Modélisation stochastique et simulation - Cours et by Bernard Bercu and Djalil Chafaï

By Bernard Bercu and Djalil Chafaï

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I{Un 0 Ø ØÓÙØ ×Ù Ø (Un ) Ú Ö Ð × Ð ØÓ Ö × Ò Ô Ò ÒØ × ÐÓ ÙÒ ÓÖÑ ×ÙÖ [0, 1]¸ Ð Ú Ö Ð Ð ØÓ Ö N = max{n ∈ N∗ ×Ø Ò ÔÖ ×ÕÙ × Ö Ñ ÒØ Ø ×Ù Ø Ð ÐÓ ÓÔÝÖ Ø ¾¼¼ ¸ ¾¼¼ ¸ º Ö
Ù ² º º ¾º ¼ ¾ ¾ Ø Ð ÕÙ U1 · · · Un > exp(−λ)} ÈÓ ××ÓÒ P(λ)º ´¾¼¼ ¹¼ ¹¼ µº Ó
ÙÑ ÒØ Ð ØÖÓÒ ÕÙ Ö Ð Ù ÐÐ Ø ¾¼¼ ¸ ¾¿ ¼ º È º À ÑÓÒ×ØÖ Ø ÓÒº ÜÔÓÒ ÒØ Ä × Ú Ö E(λ)º ÐÐ (Vn ) N0 = 0 Ð × ÇÒ ÔÓ× Ò ÈÁÌÊ ½º Éͳ Ë̹ Vn = − log(Un )/λ ØÓÙØ Ö Ð t > 0¸ × Ô Ö Ø¸ ÔÓÙÖ Nt = Card{n ∈ N∗ ; − log(U1 · · · Un ) ∗ = Card{n ∈ N ; V1 + · · · + Vn Ä Ô Ð (Nt )t 0 ÓÒ×Ø ØÙ ÙÒ ÔÖÓ ××Ù× ÈÓ ××ÓÒ × ÑÔÐ ÖØ ÙÐ Ö¸ Nt ∼ P(λt)¸ Ø ÓÒ N = N1 ∼ P(λ)º ij Ú Ò Ñ ÔÖÓ ××Ù× × ÙØ n Ó × ×ÙÖ Ð³ ÒØ ÖÚ ÐÐ Ø ÑÔ× [0, t]º Ñ ÐÐ ÁÐ Ð Ü ×Ø ÔÐÙ× ÙÖ× Ñ Ò (Un )º log Ø Ò ×Ù Ø ÓÒ
Ø ÓÒ Ä Ö × Ð ³ ÜÔÖ Ñ Ö Ð Ø ÒØ ÖÚ Ò Ö ÕÙ³ÙÒ Ö Ô º ÉÙ³ Ò ³Ó Ø Ò Ö ×ع Ð ½º Ð × Ñ ØÖ × ÖÔÓ ×× Ó × ÙÐ Ú Ð ÔÖ × ÓÒ Ð Ä ×ÓÒØ ËÁÅÍÄ Ò Ô Ò ÌÁÇÆ ÒØ × ÐÓ λt} t}.

