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Stochastic Processes and Stochastic Integration - Marcus Pivato

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Stochastic Processes and Stochastic Integration - Marcus Pivato

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Page 1: Stochastic Processes and Stochastic Integration - Marcus Pivato

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Page 2: Stochastic Processes and Stochastic Integration - Marcus Pivato

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Page 3: Stochastic Processes and Stochastic Integration - Marcus Pivato

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Page 4: Stochastic Processes and Stochastic Integration - Marcus Pivato

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§Dprk�giqcxyk{¨cq©z^k µ µh¼ ½�¾ sipre_w^¨cn²p^m�Ç�k|gµfÀeyÈ^m_siqcv�¬r³�e_¬)k{³igiqck+s+Éa�b µ µ.¼ ½C¾ s�pre�wr¨tn3È6k�¨©qczrk9m�³�¨©´�e_³«nFk{³�k+neX�sielg�p^mXg/f�k�v{m�z s�k+z^s�qcÈr¨c´3s�¬)k+m_Æ�e_¯FgiprqczrØ�s¨©qcÆ_k�Zµg�prk�¯ÒwFgiw^³ikE\�m�z^n�Zµg�prk3¬^m_s�g]\^b~rb µ µ ¼ ½C¾ sipre_w^¨cn1p^m�Ç_k3sie_xyk3s�e�³�gle_¯/m�nrnFq©giqcÇ_k$s�gi³�w^v½g�wr³ik�Xrs�e�g�p^mXg|fÀk3v{m_z�sik{zrìs�qcÈr¨c´ysi¬)k+m�Æ�e_¯7Z�xyeXÇZqczrØ3¯Òe_³�fDm�³«nrsm\�í�e_³8Z�È6m_v«ÆZfÀm_³�n^sm\-î�È-´�sie_xykVm�xye_wrz-g9bÏ�e_xyxye_z�v«pre�qcv{k+s�¯Òe_³ µ µ.¼ ½C¾ m�³�k_ÉN §DprkV³ik9m�¨�¨cq©zrk�X � � bN §DprkV³ik9m�¨AZ�p6m�¨©¯±ì\¨cq©z^k�\wX � äw�g¿ËîN §DprkVÈ6e�wrz^nFk9nog�q©xyk3qcz-gik{³�ÇXm�¨ � äw�+a.�N §DprkVq©z-gik+Ø_k+³�s�X � � bN §DprkVz^mXg�wr³�m_¨�zZwrx"È)k{³«shX � � bN §DprkV³�m�giqce_z^m_¨)zZwrx È6k+³�s�X �� b�#e_gik1gip^m�g�fÀk1nre zre_g�³�k k wrqc³�këgip6mXg µ µ.¼ ½C¾ È6k¥v{¨©e-s�k9nÕwrz^nFk+³�m_nrnFq©giqce_zR_�g�prqcsfÀe_wr¨tnk9sisik{z-giqtm�¨c¨c´²qcxy¬r¨c´m�zvZiq©zrê^zrq©gik ¯ÒwFgiwr³�k�\`X�flprqtv«p�qts�z^e�g3m�¨cfÀm�´Fs#¬)e�s�siq©Èr¨ck_ba k$m�¨ts�eynFe_z�ç g#³�k k wrqc³ik�gip^k3k�ÔFqcs�gik+z^v�kVe_¯�m�³�Èrq©gi³«m�³�´yq©zZÇ_k+³�sik+s�_-giprqtslfÀe_wr¨tn1zrk+v{k+s�ìsiq°g«mXgik²m�z®Z�qczFê^zrq©gik1¬^m_s�g]\`X<flprqcv«pèx�m�´ m_¨csie�È)k1qcxy¬6e-sisiqcÈr¨©k�b Ê�qcz^m�¨c¨c´CXÂfÀkoflqc¨©¨sie_xyk�g�q©xyk+sVv«pre-s�k µ µ.¼ ½�¾ gie²È)k ����¤9]�UXW-�+W í�q©k�b � � e�³ � � î�X�m�z^n s�e�xyk�giqcxyk+s ]_�)�R��+�5�����)��¤ íÒqck_b � � X � äw�g¿Ëî.XFk�g«v�b�î²b§Dprk"si¬^m�v�k$e_¯�m�¨c¨�¬)e�s�s�qcÈr¨ck^Z�p^qcs�gie�³iqck+s]\�e�¯Âe_w^³�si´Zs�gik+x�qts|giprk+z¥giprk�s�¬6m_v�k"e�¯m�¨c¨D¬6e-sisiqcÈr¨©k µ µ ¼ ½C¾ ° qcz^nFk{ÔZk9n¢¬^m�gipègipr³�e_w^Ø_pËsi¬^m_v{kC_Dqczèe_giprk+³ fÀe_³«nrs�X<giprk¥si¬^m_v{ke�¯�m�¨c¨�¯Òwrz6v½giqce_z6s"¯Ò³�e_x µ µh¼ ½C¾ q©z-gie Æ� ÍônF´Zz^m�xyqtv{m�¨lsi´Fsµg�k{x�v{m_zËÈ)k1³�k{Ø-m�³«nFk+nm_s v«pre�siqczrØ�m�¯Òk+f(X�s�¬)k+v{q°ê)vë¬^m�gip^s"¯Ò³ie�xÎg�prqts v{e_¨c¨©k9v½giqce_zhí�v{m_¨©¨ck+n �^U�t/�Ò�+¤ î$m_z^nnFk9v�¨tm�³�q©zrØ�gip^k{x g�e3È6kÀg�prkle_zr¨c´Z�m�¨c¨©eXfÀk+nw\|¬^mXg�p^sMR ¯Òwr³igiprk+³ixye_³�kCXXz^e_zrkle_¯^gip^k+sik¬^m�gip^s�v{m_z3q©z-gik+³�sik+v�g+b�ÍËs�gieFv«p^m_s�giqtv<¬r³�eFv�k9sis�q©xy¬)e�sik+s�m|x"w^v«p3fÀk+m_Æ_k+³�v�e�z^sµg�³�m_q©z-gE_q©g�qczZÇ_e_¨cÇ_k9s#siq©xy¬r¨c´�¬r¨cm�v�qczrØom b UX�Mt�[Ht/�Çu5�Ò�UÌ@¸¢WZ[r¤9��UXW e_z¥g�prk�si¬^m�v�k$e_¯Â¬^mXg�p^shXnFk9v�¨tm�³�q©zrØysie_xykVprqcs�gie�³iqck+sÀg�eyÈ6k$xye�³ik�e_³l¨ck+s�s8Z�¨cq©Æ�k{¨c´H\$g�p^m�z�e�g�prk{³«s+bÊre�³ix�m_¨©¨c´CXZfÀk$nFk�ê^z^kVm�s�gieFv«p^m�sµg�qcv�¬r³�eFv�k+s�slm_sÀ¯Òe�¨©¨ceXf#s+b©bcb©b

