɬɚɬɨɜ ɪɚɛɨɬ [5], [6] – ɷɬɨ ɨɛɨɫɧɨɜɚɧɢɟ ɬɨɝɨ, ɱɬɨ ɡɚɞɚɱɭ
ɨɩɬɢɦɚɥɶɧɨɣ ɮɢɥɶɬɪɚɰɢɢ ɫɢɝɧɚɥɨɜ ɦɨɠɧɨ ɫɜɟɫɬɢ ɤ ɡɚɞɚɱɟ ɫɬɚɬɢɫɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ
ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɫɨ ɫɥɭɱɚɣɧɵɦɢ ɨɛɪɵɜɚɦɢ ɢ ɜɟɬɜɥɟɧɢɹɦɢ ɬɪɚɟɤɬɨɪɢɣ,
ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɣ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɦɟɬɨɞɨɦ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɢɫɩɵɬɚɧɢɣ ɢɥɢ ɦɟɬɨɞɨɦ ɆɨɧɬɟɄɚɪɥɨ. ȿɫɬɶ ɞɜɚ ɜɚɪɢɚɧɬɚ ɩɨɞɨɛɧɵɯ ɦɟɬɨɞɨɜ ɩɨɥɭɱɟɧɢɹ ɨɩɬɢɦɚɥɶɧɨɣ ɨɰɟɧɤɢ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ
ɩɨ ɪɟɡɭɥɶɬɚɬɚɦ ɢɡɦɟɪɟɧɢɣ: ɩɟɪɜɵɣ ɨɫɧɨɜɚɧ ɧɚ ɤɥɚɫɫɢɱɟɫɤɨɦ, ɚ ɜɬɨɪɨɣ – ɧɚ ɪɨɛɚɫɬɧɨɦ ɭɪɚɜɧɟɧɢɢ
Ⱦɭɧɤɚɧɚ–Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ. Ɉɛɚ ɜɚɪɢɚɧɬɚ ɦɟɬɨɞɨɜ ɚɩɪɨɛɢɪɨɜɚɧɵ ɧɚ ɪɟɲɟɧɢɢ ɦɨɞɟɥɶɧɵɯ ɡɚɞɚɱ,
ɨɧɢ ɢɦɟɸɬ ɫɜɨɢ ɞɨɫɬɨɢɧɫɬɜɚ ɢ ɧɟɞɨɫɬɚɬɤɢ, ɜ ɧɟɤɨɬɨɪɨɦ ɫɦɵɫɥɟ ɞɨɩɨɥɧɹɹ ɞɪɭɝ ɞɪɭɝɚ.
Ⱦɥɹ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɞɢɮɮɭɡɢɨɧɧɨ-ɫɤɚɱɤɨɨɛɪɚɡɧɨɝɨ ɬɢɩɚ, ɤɨɝɞɚ ɦɨɞɟɥɶ ɨɛɴɟɤɬɚ
ɧɚɛɥɸɞɟɧɢɹ ɨɩɢɫɵɜɚɟɬɫɹ ɫɬɨɯɚɫɬɢɱɟɫɤɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɫ ɩɭɚɫɫɨɧɨɜɫɤɨɣ
ɫɨɫɬɚɜɥɹɸɳɟɣ, ɦɨɠɧɨ ɩɪɢɦɟɧɢɬɶ ɚɧɚɥɨɝɢɱɧɵɣ ɩɨɞɯɨɞ. Ʉɚɤɢɦ ɨɛɪɚɡɨɦ ɫɜɨɞɢɬɶ ɡɚɞɚɱɭ
ɨɩɬɢɦɚɥɶɧɨɣ ɮɢɥɶɬɪɚɰɢɢ ɤ ɡɚɞɚɱɟ ɚɧɚɥɢɡɚ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɫ ɨɛɪɵɜɚɦɢ,
ɪɚɡɪɵɜɚɦɢ ɢ ɜɟɬɜɥɟɧɢɹɦɢ ɬɪɚɟɤɬɨɪɢɣ, ɛɟɪɹ ɡɚ ɨɫɧɨɜɭ ɤɥɚɫɫɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ Ⱦɭɧɤɚɧɚ–
Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ, ɛɵɥɨ ɩɨɤɚɡɚɧɨ ɜ ɪɚɛɨɬɟ [8]. ȼ ɧɚɫɬɨɹɳɟɣ ɪɚɛɨɬɟ ɩɪɟɞɥɚɝɚɟɬɫɹ ɩɟɪɟɣɬɢ ɤ
ɡɚɞɚɱɟ ɚɧɚɥɢɡɚ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɧɚ ɛɚɡɟ ɪɨɛɚɫɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ Ⱦɭɧɤɚɧɚ–
Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ. ɉɪɢɱɟɦ, ɤɚɤ ɢ ɪɚɧɟɟ [8], ɪɚɡɪɵɜɵ ɬɪɚɟɤɬɨɪɢɣ ɞɥɹ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ
ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɨɛɭɫɥɨɜɥɟɧɵ ɢɫɯɨɞɧɨɣ ɩɨɫɬɚɧɨɜɤɨɣ ɡɚɞɚɱɢ, ɨɞɧɚɤɨ ɩɚɪɚɦɟɬɪɵ ɪɚɡɪɵɜɨɜ
ɦɟɧɹɸɬɫɹ ɢ ɧɚɪɭɲɚɟɬɫɹ «ɛɚɥɚɧɫ» ɦɟɠɞɭ ɩɨɝɥɨɳɟɧɢɟɦ ɢ ɜɨɫɫɬɚɧɨɜɥɟɧɢɟɦ ɬɪɚɟɤɬɨɪɢɣ. ɗɬɢ
ɩɚɪɚɦɟɬɪɵ, ɚ ɬɚɤɠɟ ɜɟɪɨɹɬɧɨɫɬɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢɧɬɟɪɜɚɥɨɜ ɜɪɟɦɟɧɢ ɦɟɠɞɭ
ɜɟɬɜɥɟɧɢɹɦɢ ɢ ɞɨ ɨɛɪɵɜɚ ɡɚɜɢɫɹɬ ɨɬ ɬɟɤɭɳɢɯ ɢɡɦɟɪɟɧɢɣ ɨɰɟɧɢɜɚɟɦɨɝɨ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ.
1
Ɋɚɛɨɬɚ ɜɵɩɨɥɧɟɧɚ ɩɪɢ ɮɢɧɚɧɫɨɜɨɣ ɩɨɞɞɟɪɠɤɟ ɊɎɎɂ (ɩɪɨɟɤɬ ʋ 13-08-00323-ɚ).
74
Ʉ.Ⱥ. Ɋɵɛɚɤɨɜ
Ⱥɤɬɭɚɥɶɧɨɫɬɶ ɷɬɨɣ ɪɚɛɨɬɵ ɨɛɭɫɥɨɜɥɟɧɚ ɛɨɥɶɲɢɦ ɱɢɫɥɨɦ ɩɪɢɤɥɚɞɧɵɯ ɡɚɞɚɱ ɜ
ɪɚɞɢɨɬɟɯɧɢɤɟ, ɧɚɜɢɝɚɰɢɢ, ɭɩɪɚɜɥɟɧɢɢ ɞɜɢɠɭɳɢɦɢɫɹ ɨɛɴɟɤɬɚɦɢ, ɬɪɟɛɭɸɳɢɯ ɷɮɮɟɤɬɢɜɧɵɯ
ɦɟɬɨɞɨɜ ɢ ɚɥɝɨɪɢɬɦɨɜ ɨɰɟɧɢɜɚɧɢɹ ɩɚɪɚɦɟɬɪɨɜ ɜ ɪɟɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ ɩɨ ɪɟɡɭɥɶɬɚɬɚɦ ɤɨɫɜɟɧɧɵɯ
ɢɡɦɟɪɟɧɢɣ ɧɚ ɮɨɧɟ ɩɨɦɟɯ.