Ud−1 )º Ð ÓÙÖÒ Ø ÙÒ Ð ÐÓ µ Ø ÓÒÒ Ð ÓÒ×ØÖÙ
Ø ÓÒ Ð ÓÒ
Ø ÓÒ fµ ÙÒ Ò× Ñ Ð Ð × Ø ÓÒ (x1 , . . , xd ) Ö Ð × Ø ÓÒ Ö Ä × Ù ¹Ò Ä Ð Ð Ô ØÖ ÓÙÖÒ ×× ÒØ Ò ÐÓ ÙÒ ÓÖÑ ×ÙÖ Ð Ø× {0, 1}º Ä ÑÔ Ö Ø × ÁÐ Ò× Ð Ø × ÑÔÐ × ÙØ Ð × × ÑÙÐ Ø ÓÒ Ð Ð × ÔÓÙÖ Ð ÐÓ ÙÒ ÓÖÑ Ú ÐÓÔÔ ¸ Ö Ò׸ Ø × Øº ×Ù [0, 1]º ÒÒ Ð ÖØ × ½ ÒÓÑ Ö ÙÜ Ð Ø ÙÖ Ð Ö Ø ÊÓ×× ½¾ ℄ ÔÖÓ Ð Ñ ×ÙÖ Ò× Ð × Ø Öº Ä × Ø ÓÒ ½º½¿ ÑÓÒØÖ × Ö ×ÙÖ Ð ×Ø ÔÓ×× Ð ÐÐ Ñ ÒØ ÙØ Ð × Ø Ò ÒÓÑ Ö Å Ø Ó × ÑÔÖ ÔÖÓÚ Ò ÒØ Ò × ¼º ÐÐ × ÙØ ÙÖ׸ ÓÒØ Ú ÔÖÓ Ø Ð × Ð ÚÖ × Ò Ò Ö Ð × ÖÒÓÙÐÐ × Ö Ð × Ø ÓÒ× Ò ÙÒ ×Ù Ø Ð Ô Ò Ö Ñ Ù Ø Ø ÕÙ Ò× Ò ÒØ × Ø [0, 1] Ù × Ù ÔÖÓ Ð Ñ ÕÙ Ð Ò Ö × ×Ù Ø × × Ö ×ÙÖ ×ØÖ B(1/2) = (δ0 +δ1 )/2º Ô Ò Ò Ö Ø ÓÒ ÐÓ ÙÒ ÓÖÑ ÒØ× × Ô Ø ×ØÖ Ð ÓÑÑ ÒØ Ù × × ÐÓ ÑÓÒÒ º Ò × Ö Ô Ò Ð × ÒØ º Ø Ð × Ò× Ð × ÓÖ × Ò ÙÜ Ö Ò × Ø × ÓÒ× ×Ø ÒØ {0, 1} × ×Ù Ø × ÓÒÒ Ò Ø ÙÖ× ÔÓÙÖ × ÑÙÐ Ö Ð Ò Ô ÖØ Ö ×× ÒØ Ð × Ú Ð ÙÖ× ÓÖ × ÐÓ ÙÒ ÓÖÑ × Ð ÓÖ Ø Ñ × Ú Ð ÙÖ× Ò Ø Ò Ø ×ÙÖ Ð × ÙØ Ð × Ø ÖÑ Ò ×Ø × Õ٠Р׺ ׺ {0, 1} × ×Ù Ø × ×ÓÒØ × ×Ù Ø × Ô ×× ÒØ Ø ×Ø× ×Ø Ø ×Ø ÕÙ × Ø Ð ×º ³ÙÒ ×Ý×Ø Ñ ¾¼¼ ¸ ¾¼¼ ¸ º Ö
Ù ² º X ÕÙ ×ÓÒØ ÐÓ × Ñ Ø Ó ÔÖ Ø Ð × 2 ×Ø Ð ÓÒ×ØÖÙ Ö ØÖ × Ö Ò ÔÖÓ Ù × ÒØ Ö ÙÖ× Ú Ñ ÒØ × Ú Ø ×× × Ø Ð ×º Ò × ÑÙÐ Ø ÓÒ {0, 1}¸ ÕÙ ÕÙ ÔÖÓ Ù Ø Ø Ô ÙÚ ÒØ Ö ÔÖÓ Ù
Ø Ð × X ÔÖÓ Ð Ñ Ð³ Ò