1. ɉɈɋɌȺɇɈȼɄȺ ɁȺȾȺɑɂ
Ʉɚɤ ɢ ɜ [8], ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɦɨɞɟɥɶ ɨɛɴɟɤɬɚ ɧɚɛɥɸɞɟɧɢɹ, ɨɩɢɫɵɜɚɟɦɭɸ
ɫɬɨɯɚɫɬɢɱɟɫɤɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɂɬɨ ɫ ɩɭɚɫɫɨɧɨɜɫɤɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ:
dX (t ) = f (t , X (t )) dt + σ(t , X (t )) dW (t ) + dQ(t ), X (t0 ) = X 0 ,
(1)
ɝɞɟ X ∈ R n – ɜɟɤɬɨɪ ɫɨɫɬɨɹɧɢɹ; t ∈ T , T = [t0 , t1 ] – ɨɬɪɟɡɨɤ ɜɪɟɦɟɧɢ ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ ɫɢɫɬɟɦɵ;
f (t , x) : T × R n → R n – ɜɟɤɬɨɪ-ɮɭɧɤɰɢɹ n × 1 , σ(t , x) : T × R n → R n×s – ɦɚɬɪɢɱɧɚɹ ɮɭɧɤɰɢɹ n × s ;
W (t ) – s -ɦɟɪɧɵɣ ɫɬɚɧɞɚɪɬɧɵɣ ɜɢɧɟɪɨɜɫɤɢɣ ɩɪɨɰɟɫɫ, ɧɟ ɡɚɜɢɫɹɳɢɣ ɨɬ ɧɚɱɚɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɹ
X 0 , ɤɨɬɨɪɨɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɥɨɬɧɨɫɬɶɸ ɜɟɪɨɹɬɧɨɫɬɢ ϕ0 ( x) . Ⱦɚɥɟɟ, Q(t ) – ɨɛɳɢɣ ɩɭɚɫɫɨɧɨɜɫɤɢɣ
ɩɪɨɰɟɫɫ, ɡɚɞɚɧɧɵɣ ɜ ɮɨɪɦɟ
P (t )
Q(t ) = ¦ Δ k .
k =1
ȼ ɩɪɚɜɨɣ ɱɚɫɬɢ ɩɨɫɥɟɞɧɟɝɨ ɪɚɜɟɧɫɬɜɚ P (t ) – ɩɭɚɫɫɨɧɨɜɫɤɢɣ ɩɪɨɰɟɫɫ, Δ k – ɧɟɡɚɜɢɫɢɦɵɟ
ɫɥɭɱɚɣɧɵɟ ɜɟɤɬɨɪɵ ɢɡ R n , ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɤɨɬɨɪɵɯ ɡɚɞɚɧɨ ɩɥɨɬɧɨɫɬɶɸ ɜɟɪɨɹɬɧɨɫɬɢ ψ (τk , Δ) ,
ɬ.ɟ. ɜɟɤɬɨɪ ɫɨɫɬɨɹɧɢɹ X ɩɨɥɭɱɚɟɬ ɫɥɭɱɚɣɧɵɟ ɩɪɢɪɚɳɟɧɢɹ ɜ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ τ1 , τ2 ,... ∈ T ,
ɨɛɪɚɡɭɸɳɢɟ ɩɭɚɫɫɨɧɨɜɫɤɢɣ ɩɨɬɨɤ ɫɨɛɵɬɢɣ:
X (τk ) = X (τk − 0) + Δ k .
(2)
ȿɫɥɢ ɜɟɥɢɱɢɧɚ ɩɪɢɪɚɳɟɧɢɹ ɡɚɜɢɫɢɬ ɨɬ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ, ɬɨ ɢɫɩɨɥɶɡɭɟɬɫɹ
ɭɫɥɨɜɧɚɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ψ (τk , Δ | ξ) , ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɪɚɫɩɪɟɞɟɥɟɧɢɟ Δ k ɩɪɢ
ɭɫɥɨɜɢɢ X (τk − 0) = ξ . ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ψ (τk , Δ | ξ) = ψ (τk , Δ) . ɇɚɪɹɞɭ ɫ ψ (τk , Δ | ξ) ɜɜɟɞɟɦ
ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ η(τk , x | ξ) , ɯɚɪɚɤɬɟɪɢɡɭɸɳɭɸ ɪɚɫɩɪɟɞɟɥɟɧɢɟ X (τk ) ɩɪɢ ɭɫɥɨɜɢɢ
X (τk − 0) = ξ .
ɉɭɚɫɫɨɧɨɜɫɤɢɣ ɩɨɬɨɤ ɫɨɛɵɬɢɣ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ τ1 , τ2 ,... ,
ɚ ɬɚɤɠɟ ɩɭɚɫɫɨɧɨɜɫɤɢɣ ɩɪɨɰɟɫɫ P (t ) ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ λ(t , x) , ɬ.ɟ.
ɭɫɥɨɜɧɚɹ ɜɟɪɨɹɬɧɨɫɬɶ ɫɨɛɵɬɢɹ (2) ɩɪɢ X (t ) = x ɧɚ ɩɪɨɦɟɠɭɬɤɟ [t , t + Δt ] ɡɚɞɚɟɬɫɹ
ɪɚɜɟɧɫɬɜɨɦ
P(t , t + Δt ) = Pr( P(t + Δt ) − P(t ) = 1| X (t ) = x) = λ(t, x) Δt + o( Δt ).
Ɇɨɞɟɥɶ ɢɡɦɟɪɢɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ ɩɪɨɳɟ, ɱɟɦ ɜ [8], ɚ ɢɦɟɧɧɨ ɡɞɟɫɶ, ɤɚɤ ɢ ɜ [6], ɛɭɞɟɦ
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɬɚɰɢɨɧɚɪɧɭɸ ɦɨɞɟɥɶ:
dY (t ) = c( X (t )) dt + dV (t ), Y (t0 ) = Y0 = 0,
(3)
Ɏɢɥɶɬɪɚɰɢɹ ɫɢɝɧɚɥɨɜ ɜ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɫɢɫɬɟɦɚɯ…
ɢɥɢ
75
Z (t ) = c( X (t )) + N (t ),
(4)
ɝɞɟ Y , Z ∈ R m – ɜɟɤɬɨɪɵ ɢɡɦɟɪɟɧɢɣ; c( x) : R n → R m – ɜɟɤɬɨɪ-ɮɭɧɤɰɢɹ m × 1 , V (t ) – m -ɦɟɪɧɵɣ
ɫɬɚɧɞɚɪɬɧɵɣ ɜɢɧɟɪɨɜɫɤɢɣ ɩɪɨɰɟɫɫ, ɧɟ ɡɚɜɢɫɹɳɢɣ ɨɬ W (t ) ɢ ɨɬ ɧɚɱɚɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɹ X 0 , N (t ) – m ɦɟɪɧɵɣ ɫɬɚɧɞɚɪɬɧɵɣ ɝɚɭɫɫɨɜɫɤɢɣ ɛɟɥɵɣ ɲɭɦ. Ɉɬɦɟɬɢɦ, ɱɬɨ ɫɬɚɰɢɨɧɚɪɧɨɫɬɶ ɦɨɞɟɥɢ (3) ɢɥɢ (4) ɧɟ
ɢɦɟɟɬ ɩɪɢɧɰɢɩɢɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ, ɧɨ ɩɨɡɜɨɥɹɟɬ ɡɚɩɢɫɵɜɚɬɶ ɦɧɨɝɢɟ ɫɨɨɬɧɨɲɟɧɢɹ ɛɨɥɟɟ ɤɨɦɩɚɤɬɧɨ.
Ɂɚɞɚɱɚ ɨɩɬɢɦɚɥɶɧɨɣ ɮɢɥɶɬɪɚɰɢɢ ɫɨɫɬɨɢɬ ɜ ɧɚɯɨɠɞɟɧɢɢ ɨɰɟɧɤɢ Xˆ (t ) ɬɪɚɟɤɬɨɪɢɢ ɫɥɭɱɚɣɧɨɝɨ
ɩɪɨɰɟɫɫɚ X (t ) ɩɨ ɪɟɡɭɥɶɬɚɬɚɦ ɢɡɦɟɪɟɧɢɣ Y0t = {Y (τ), τ ∈ [t0 , t )} ɢɥɢ Z 0t = {Z (τ), τ ∈ [t0 , t )} ɜ
ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɞɚɧɧɵɦ ɤɪɢɬɟɪɢɟɦ ɨɩɬɢɦɚɥɶɧɨɫɬɢ. Ⱦɚɥɟɟ ɩɪɢ ɡɚɩɢɫɢ ɫɨɨɬɧɨɲɟɧɢɣ ɛɭɞɭɬ
ɢɫɩɨɥɶɡɨɜɚɧɵ ɢɡɦɟɪɟɧɢɹ Y (t ) ɢɥɢ Z (t ) ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɭɞɨɛɫɬɜɚ. Ɂɚɞɚɱɭ ɨɩɬɢɦɚɥɶɧɨɣ
ɮɢɥɶɬɪɚɰɢɢ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɲɢɪɟ: ɤɚɤ ɧɚɯɨɠɞɟɧɢɟ ɚɩɨɫɬɟɪɢɨɪɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ
p (t , x | Y0t ) ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ X .
ɉɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɤɪɢɬɟɪɢɹ ɦɢɧɢɦɭɦɚ ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɟɫɤɨɣ ɨɲɢɛɤɢ ɨɰɟɧɢɜɚɧɢɹ
ɢɦɟɟɦ [10]:
Xˆ (t ) = M ª¬ X (t ) | Y0t º¼ = ³ n xp(t , x | Y0t ) dx.
R
(5)
2. ɍɊȺȼɇȿɇɂə ȾɅə ɇȿɇɈɊɆɂɊɈȼȺɇɇɕɏ ȺɉɈɋɌȿɊɂɈɊɇɕɏ ɉɅɈɌɇɈɋɌȿɃ ȼȿɊɈəɌɇɈɋɌɂ
ɉɪɢɜɟɞɟɦ ɭɪɚɜɧɟɧɢɟ Ⱦɭɧɤɚɧɚ–Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ [3], [4], [10] ɞɥɹ ɧɟɧɨɪɦɢɪɨɜɚɧɧɨɣ
ɚɩɨɫɬɟɪɢɨɪɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ ϕ(t , x | Y0t ) , ɤɨɬɨɪɚɹ ɫɜɹɡɚɧɚ ɫ ɮɭɧɤɰɢɟɣ p (t , x | Y0t )
ɫɨɨɬɧɨɲɟɧɢɟɦ
p (t , x | Y0t ) =
³
R
ϕ(t , x | Y0t )
ϕ(t , x | Y0t ) dx
n
,
(6)
ɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɧɟɤɨɬɨɪɵɦɢ ɪɟɡɭɥɶɬɚɬɚɦɢ, ɩɨɥɭɱɟɧɧɵɦɢ ɜ [8].
ɍɪɚɜɧɟɧɢɟ Ⱦɭɧɤɚɧɚ–Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ ɨɬɧɨɫɢɬɫɹ ɤ ɤɥɚɫɫɭ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ. ɂɫɩɨɥɶɡɭɹ ɮɨɪɦɭ ɋɬɪɚɬɨɧɨɜɢɱɚ ɢ
ɮɢɤɫɢɪɭɹ ɢɡɦɟɪɟɧɢɹ, ɟɝɨ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
∂ϕ(t , x | Y0t )
= $ϕ(t , x | Y0t ) − λ(t , x)ϕ(t , x | Y0t ) + ³ n λ(t , ξ)η(t , x | ξ)ϕ(t , ξ | Y0t ) d ξ +
R
∂t
m
dY (t )
+ ¦ cα ( x) α ϕ(t , x | Y0t ), ϕ(t0 , x) = ϕ0 ( x),
dt
α=1
ɝɞɟ
$ϕ(t , x | Y0t ) = ϕ(t , x | Y0t ) −
n
ϕ(t , x | Y0t ) = −¦
i =1
1 m 2
cα ( x)ϕ(t , x | Y0t ),
¦
2 α=1
∂
1 n n ∂2
ª¬ f i (t , x)ϕ(t , x | Y0t ) º¼ + ¦¦
ª gij (t , x)ϕ(t , x | Y0t ) ¼º ,
∂xi
2 i =1 j =1 ∂xi ∂x j ¬
g (t , x) = σ(t , x)σT (t , x),
(7)
76
Ʉ.Ⱥ. Ɋɵɛɚɤɨɜ
ɢɥɢ
∂ϕ(t , x | Y0t )
= ϕ(t , x | Y0t ) − λ(t , x)ϕ(t , x | Y0t ) + ³ n λ(t , ξ)η(t , x | ξ)ϕ(t , ξ | Y0t ) d ξ −
R
∂t
dY (t )
, ϕ(t0 , x) = ϕ0 ( x)
− μ − ( x, Z (t ))ϕ(t , x | Y0t ) + μ + ( x, Z (t ))ϕ(t , x | Y0t ), Z (t ) =
dt
(8)
ɝɞɟ
­−μ( x, z ), μ( x, z ) < 0,
μ − ( x, z ) = ®
¯0, μ( x, z ) ≥ 0,
­μ( x, z ), μ( x, z ) > 0,
μ + ( x, z ) = ®
¯0, μ( x, z ) ≤ 0;
m
c ( x) ·
§
μ( x, z ) = ¦ cα ( x) ¨ zα − α
¸.
2 ¹
©
α =1
ɍɪɚɜɧɟɧɢɟ (8) ɩɨ ɫɬɪɭɤɬɭɪɟ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɨɛɨɛɳɟɧɧɨɦɭ ɭɪɚɜɧɟɧɢɸ Ɏɨɤɤɟɪɚ–ɉɥɚɧɤɚ–
Ʉɨɥɦɨɝɨɪɨɜɚ, ɩɪɢ ɷɬɨɦ ɮɭɧɤɰɢɢ μ − ( X (t ), Z (t )) ɢ μ + ( X (t ), Z (t )) – ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɨɛɪɵɜɨɜ ɢ
ɜɟɬɜɥɟɧɢɣ ɬɪɚɟɤɬɨɪɢɣ ɫɥɭɱɚɣɧɨɝɨ ɩɪɨɰɟɫɫɚ X (t ) , ɚ ɩɪɨɢɡɜɟɞɟɧɢɹ μ − ( x, Z (t ))ϕ(t , x | Y0t ) ɢ
μ + ( x, Z (t ))ϕ(t , x | Y0t ) – ɮɭɧɤɰɢɢ ɩɨɝɥɨɳɟɧɢɹ ɢ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ [1].
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɫɥɨɜɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ ɨɛɪɵɜɨɜ ɢ ɜɟɬɜɥɟɧɢɣ ɩɪɢ X (t ) = x ɢ Z (t ) = z ɧɚ
ɩɪɨɦɟɠɭɬɤɟ [t , t + Δt ] ɨɩɪɟɞɟɥɹɸɬɫɹ ɪɚɜɟɧɫɬɜɚɦɢ:
P − (t , t + Δt ) = μ − ( x, z )Δt + o(Δt ), P + (t , t + Δt ) = μ + ( x, z )Δt + o(Δt ).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɮɭɧɤɰɢɢ p (t , x | Y0t ) ɢ ϕ(t , x | Y0t ) ɯɚɪɚɤɬɟɪɢɡɭɸɬ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɤɬɨɪɚ
X – ɫɨɫɬɨɹɧɢɹ ɨɛɴɟɤɬɚ ɧɚɛɥɸɞɟɧɢɹ, ɨɩɢɫɵɜɚɟɦɨɝɨ ɭɪɚɜɧɟɧɢɟɦ (1), – ɫ ɭɱɟɬɨɦ ɬɨɝɨ, ɱɬɨ
ɬɪɚɟɤɬɨɪɢɢ ɫɥɭɱɚɣɧɨɝɨ ɩɪɨɰɟɫɫɚ X (t ) ɩɨɥɭɱɚɸɬ ɫɥɭɱɚɣɧɵɟ ɩɪɢɪɚɳɟɧɢɹ, ɜɟɬɜɹɬɫɹ ɢɥɢ
ɨɛɪɵɜɚɸɬɫɹ. ɇɚɩɨɦɧɢɦ [8], ɱɬɨ ɩɟɪɟɱɢɫɥɟɧɧɵɟ ɫɨɛɵɬɢɹ ɨɛɪɚɡɭɸɬ ɧɟɨɞɧɨɪɨɞɧɵɟ
ɩɭɚɫɫɨɧɨɜɫɤɢɟ ɩɨɬɨɤɢ ɫ ɢɡɜɟɫɬɧɵɦɢ ɢɧɬɟɧɫɢɜɧɨɫɬɹɦɢ, ɩɪɢ ɜɟɬɜɥɟɧɢɢ ɜ ɮɢɤɫɢɪɨɜɚɧɧɵɣ
ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɦɨɠɟɬ ɩɨɹɜɢɬɶɫɹ ɬɨɥɶɤɨ ɨɞɧɚ ɧɨɜɚɹ ɜɟɬɜɶ, ɩɪɢ ɨɛɪɵɜɟ ɩɪɟɤɪɚɳɚɟɬɫɹ
ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɬɨɥɶɤɨ ɨɞɧɨɣ ɜɟɬɜɢ. Ⱦɥɹ ɭɞɨɛɫɬɜɚ ɦɨɞɟɥɢɪɨɜɚɧɢɹ, ɤɚɤ ɢ ɜ ɫɥɭɱɚɟ ɫɢɫɬɟɦ
ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɬɢɩɚ [5; 6], ɤɚɠɞɚɹ ɢɡ ɧɨɜɵɯ ɜɟɬɜɟɣ ɞɨɥɠɧɚ ɪɚɫɫɦɚɬɪɢɜɚɬɶɫɹ ɤɚɤ
ɫɚɦɨɫɬɨɹɬɟɥɶɧɚɹ ɬɪɚɟɤɬɨɪɢɹ.
Ɉɞɧɚɤɨ, ɤɚɤ ɢ ɞɥɹ ɫɢɫɬɟɦ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɬɢɩɚ, ɭɪɚɜɧɟɧɢɟ Ⱦɭɧɤɚɧɚ–Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ
ɫɨɞɟɪɠɢɬ ɦɧɨɠɢɬɟɥɶ ɬɢɩɚ ɛɟɥɨɝɨ ɲɭɦɚ – ɫɥɭɱɚɣɧɵɣ ɩɪɨɰɟɫɫ N (t ) , ɧɚɥɢɱɢɟ ɤɨɬɨɪɨɝɨ
ɭɫɥɨɠɧɹɟɬ ɟɝɨ ɪɟɲɟɧɢɟ ɫ ɩɨɦɨɳɶɸ ɦɟɬɨɞɚ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɢɫɩɵɬɚɧɢɣ. Ɉɫɧɨɜɧɚɹ
ɫɥɨɠɧɨɫɬɶ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ N (t ) ɜɯɨɞɢɬ ɜ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɨɛɪɵɜɨɜ ɢ ɜɟɬɜɥɟɧɢɣ
ɬɪɚɟɤɬɨɪɢɣ ɫɥɭɱɚɣɧɨɝɨ ɩɪɨɰɟɫɫɚ X (t ) , ɞɟɥɚɹ ɢɯ, ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɧɟɨɝɪɚɧɢɱɟɧɧɵɦɢ
(ɫɦ. (4) ɢ (8)). ȼ ɪɟɡɭɥɶɬɚɬɟ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɩɪɢɦɟɪɟ ɜ ɪɚɛɨɬɟ [6], ɞɥɹ
ɢɧɬɟɧɫɢɜɧɨɫɬɟɣ
μ − (t ) = μ − ( X (t ), Z (t )) ɢ μ + (t ) = μ + ( X (t ), Z (t ))
ɯɚɪɚɤɬɟɪɧɨ ɛɵɫɬɪɨɟ ɜɨɡɪɚɫɬɚɧɢɟ ɢ ɭɛɵɜɚɧɢɟ, ɚ ɬɚɤɠɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɛɨɥɶɲɢɟ ɦɚɤɫɢɦɚɥɶɧɵɟ
ɡɧɚɱɟɧɢɹ, ɨɩɪɟɞɟɥɹɸɳɢɟ ɫɪɟɞɧɟɟ ɱɢɫɥɨ ɨɛɪɵɜɨɜ ɢ ɜɟɬɜɥɟɧɢɣ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ.
Ɏɢɥɶɬɪɚɰɢɹ ɫɢɝɧɚɥɨɜ ɜ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɫɢɫɬɟɦɚɯ…
77
ɑɬɨɛɵ ɦɢɧɢɦɢɡɢɪɨɜɚɬɶ ɜɥɢɹɧɢɟ ɭɤɚɡɚɧɧɨɝɨ ɧɟɞɨɫɬɚɬɤɚ,
ɧɟɧɨɪɦɢɪɨɜɚɧɧɨɣ ɚɩɨɫɬɟɪɢɨɪɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ [9]:
ϕ(t , x | Y0t ) = eY
T
(t ) c ( x )
ɩɪɨɜɟɞɟɦ
ɡɚɦɟɧɭ
ρ(t , x | Y0t ).
(9)
Ɍɨɝɞɚ [8]
T
∂ρ(t , x | Y0t )
= $ρ(t , x | Y0t ) − λ(t , x)ρ(t , x | Y0t ) + ³ n λ(t , ξ)η(t , x | ξ) eY (t )( c ( ξ ) −c ( x ) ) ρ(t , ξ | Y0t ) d ξ −
R
∂t
m
m
m
− ¦ Yα (t )$α ρ(t , x | Y ) + ¦¦ Yα (t )Yβ (t )$αβρ(t , x | Y ).
t
0
α=1
ȼ
ɭɪɚɜɧɟɧɢɢ
(10)
(10)
t
0
α=1 β=1
$α = [α , $] ,
$αβ = 12 [α ,[β , $]] ,
ɝɞɟ
[α , $]
ɢ
[α , $β ]
–
ɤɨɦɦɭɬɚɬɨɪɵ ɭɤɚɡɚɧɧɵɯ ɜ ɫɤɨɛɤɚɯ ɨɩɟɪɚɬɨɪɨɜ, α – ɨɩɟɪɚɬɨɪɵ ɭɦɧɨɠɟɧɢɹ ɧɚ ɮɭɧɤɰɢɢ cα ( x) ,
α, β = 1, 2, …, m .
ɇɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ ɞɥɹ ɭɪɚɜɧɟɧɢɹ (10) ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɟɦ ɞɥɹ ɭɪɚɜɧɟɧɢɹ
T
(7) ɢɥɢ (8), ɩɨɫɤɨɥɶɤɭ eY ( t ) c ( x ) = 1 ɩɪɢ t = t0 , ɬ.ɟ. ρ(t0 , x) = ϕ0 ( x) .
ɍɪɚɜɧɟɧɢɟ (8) ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɪɨɛɚɫɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ Ⱦɭɧɤɚɧɚ–Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ ɞɥɹ
ɫɢɫɬɟɦ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɬɢɩɚ ɧɚɥɢɱɢɟɦ ɫɥɚɝɚɟɦɵɯ
− λ (t , x)ρ(t , x | Y0t ) + ³ n λ(t , ξ)η(t , x | ξ) eY
T
( t )( c ( ξ ) − c ( x ) )
R
ρ(t , ξ | Y0t ) d ξ,
ɩɨɷɬɨɦɭ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɪɟɡɭɥɶɬɚɬɚɦɢ ɪɚɛɨɬɵ [6], ɝɞɟ ɩɨɞɪɨɛɧɨ ɩɨɤɚɡɚɧɨ, ɤɚɤ ɩɨɥɭɱɢɬɶ
ɞɪɭɝɭɸ ɮɨɪɦɭ ɡɚɩɢɫɢ ɪɨɛɚɫɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ Ⱦɭɧɤɚɧɚ–Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ ɧɚ ɨɫɧɨɜɟ
ɬɨɠɞɟɫɬɜɟɧɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɜɵɪɚɠɟɧɢɹ
m
m
m
$ρ(t , x | Y0t ) − ¦ Yα (t )$α ρ(t , x | Y0t ) + ¦¦ Yα (t )Yβ (t )$αβρ(t , x | Y0t ),
α=1
α=1 β=1
ɤɨɬɨɪɨɟ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ
ρ(t , x | Y0t ) + ν(t , x, Y (t ))ρ(t , x | Y0t ),
ɝɞɟ
n
∂ 1 n n ∂2
t
t
ª
º
ª¬ gij (t , x)ρ(t , x | Y0t ) º¼,
ρ(t , x | Y0 ) = −¦
fi (t , x, Y (t ))ρ(t , x | Y0 ) ¼ + ¦¦
¬
2 i =1 j =1 ∂xi ∂x j
i =1 ∂xi
T
ª ∂c( x) º
f (t , x, Y (t )) = f (t , x) − g (t , x) «
Y (t ),
¬ ∂x »¼
∂c( x)
1
f (t , x) − tr ( g (t , x)∇∇ T (Y T (t )c( x)) ) −
ν(t , x, Y (t )) = −Y T (t )
∂x
2
T
1
1
∂c( x)
ª ∂c( x) º
− c T ( x)c( x) + Y T (t )
g (t , x ) «
Y (t ).
2
2
∂x
¬ ∂x »¼
78
Ʉ.Ⱥ. Ɋɵɛɚɤɨɜ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɦɟɟɦ
T
∂ρ(t , x | Y0t ) = ρ(t , x | Y0t ) − λ (t , x)ρ(t , x | Y0t ) + ³ n λ (t , ξ)η(t , x | ξ) eY (t )( c ( ξ )−c ( x )) ρ(t , ξ | Y0t ) d ξ +
R
(11)
∂t
t
+ ν(t , x, Y (t ))ρ(t , x | Y0 ), ρ(t0 , x) = ϕ0 ( x).