ÓÑ Ö Ñ Òظ Ð Å Ø Ó × ÔÖ ³ ÒØ Ö Øº Ä × × Ø ÓÒ× ×Ù Ú ÒØ × Ú ÐÓÔÔ Ñ ÒØ× Ñ Ò × Ô Ö Ð Ø Ð × Ø ÑÔÖ Ä × Ñ Ø Ó ÓÔÝÖ Ø Ø Å Ö× Ô Ù ÚÓÒ Æ ÙÑ ÒÒ Ð³ Ö ØÙÖ Ñ Ø Ó {0, 1} Òظ ÓÙØÖ ÙÒ ÖØ Ð ℄¸ Ê ÔÐ Ý ½¾ ℄¸ ÐÓ ÙÒ ÓÖÑ Å Ø Ó × ÔÖ ×ÓÒØ × Ò Ñ ÒØ× Ð × Ù ÔÓ ÒØ ÙÒ ×ÙÖ ÜÔÐ Ø × Ô ÖØ Ö × ×ÓÒÒ Ñ ÒØ ÔÖ × ÒØ × ÑÙÐ Ø ÓÒ Ô Ò ÜÔÐ Ø Ø ÓÒ Ð ÐÓ ÙÒ ÓÖÑ ×ÙÖ ß¼¸½ Ø × ÙÐ Ñ ÒØ × ÙÒ ÓÖÑ ÓÒ
Ø ÓÒ× ÓÙØ ℄¸ ÃÒÙØ Ê Ù
Ø ÓÒ Ñ ØØÖ Ö Ö × ×ÓÒØ × Ò× ÚÖÓÝ ÍÒ Ö × Ò × ÐÓ × Ù×Ù ÐÐ × × Ö ÔÓ× ÒØ ×ÙÖ Ò×Ù Ø Ð × ÔÐÙ× Ð ÔÖ ×º Ò³ ×Ø Ô × ÙÒ ÕÙ º ËÓÒ ÖØ × Ñ Ø Ó ÓÒØ fµ ÓÒ
Ø ÓÒ × ÑÙÐ Ø ÓÒ × ÔÖ ÙÚ ×ÓÒÒ d > 2¸ ÄÓÖ×ÕÙ Ð Ä ½º Ñ Ð × ÑÙÐ Ø ÓÒµº ÈÓÙÖ ØÓÙØ ÐÓ µ ×ÙÖ Rd ¸ Ð Ü ×Ø ÓÒØ Ð³ Ò× Ñ Ð × ÔÓ ÒØ× ×
ÓÒØ ÒÙ Ø ×Ø Ä × Ù ¹ ØÓ Ö fµ (U1 , .

Xd ) Ö Ð × Ø ÓÒ Ö Ä × Ù ¹Ò Ä Ð Ð Ô ØÖ ÓÙÖÒ ×× ÒØ Ò ÐÓ ÙÒ ÓÖÑ ×ÙÖ Ð Ø× {0, 1}º Ä ÑÔ Ö Ø × ÁÐ Ò× Ð Ø × ÑÔÐ × ÙØ Ð × × ÑÙÐ Ø ÓÒ Ð Ð × ÔÓÙÖ Ð ÐÓ ÙÒ ÓÖÑ Ú ÐÓÔÔ ¸ Ö Ò׸ Ø × Øº ×Ù [0, 1]º ÒÒ Ð ÖØ × ½ ÒÓÑ Ö ÙÜ Ð Ø ÙÖ Ð Ö Ø ÊÓ×× ½¾ ℄ ÔÖÓ Ð Ñ ×ÙÖ Ò× Ð × Ø Öº Ä × Ø ÓÒ ½º½¿ ÑÓÒØÖ × Ö ×ÙÖ Ð ×Ø ÔÓ×× Ð ÐÐ Ñ ÒØ ÙØ Ð × Ø Ò ÒÓÑ Ö Å Ø Ó × ÑÔÖ ÔÖÓÚ Ò ÒØ Ò × ¼º ÐÐ × ÙØ ÙÖ׸ ÓÒØ Ú ÔÖÓ Ø Ð × Ð ÚÖ × Ò Ò Ö Ð × ÖÒÓÙÐÐ × Ö Ð × Ø ÓÒ× Ò ÙÒ ×Ù Ø Ð Ô Ò Ö Ñ Ù Ø Ø ÕÙ Ò× Ò ÒØ × Ø [0, 1] Ù × Ù ÔÖÓ Ð Ñ ÕÙ Ð Ò Ö × ×Ù Ø × × Ö ×ÙÖ ×ØÖ B(1/2) = (δ0 +δ1 )/2º Ô Ò Ò Ö Ø ÓÒ ÐÓ ÙÒ ÓÖÑ ÒØ× × Ô Ø ×ØÖ Ð ÓÑÑ ÒØ Ù × × ÐÓ ÑÓÒÒ º Ò × Ö Ô Ò Ð × ÒØ º Ø Ð × Ò× Ð × ÓÖ × Ò ÙÜ Ö Ò × Ø × ÓÒ× ×Ø ÒØ {0, 1} × ×Ù Ø × ÓÒÒ Ò Ø ÙÖ× ÔÓÙÖ × ÑÙÐ Ö Ð Ò Ô ÖØ Ö ×× ÒØ Ð × Ú Ð ÙÖ× ÓÖ × ÐÓ ÙÒ ÓÖÑ × Ð ÓÖ Ø Ñ × Ú Ð ÙÖ× Ò Ø Ò Ø ×ÙÖ Ð × ÙØ Ð × Ø ÖÑ Ò ×Ø × Õ٠Р׺ ׺ {0, 1} × ×Ù Ø × ×ÓÒØ × ×Ù Ø × Ô ×× ÒØ Ø ×Ø× ×Ø Ø ×Ø ÕÙ × Ø Ð ×º ³ÙÒ ×Ý×Ø Ñ ¾¼¼ ¸ ¾¼¼ ¸ º Ö
Ù ² º X ÕÙ ×ÓÒØ ÐÓ × Ñ Ø Ó ÔÖ Ø Ð × 2 ×Ø Ð ÓÒ×ØÖÙ Ö ØÖ × Ö Ò ÔÖÓ Ù × ÒØ Ö ÙÖ× Ú Ñ ÒØ × Ú Ø ×× × Ø Ð ×º Ò × ÑÙÐ Ø ÓÒ {0, 1}¸ ÕÙ ÕÙ ÔÖÓ Ù Ø Ø Ô ÙÚ ÒØ Ö ÔÖÓ Ù
Ø Ð × X ÔÖÓ Ð Ñ Ð³ Ò
ÓÑ Ö Ñ Òظ Ð Å Ø Ó × ÔÖ ³ ÒØ Ö Øº Ä × × Ø ÓÒ× ×Ù Ú ÒØ × Ú ÐÓÔÔ Ñ ÒØ× Ñ Ò × Ô Ö Ð Ø Ð × Ø ÑÔÖ Ä × Ñ Ø Ó ÓÔÝÖ Ø Ø Å Ö× Ô Ù ÚÓÒ Æ ÙÑ ÒÒ Ð³ Ö ØÙÖ Ñ Ø Ó {0, 1} Òظ ÓÙØÖ ÙÒ ÖØ Ð ℄¸ Ê ÔÐ Ý ½¾ ℄¸ ÐÓ ÙÒ ÓÖÑ Å Ø Ó × ÔÖ ×ÓÒØ × Ò Ñ ÒØ× Ð × Ù ÔÓ ÒØ ÙÒ ×ÙÖ ÜÔÐ Ø × Ô ÖØ Ö × ×ÓÒÒ Ñ ÒØ ÔÖ × ÒØ × ÑÙÐ Ø ÓÒ Ô Ò ÜÔÐ Ø Ø ÓÒ Ð ÐÓ ÙÒ ÓÖÑ ×ÙÖ ß¼¸½ Ø × ÙÐ Ñ ÒØ × ÙÒ ÓÖÑ ÓÒ
Ø ÓÒ× ÓÙØ ℄¸ ÃÒÙØ Ê Ù
Ø ÓÒ Ñ ØØÖ Ö Ö × ×ÓÒØ × Ò× ÚÖÓÝ ÍÒ Ö × Ò × ÐÓ × Ù×Ù ÐÐ × × Ö ÔÓ× ÒØ ×ÙÖ Ò×Ù Ø Ð × ÔÐÙ× Ð ÔÖ ×º Ò³ ×Ø Ô × ÙÒ ÕÙ º ËÓÒ ÖØ × Ñ Ø Ó ÓÒØ fµ ÓÒ
Ø ÓÒ × ÑÙÐ Ø ÓÒ × ÔÖ ÙÚ ×ÓÒÒ d > 2¸ ÄÓÖ×ÕÙ Ð Ä ½º Ñ Ð × ÑÙÐ Ø ÓÒµº ÈÓÙÖ ØÓÙØ ÐÓ µ ×ÙÖ Rd ¸ Ð Ü ×Ø ÓÒØ Ð³ Ò× Ñ Ð × ÔÓ ÒØ× ×
ÓÒØ ÒÙ Ø ×Ø Ä × Ù ¹ ØÓ Ö fµ (U1 , .

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