3. ɋȼȿȾȿɇɂȿ Ʉ ɁȺȾȺɑȿ ȺɇȺɅɂɁȺ ɋɂɋɌȿɆ ɋ ɈȻɊɕȼȺɆɂ ɂ ȼȿɌȼɅȿɇɂəɆɂ ɌɊȺȿɄɌɈɊɂɃ
ɉɪɢɦɟɧɢɦ ɩɨɞɯɨɞ, ɢɫɩɨɥɶɡɨɜɚɧɧɵɣ ɪɚɧɟɟ ɞɥɹ ɫɢɫɬɟɦ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɬɢɩɚ [6], ɚ ɢɦɟɧɧɨ
ɫɜɟɞɟɦ ɡɚɞɚɱɭ ɨɩɬɢɦɚɥɶɧɨɣ ɮɢɥɶɬɪɚɰɢɢ ɤ ɡɚɞɚɱɟ ɚɧɚɥɢɡɚ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ
ɫɢɫɬɟɦɵ ɫ ɨɛɪɵɜɚɦɢ ɢ ɜɟɬɜɥɟɧɢɹɦɢ ɬɪɚɟɤɬɨɪɢɣ. ɑɬɨɛɵ ɩɪɚɜɢɥɶɧɨ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ ɜɯɨɞɹɳɢɟ
ɜ ɭɪɚɜɧɟɧɢɟ (11) ɫɥɚɝɚɟɦɵɟ ɩɪɢ ɮɨɪɦɢɪɨɜɚɧɢɢ ɚɥɝɨɪɢɬɦɚ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɬɪɚɟɤɬɨɪɢɣ
ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɝɨ ɫɥɭɱɚɣɧɨɝɨ ɩɪɨɰɟɫɫɚ ɫ ɨɛɪɵɜɚɦɢ, ɪɚɡɪɵɜɚɦɢ ɢ ɜɟɬɜɥɟɧɢɹɦɢ, ɜɨ-ɩɟɪɜɵɯ,
ɩɪɟɞɫɬɚɜɢɦ ɮɭɧɤɰɢɸ ν(t , x, y ) ɜ ɜɢɞɟ:
ν(t , x, y ) = −ν − (t , x, y ) + ν + (t , x, y),
ɝɞɟ
­−ν(t , x, y ), ν(t , x, y ) < 0,
ν − (t , x, y ) = ®
¯0, ν(t , x, y ) ≥ 0,
­ν(t , x, y), ν(t , x, y) > 0,
ν + (t , x, y ) = ®
¯0, ν(t , x, y ) ≤ 0,
ɚ ɜɨ-ɜɬɨɪɵɯ, ɨɩɪɟɞɟɥɢɦ ɮɭɧɤɰɢɢ
yT c ( ξ ) −c ( x ) )
λ (t , ξ, y ) = λ(t , ξ) ³ n η(t , x | ξ) e (
dx, η (t , x | ξ, y) =
R
η(t , x | ξ) e
³
R
yT ( c ( ξ ) −c ( x ) )
η(t , x | ξ) e y
n
T
(c ( ξ)−c ( x ))
dx
.
Ɍɨɝɞɚ
∂ρ(t , x | Y0t ) = ρ(t , x | Y0t ) − λ(t , x)ρ(t , x | Y0t ) + ³ n λ (t , ξ, Y (t ))η (t , x | ξ, Y (t ))ρ(t , ξ | Y0t ) d ξ −
R
∂t
− ν − (t , x, Y (t ))ρ(t , x | Y0t ) + ν + (t , x, Y (t ))ρ(t , x | Y0t ), ρ(t0 , x) = ϕ0 ( x),
(12)
ɝɞɟ
λ (t , x, y ) ≥ 0, η (t , x | ξ, y) ≥ 0 ɢ
³
Rn
η (t , x | ξ, y) dx = 1, t ∈ T , x, ξ ∈ R n , y ∈ R m ,
ɬ.ɟ. ɩɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɪɨɛɚɫɬɧɨɟ ɭɪɚɜɧɟɧɢɟ Ⱦɭɧɤɚɧɚ–Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ ɩɪɟɞɫɬɚɜɥɟɧɨ ɬɚɤ,
ɱɬɨ ɟɝɨ ɫɬɪɭɤɬɭɪɚ ɫɨɜɩɚɞɚɟɬ ɫɨ ɫɬɪɭɤɬɭɪɨɣ ɭɪɚɜɧɟɧɢɹ (8).
ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɨɩɢɫɵɜɚɟɬ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɧɟɧɨɪɦɢɪɨɜɚɧɧɨɣ ɚɩɨɫɬɟɪɢɨɪɧɨɣ ɩɥɨɬɧɨɫɬɢ
ɜɟɪɨɹɬɧɨɫɬɢ ρ(t , x | Y0t ) ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ X = X (t ) ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ɞɥɹ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ
ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɜɟɤɬɨɪ ɫɧɨɫɚ – f (t , x, Y (t )) , ɚ ɦɚɬɪɢɰɚ ɞɢɮɮɭɡɢɢ
ɫɨɜɩɚɞɚɟɬ ɫ ɦɚɬɪɢɰɟɣ ɞɢɮɮɭɡɢɢ ɢɫɯɨɞɧɨɝɨ ɨɛɴɟɤɬɚ ɧɚɛɥɸɞɟɧɢɹ (1) – g (t , x) . ɇɚɱɚɥɶɧɨɟ
ɫɨɫɬɨɹɧɢɟ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɱɚɥɶɧɵɦ ɫɨɫɬɨɹɧɢɟɦ X 0
Ɏɢɥɶɬɪɚɰɢɹ ɫɢɝɧɚɥɨɜ ɜ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɫɢɫɬɟɦɚɯ…
79
ɢɫɯɨɞɧɨɣ ɫɢɫɬɟɦɵ (1). Ʉɪɨɦɟ ɬɨɝɨ, ɜɨ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɜɨɡɦɨɠɧɵ
ɫɥɟɞɭɸɳɢɟ ɫɨɛɵɬɢɹ, ɨɛɪɚɡɭɸɳɢɟ ɧɟɨɞɧɨɪɨɞɧɵɟ ɩɭɚɫɫɨɧɨɜɫɤɢɟ ɩɨɬɨɤɢ:
1) ɨɛɪɵɜ ɬɪɚɟɤɬɨɪɢɢ ɫ ɫɭɦɦɚɪɧɨɣ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ λ(t , X (t )) + ν − (t , X (t ), Y (t )) , ɬ.ɟ.
ɜɟɪɨɹɬɧɨɫɬɶ ɨɛɪɵɜɚ ɬɪɚɟɤɬɨɪɢɢ ɩɪɢ X (t ) = x ɢ Y (t ) = y ɡɚɞɚɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ
P − (t , t + Δt ) = (λ(t , x) + ν − (t , x, y ))Δt + o(Δt );
2) ɜɟɬɜɥɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɫ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ ν + (t , X (t ), Y (t )) , ɬ.ɟ. ɜɟɪɨɹɬɧɨɫɬɶ ɜɟɬɜɥɟɧɢɹ
ɬɪɚɟɤɬɨɪɢɢ ɩɪɢ X (t ) = x ɢ Y (t ) = y ɡɚɞɚɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ
P + (t , t + Δt ) = ν + (t , x, y )Δt + o(Δt );
3) ɜɟɬɜɥɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɫɨ ɫɤɚɱɤɨɦ ɫ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ λ (t , X (t ), Y (t )) , ɬ.ɟ. ɜɟɪɨɹɬɧɨɫɬɶ
ɬɚɤɨɝɨ ɫɨɛɵɬɢɹ ɩɪɢ X (t ) = x ɢ Y (t ) = y ɡɚɞɚɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ
P* (t , t + Δt ) = λ (t , x, y )Δt + o(Δt ),
ɚ ɜɟɥɢɱɢɧɚ X (τk ) ɩɪɢ ɜɟɬɜɥɟɧɢɢ ɫɨ ɫɤɚɱɤɨɦ ɞɥɹ ɧɨɜɨɣ ɜɟɬɜɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɫɥɨɜɧɨɣ
ɩɥɨɬɧɨɫɬɶɸ ɜɟɪɨɹɬɧɨɫɬɢ η (τk , x | ξ, y ) , ɟɫɥɢ X (τk − 0) = ξ , Y (τk ) = y , τk – ɦɨɦɟɧɬ ɜɟɬɜɥɟɧɢɹ.
ȼɵɪɚɠɟɧɢɟ
( λ(t, x) + ν
−
(t , x, Y (t )) ) ρ(t , x | Y0t )
ɨɩɪɟɞɟɥɹɟɬ ɮɭɧɤɰɢɸ ɩɨɝɥɨɳɟɧɢɹ, ɚ ɜɵɪɚɠɟɧɢɟ
³
Rn
λ (t , ξ, Y (t ))η (t , x | ξ, Y (t ))ρ(t , ξ | Y0t ) d ξ + ν + (t , x, Y (t ))ρ(t , x | Y0t )
– ɮɭɧɤɰɢɸ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɬɪɚɟɤɬɨɪɢɣ ɫɥɭɱɚɣɧɨɝɨ ɩɪɨɰɟɫɫɚ X (t ) [1].
Ⱦɥɹ ɷɬɨɣ ɫɢɫɬɟɦɵ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɦɨɦɟɧɬɨɜ ɜɪɟɦɟɧɢ ɩɨɹɜɥɟɧɢɹ ɨɛɪɵɜɨɜ ɢ ɜɟɬɜɥɟɧɢɣ
ɭɩɪɚɜɥɹɟɬ ɩɪɨɰɟɫɫ Y (t ) , ɚ ɧɟ Z (t ) , ɤɚɤ ɷɬɨ ɩɪɨɢɫɯɨɞɢɬ ɫɨ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ
ɫɢɫɬɟɦɨɣ, ɩɨɫɬɪɨɟɧɧɨɣ ɧɚ ɨɫɧɨɜɟ ɤɥɚɫɫɢɱɟɫɤɨɝɨ, ɚ ɧɟ ɪɨɛɚɫɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ Ⱦɭɧɤɚɧɚ–
Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ. ɗɬɨɬ ɩɪɨɰɟɫɫ ɜɥɢɹɟɬ ɢ ɧɚ ɩɨɜɟɞɟɧɢɟ ɬɪɚɟɤɬɨɪɢɣ ɦɟɠɞɭ
ɦɨɦɟɧɬɚɦɢ ɜɟɬɜɥɟɧɢɣ, ɢ ɧɚ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɜɟɬɜɥɟɧɢɣ ɫɨ ɫɤɚɱɤɨɦ, ɢ ɧɚ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɥɢɱɢɧɵ
ɷɬɨɝɨ ɫɤɚɱɤɚ. Ɉɫɧɨɜɵɜɚɹɫɶ ɧɚ ɪɟɡɭɥɶɬɚɬɚɯ ɦɨɞɟɥɢɪɨɜɚɧɢɹ, ɦɨɠɧɨ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ
ɢɧɬɟɧɫɢɜɧɨɫɬɢ
ν − (t ) = ν − (t , X (t ), Y (t )) ɢ ν + (t ) = ν + (t , X (t ), Y (t ))
ɦɟɧɹɸɬɫɹ ɡɧɚɱɢɬɟɥɶɧɨ ɦɟɞɥɟɧɧɟɟ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɢɧɬɟɧɫɢɜɧɨɫɬɹɦɢ
μ − (t ) = μ − ( X (t ), Z (t )) ɢ μ + (t ) = μ + ( X (t ), Z (t )),
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɪɟɞɧɟɟ ɱɢɫɥɨ ɨɛɪɵɜɨɜ ɢ ɜɟɬɜɥɟɧɢɣ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɨɤɚɡɵɜɚɟɬɫɹ
ɫɭɳɟɫɬɜɟɧɧɨ ɦɟɧɶɲɟ. ɇɚɪɹɞɭ ɫ ɷɬɢɦ ɟɫɬɶ ɢ ɧɟɞɨɫɬɚɬɨɤ, ɤɨɬɨɪɵɣ ɩɪɨɹɜɥɹɟɬɫɹ ɜ ɧɚɪɭɲɟɧɢɢ
80
Ʉ.Ⱥ. Ɋɵɛɚɤɨɜ
«ɛɚɥɚɧɫɚ» ɦɟɠɞɭ ɩɨɝɥɨɳɟɧɢɟɦ ɢ ɜɨɫɫɬɚɧɨɜɥɟɧɢɟɦ ɬɪɚɟɤɬɨɪɢɣ ɩɪɢ ɪɚɡɪɵɜɟ ɬɪɚɟɤɬɨɪɢɢ. ȿɫɥɢ
ɞɥɹ ɢɫɯɨɞɧɨɣ ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɢɧɬɟɧɫɢɜɧɨɫɬɢ, ɜɯɨɞɹɳɢɟ ɜ ɮɭɧɤɰɢɢ ɩɨɝɥɨɳɟɧɢɹ ɢ
ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ, ɨɞɢɧɚɤɨɜɵ – ɨɧɢ ɫɨɜɩɚɞɚɸɬ ɫ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ λ (t , x) ɪɚɡɪɵɜɚ ɬɪɚɟɤɬɨɪɢɣ
ɫɥɭɱɚɣɧɨɝɨ ɩɪɨɰɟɫɫɚ X (t ) , ɬɨ ɞɥɹ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɨɧɢ ɪɚɡɧɵɟ. Ⱦɚɠɟ
ɟɫɥɢ ɜ ɢɫɯɨɞɧɨɣ ɩɨɫɬɚɧɨɜɤɟ ɡɚɞɚɱɢ λ (t , x) = λ = const , ɬɨ ɞɥɹ ɫɥɭɱɚɣɧɨɝɨ ɩɪɨɰɟɫɫɚ X (t ) ɩɪɢ
ɜɨɫɫɬɚɧɨɜɥɟɧɢɢ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɛɭɞɟɬ ɩɟɪɟɦɟɧɧɨɣ ɩɨ ɜɪɟɦɟɧɢ ɢ ɛɭɞɟɬ ɨɩɪɟɞɟɥɹɬɶɫɹ ɬɟɤɭɳɢɦ
ɫɨɫɬɨɹɧɢɟɦ, ɩɨɫɤɨɥɶɤɭ ɨɧɚ ɞɨɥɠɧɚ ɜɵɱɢɫɥɹɬɶɫɹ ɧɚ ɬɪɚɟɤɬɨɪɢɹɯ X (t ) ɢ Y (t ) . ɗɬɚ ɢɧɬɟɧɫɢɜɧɨɫɬɶ
ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɤɨɷɮɮɢɰɢɟɧɬ ɫɧɨɫɚ c( x) ɜ ɭɪɚɜɧɟɧɢɢ (3) ɢɥɢ (4) ɦɨɞɟɥɢ ɢɡɦɟɪɢɬɟɥɶɧɨɣ
ɫɢɫɬɟɦɵ ɢ ɹɜɧɵɦ ɨɛɪɚɡɨɦ ɡɚɜɢɫɢɬ ɨɬ Y (t ) .
ɇɟɧɨɪɦɢɪɨɜɚɧɧɭɸ ɚɩɨɫɬɟɪɢɨɪɧɭɸ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ρ(t , x | Y0t ) ɦɨɠɧɨ ɧɚɣɬɢ
ɩɪɢɛɥɢɠɟɧɧɨ, ɦɨɞɟɥɢɪɭɹ ɬɪɚɟɤɬɨɪɢɢ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɝɨ ɫɥɭɱɚɣɧɨɝɨ ɩɪɨɰɟɫɫɚ X (t ) ɫ ɭɱɟɬɨɦ
ɨɛɪɵɜɨɜ, ɪɚɡɪɵɜɨɜ ɢ ɜɟɬɜɥɟɧɢɣ. Ʉɚɤ ɢ ɜ ɚɧɚɥɨɝɢɱɧɵɯ ɚɥɝɨɪɢɬɦɚɯ, ɪɚɡɪɚɛɨɬɚɧɧɵɯ ɞɥɹ ɛɨɥɟɟ
ɩɪɨɫɬɵɯ ɦɨɞɟɥɟɣ ɫɢɫɬɟɦ ɧɚɛɥɸɞɟɧɢɹ [5], [6], ɩɨ ɚɧɫɚɦɛɥɸ ɬɪɚɟɤɬɨɪɢɣ, ɩɨɥɭɱɟɧɧɨɦɭ ɜ
ɪɟɡɭɥɶɬɚɬɟ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɨɜ ɱɢɫɥɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ
ɭɪɚɜɧɟɧɢɣ ɢ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɧɟɨɞɧɨɪɨɞɧɵɯ ɩɭɚɫɫɨɧɨɜɫɤɢɯ ɩɨɬɨɤɨɜ ɫɨɛɵɬɢɣ [2], [3], ɦɨɠɧɨ
ɨɰɟɧɢɬɶ ɮɭɧɤɰɢɸ ρ(t , x | Y0t ) , ɧɚɩɪɢɦɟɪ, ɫ ɩɨɦɨɳɶɸ ɩɨɫɬɪɨɟɧɢɹ ɝɢɫɬɨɝɪɚɦɦɵ, ɢ ɩɨ ɷɬɨɣ ɨɰɟɧɤɟ
ɧɚɣɬɢ ɚɩɨɫɬɟɪɢɨɪɧɭɸ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ p (t , x | Y0t ) ɫɨɝɥɚɫɧɨ ɮɨɪɦɭɥɚɦ (6) ɢ (9). ɉɨ
ɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ p (t , x | Y0t ) ɦɨɠɟɬ ɛɵɬɶ ɧɚɣɞɟɧɚ ɢ ɨɩɬɢɦɚɥɶɧɚɹ ɨɰɟɧɤɚ Xˆ (t ) , ɩɪɢɱɟɦ ɦɨɠɧɨ
ɢɫɩɨɥɶɡɨɜɚɬɶ ɧɟ ɬɨɥɶɤɨ ɜɵɪɚɠɟɧɢɟ (5), ɟɫɥɢ ɦɨɞɢɮɢɰɢɪɨɜɚɬɶ ɩɨɫɬɚɧɨɜɤɭ ɡɚɞɚɱɢ ɨɩɬɢɦɚɥɶɧɨɣ
ɮɢɥɶɬɪɚɰɢɢ, ɡɚɦɟɧɢɜ ɤɪɢɬɟɪɢɣ ɦɢɧɢɦɭɦɚ ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɟɫɤɨɣ ɨɲɢɛɤɢ ɨɰɟɧɢɜɚɧɢɹ ɤɚɤɢɦɥɢɛɨ ɞɪɭɝɢɦ ɤɪɢɬɟɪɢɟɦ [3].
Ʉɚɤ ɨɬɦɟɱɚɥɨɫɶ ɩɪɢ ɮɨɪɦɭɥɢɪɨɜɚɧɢɢ ɩɨɫɬɚɧɨɜɤɢ ɡɚɞɚɱɢ, ɫɬɚɰɢɨɧɚɪɧɨɫɬɶ ɦɨɞɟɥɢ
ɢɡɦɟɪɢɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ ɧɟ ɢɦɟɟɬ ɩɪɢɧɰɢɩɢɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ ɢ ɩɪɟɞɥɚɝɚɟɦɚɹ ɦɟɬɨɞɢɤɚ ɪɟɲɟɧɢɹ
ɡɚɞɚɱɢ ɨɩɬɢɦɚɥɶɧɨɣ ɮɢɥɶɬɪɚɰɢɢ ɦɨɠɟɬ ɩɪɢɦɟɧɹɬɶɫɹ ɢ ɞɥɹ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ ɦɨɞɟɥɟɣ. ɉɪɢ
ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɜɪɟɦɟɧɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɫɧɨɫɚ ɜ ɭɪɚɜɧɟɧɢɢ (3) ɢɥɢ (4) ɦɨɞɟɥɢ ɢɡɦɟɪɢɬɟɥɶɧɨɣ
ɫɢɫɬɟɦɵ ɜ ɪɨɛɚɫɬɧɨɦ ɭɪɚɜɧɟɧɢɢ Ⱦɭɧɤɚɧɚ–Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ ɩɨɹɜɢɬɫɹ ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ
ɫɥɚɝɚɟɦɨɟ, ɤɨɬɨɪɨɟ ɛɭɞɟɬ ɨɬɧɨɫɢɬɶɫɹ ɤ ɢɧɬɟɧɫɢɜɧɨɫɬɹɦ ɨɛɪɵɜɨɜ ɢ ɜɟɬɜɥɟɧɢɣ ɬɪɚɟɤɬɨɪɢɣ
ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɝɨ ɫɥɭɱɚɣɧɨɝɨ ɩɪɨɰɟɫɫɚ [7]. Ʉɪɨɦɟ ɬɨɝɨ, ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɜɚɪɢɚɧɬ ɡɚɞɚɱɢ
ɨɩɬɢɦɚɥɶɧɨɣ ɮɢɥɶɬɪɚɰɢɢ, ɤɨɝɞɚ ɪɚɡɦɟɪɧɨɫɬɢ ɜɟɤɬɨɪɚ ɢɡɦɟɪɟɧɢɣ ɢ ɜɟɤɬɨɪɚ ɲɭɦɚ ɜ ɭɪɚɜɧɟɧɢɢ
ɢɡɦɟɪɢɬɟɥɶɧɨɣ ɫɢɫɬɟɦɵ ɧɟ ɫɨɜɩɚɞɚɸɬ, ɚ ɜ ɫɥɭɱɚɟ ɫɨɜɩɚɞɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɢ ɲɭɦɟ
ɧɟɨɛɹɡɚɬɟɥɶɧɨ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɟɞɢɧɢɱɧɭɸ ɦɚɬɪɢɰɭ ɢ ɦɨɠɟɬ ɛɵɬɶ ɦɚɬɪɢɱɧɨɣ ɮɭɧɤɰɢɟɣ
ɜɪɟɦɟɧɢ, ɧɨ ɧɟ ɮɭɧɤɰɢɟɣ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ [3], [4].
Ⱦɥɹ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɧɟɨɞɧɨɪɨɞɧɵɯ ɩɭɚɫɫɨɧɨɜɫɤɢɯ ɩɨɬɨɤɨɜ ɫɨɛɵɬɢɣ ɪɟɤɨɦɟɧɞɭɟɬɫɹ
ɢɫɩɨɥɶɡɨɜɚɬɶ ɦɟɬɨɞ «ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɫɟɱɟɧɢɹ» [2], ɚ ɞɥɹ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɮɪɚɝɦɟɧɬɨɜ ɬɪɚɟɤɬɨɪɢɣ
ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɝɨ ɫɥɭɱɚɣɧɨɝɨ ɩɪɨɰɟɫɫɚ X (t ) ɦɟɠɞɭ ɦɨɦɟɧɬɚɦɢ ɜɟɬɜɥɟɧɢɣ ɢ ɞɨ ɦɨɦɟɧɬɚ ɨɛɪɵɜɚ
ɦɨɠɧɨ ɩɪɢɦɟɧɹɬɶ ɥɸɛɵɟ ɩɨɞɯɨɞɹɳɢɟ ɚɥɝɨɪɢɬɦɵ ɱɢɫɥɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ [3].
ɅɂɌȿɊȺɌɍɊȺ
1. Ʉɚɡɚɤɨɜ ɂ.ȿ., Ⱥɪɬɟɦɶɟɜ ȼ.Ɇ., Ȼɭɯɚɥɟɜ ȼ.Ⱥ. Ⱥɧɚɥɢɡ ɫɢɫɬɟɦ ɫɥɭɱɚɣɧɨɣ ɫɬɪɭɤɬɭɪɵ. – Ɇ.: Ɏɢɡɦɚɬɥɢɬ, 1993.
2. Ɇɢɯɚɣɥɨɜ Ƚ.Ⱥ., Ⱥɜɟɪɢɧɚ Ɍ.Ⱥ. Ⱥɥɝɨɪɢɬɦ «ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɫɟɱɟɧɢɹ» ɜ ɦɟɬɨɞɟ Ɇɨɧɬɟ-Ʉɚɪɥɨ // Ⱦɨɤɥɚɞɵ Ⱥɇ.
2009. Ɍ. 428. ʋ 2. ɋ. 163–165.
3. ɉɚɧɬɟɥɟɟɜ Ⱥ.ȼ., Ɋɭɞɟɧɤɨ ȿ.Ⱥ., Ȼɨɪɬɚɤɨɜɫɤɢɣ Ⱥ.ɋ. ɇɟɥɢɧɟɣɧɵɟ ɫɢɫɬɟɦɵ ɭɩɪɚɜɥɟɧɢɹ: ɨɩɢɫɚɧɢɟ, ɚɧɚɥɢɡ ɢ
ɫɢɧɬɟɡ. – Ɇ.: ȼɭɡɨɜɫɤɚɹ ɤɧɢɝɚ, 2008.
4. ɉɚɪɚɟɜ ɘ.ɂ. ȼɜɟɞɟɧɢɟ ɜ ɫɬɚɬɢɫɬɢɱɟɫɤɭɸ ɞɢɧɚɦɢɤɭ ɩɪɨɰɟɫɫɨɜ ɭɩɪɚɜɥɟɧɢɹ ɢ ɮɢɥɶɬɪɚɰɢɢ. – Ɇ.: ɋɨɜɟɬɫɤɨɟ
ɪɚɞɢɨ, 1976.
Ɏɢɥɶɬɪɚɰɢɹ ɫɢɝɧɚɥɨɜ ɜ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɫɢɫɬɟɦɚɯ…
81
5. Ɋɵɛɚɤɨɜ Ʉ.Ⱥ. ɋɜɟɞɟɧɢɟ ɡɚɞɚɱɢ ɧɟɥɢɧɟɣɧɨɣ ɮɢɥɶɬɪɚɰɢɢ ɤ ɡɚɞɚɱɟ ɚɧɚɥɢɡɚ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɫ
ɨɛɪɵɜɚɦɢ ɢ ɜɟɬɜɥɟɧɢɹɦɢ ɬɪɚɟɤɬɨɪɢɣ // Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɢ ɩɪɨɰɟɫɫɵ ɭɩɪɚɜɥɟɧɢɹ. 2012. ʋ 3.
ɋ. 91–110. [ɗɥɟɤɬɪɨɧɧɵɣ ɪɟɫɭɪɫ]. URL: http://www.math.spbu.ru/diffjournal.
6. Ɋɵɛɚɤɨɜ Ʉ.Ⱥ. ɉɪɢɛɥɢɠɟɧɧɨɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɨɩɬɢɦɚɥɶɧɨɣ ɧɟɥɢɧɟɣɧɨɣ ɮɢɥɶɬɪɚɰɢɢ ɞɥɹ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɫɢɫɬɟɦ ɦɟɬɨɞɨɦ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɢɫɩɵɬɚɧɢɣ // ɋɢɛɢɪɫɤɢɣ ɠɭɪɧɚɥ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ
ɦɚɬɟɦɚɬɢɤɢ. 2013. Ɍ. 16. ʋ 4. ɋ. 377–391.
7. Ɋɵɛɚɤɨɜ Ʉ.Ⱥ. Ɉ ɪɟɲɟɧɢɢ ɪɨɛɚɫɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ Ⱦɭɧɤɚɧɚ–Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ ɞɥɹ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ
ɫɢɫɬɟɦ // ɂɧɮɨɪɦɚɰɢɨɧɧɵɟ ɢ ɬɟɥɟɤɨɦɦɭɧɢɤɚɰɢɨɧɧɵɟ ɬɟɯɧɨɥɨɝɢɢ. 2014. ʋ 22. ɋ. 9–15.
8. Ɋɵɛɚɤɨɜ Ʉ.Ⱥ. ɉɪɢɛɥɢɠɟɧɧɵɣ ɦɟɬɨɞ ɮɢɥɶɬɪɚɰɢɢ ɫɢɝɧɚɥɨɜ ɜ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɫɢɫɬɟɦɚɯ ɞɢɮɮɭɡɢɨɧɧɨɫɤɚɱɤɨɨɛɪɚɡɧɨɝɨ ɬɢɩɚ // ɇɚɭɱɧɵɣ ɜɟɫɬɧɢɤ ɆȽɌɍ ȽȺ. 2014. ʋ 207. ɋ. 54–60.
9. Ɋɵɛɚɤɨɜ Ʉ.Ⱥ. Ɋɟɲɟɧɢɟ ɪɨɛɚɫɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ Ⱦɭɧɤɚɧɚ–Ɇɨɪɬɟɧɫɟɧɚ–Ɂɚɤɚɢ ɞɥɹ ɫɢɫɬɟɦ ɞɢɮɮɭɡɢɨɧɧɨɫɤɚɱɤɨɨɛɪɚɡɧɨɝɨ ɬɢɩɚ ɧɚ ɨɫɧɨɜɟ ɫɩɟɤɬɪɚɥɶɧɨɝɨ ɦɟɬɨɞɚ // ɋɢɫɬɟɦɢ ɨɛɪɨɛɤɢ ɿɧɮɨɪɦɚɰɿʀ. 2014. ȼɵɩ. 7 (123).
ɋ. 143–147.
10. ɋɢɧɢɰɵɧ ɂ.ɇ. Ɏɢɥɶɬɪɵ Ʉɚɥɦɚɧɚ ɢ ɉɭɝɚɱɟɜɚ. – Ɇ.: Ʌɨɝɨɫ, 2007.
FILTERING FOR JUMP-DIFFUSION MODELS BY STATISTICAL MODELING METHOD
Rybakov K.A.
This article develops new methods that reduce the optimal filtering problem for jump-diffusion models to the
analysis problem for the special stochastic system with jumps, branching and terminating trajectories. Earlier appropriate
methods and algorithms have been proposed and tested for diffusion models.
Keywords: branching processes, conditional density, Duncan–Mortensen–Zakai equation, jump-diffusion, Monte
Carlo method, optimal filtering problem, statistical modeling, stochastic system.
REFERENCES
1. Kazakov I.Ye., Artem’ev V.M., Bukhalev V.A. Analiz sistem sluchaynoy struktury (Analysis of Systems with
Random Structure), Moscow, Fizmatlit Publishing Company, Nauka Publishers, 1993.
2. Mikhaylov G.A., Averina T.A. The Maximal Section Algorithm in the Monte Carlo Method, Doklady
Mathematics, 2009, vol. 80, no. 2, pp. 671–673.
3. Panteleev A.V., Rudenko Ye.A., Bortakovskiy A.S. Nelineynye sistemy upravleniya: opisanie, analiz i sintez
(Nonlinear Control Systems: Description, Analysis, and Synthesis), Moscow, University Book, 2008.
4. Paraev Yu.I. Vvedenie v statisticheskuyu dinamiku protsessov upravleniya i fil'tratsii (Introduction to Statistical
Dynamics of Control and Filtering Processes), Moscow, Soviet Radio, 1976.
5. Rybakov K.A. Differentsial'nye uravneniya i protsessy upravleniya, 2012, no. 3, pp. 91–110, available at:
http://www.math.spbu.ru/diffjournal.
6. Rybakov K.A. Solving approximately an optimal nonlinear filtering problem for stochastic differential systems by
statistical modeling, Numerical Analysis and Applications, 2013, vol. 6, no. 4, pp. 324–336.
7. Rybakov K.A. Informatsionnye i telekommunikatsionnye tekhnologii, 2014, no. 22, pp. 9–15.
8. Rybakov K.A. Nauchnyi vestnik MGTU GA, 2014, no. 207, pp. 54–60.
9. Rybakov K.A. Sistemy obrabotki informatsii, 2014, no. 7 (123), pp. 143–147.
10. Sinitsyn I.N. Fil'try Kalmana i Pugacheva (Kalman and Pugachev Filters), Moscow, Logos, 2007.
ɋɜɟɞɟɧɢɹ ɨɛ ɚɜɬɨɪɟ
Ɋɵɛɚɤɨɜ Ʉɨɧɫɬɚɧɬɢɧ Ⱥɥɟɤɫɚɧɞɪɨɜɢɱ, 1979 ɝ.ɪ., ɨɤɨɧɱɢɥ ɆȺɂ (2002), ɤɚɧɞɢɞɚɬ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ
ɧɚɭɤ, ɞɨɰɟɧɬ ɤɚɮɟɞɪɵ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɤɢɛɟɪɧɟɬɢɤɢ ɮɚɤɭɥɶɬɟɬɚ «ɉɪɢɤɥɚɞɧɚɹ ɦɚɬɟɦɚɬɢɤɚ ɢ ɮɢɡɢɤɚ» ɆȺɂ, ɚɜɬɨɪ
110 ɧɚɭɱɧɵɯ ɪɚɛɨɬ, ɨɛɥɚɫɬɶ ɧɚɭɱɧɵɯ ɢɧɬɟɪɟɫɨɜ – ɚɧɚɥɢɡ ɢ ɫɢɧɬɟɡ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɭɩɪɚɜɥɟɧɢɹ, ɫɩɟɤɬɪɚɥɶɧɚɹ
ɮɨɪɦɚ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɩɢɫɚɧɢɹ ɫɢɫɬɟɦ ɭɩɪɚɜɥɟɧɢɹ, ɦɟɬɨɞɵ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɫɢɫɬɟɦ ɭɩɪɚɜɥɟɧɢɹ.
1/--страниц
Пожаловаться на содержимое документа