Page 12: Stochastic Processes and Stochastic Integration - Marcus Pivato

Àw�¦~C�'à�ÁTSáVURÁµß|áÀâ�z�ß(OÚâ�àMz#à/á�âw{Nz$Ý/á a�a�W��R���R¶²WYX����Ò�9���)�[Z LsB�DU:wQP>ULY<ÇD\ºimBED]I.>U>ò�k{gRÆöÈ)k�sie_xyk�sik�g�X µ µ�¼ ½C¾ sie_xykÂe_giprk+³�sik�g+b/ò�k{g]�ïÈ6kÀs�e�xyk�s�qcØ_x�m�ì÷m_¨©Ø�k{Èr³«m�e�zÆ�^ ^ ¼ ½�¾ b�§Dp^k{z mQ� ° ¸¢WZ[r¤9��UX[Ht u�W ¤+�+��]�£�[r¤+�+��] b UX��]_WZ¤9¤ e_zg�prk ¤+�+[F�9W-¤ b [r]_WÆÈXFeXÇ�k{³Àgiprk �9��¸¢W=u5����W µ µ.¼ ½C¾ XFqts#m�¬r³�e_È^m_Èrqc¨©q©gµ´ëxyk+m�s�wr³�k�e_z_�Nba q©gipre�wFgV¯Òwr³�g�prk{³VpZ´Z¬6e_giprk9s�k9s�m_È6e�wFgVgiprkys�g�mXg�k+si¬^m_v{k�Æ X6g�prky¬r³�e_¬)k{³igiqck+s�e�¯gip^kënFk+siq©³�k+n�¬^mXg�p^s�X�e_³�g�prkësµg�³iw6v½giw^³ikye�¯1��X�q°g$qtsVqcx�¬)e�s�siq©Èr¨ck�gie¥È)k�m�zZ´xye_³�ksi¬6k9v�q©ê6v�bÊre�³os�g�m�³igik+³�s�XÀflp^mXgosiq©Ø�x�mXì¼m�¨cØ_k{È^³�m s�p^e_wr¨tnéf�k¥w^sik`G�a k{¨c¨¸XlfÀkfDm�z-g�gip^ksiq©Ø�xym�ì÷m_¨©Ø�k{Èr³«m gieéÈ6k ê^zrk k{zre�wrØ_p@gip^m�g�X�m�g1g�prk Ç_k{³�´é¨©k9m_s�g�Xl¯Òe�³1k+Ç_k{³�´égiqcxyka � µ µ.¼ ½C¾ X/giprk�s�g�mXg�k1e_¯lgiprk�si´Zs�gik+xÙmXg giqcx�k a qtsymxyk+m�s�wr³«m�È^¨©k�¯Òw^z^v½g�q©e�z�bÕº¼ze�g�prk{³lfÀe_³«nrs�X-¯Òe_³#m�¨c¨ a � µ µh¼ ½C¾ X^f�kVfDm�z-g+É

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Page 13: Stochastic Processes and Stochastic Integration - Marcus Pivato

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Page 14: Stochastic Processes and Stochastic Integration - Marcus Pivato

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Page 18: Stochastic Processes and Stochastic Integration - Marcus Pivato

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Page 42: Stochastic Processes and Stochastic Integration - Marcus Pivato

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Page 86: Stochastic Processes and Stochastic Integration - Marcus Pivato

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Page 93: Stochastic Processes and Stochastic Integration - Marcus Pivato

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Page 94: Stochastic Processes and Stochastic Integration - Marcus Pivato

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=�<>=B  K@? F 6 = 9A7CBED <GF DIH>? ? �78HGB1D ="F @J?bIKD < 7CBL6:789M7ND�K = I = KQK@7:H y DO?PD <>=MQ ?RHTS = K^"y = H QU= ?bI >�¹U¼�V¯. 3 »�WU¹#>C+ X7:D�K = I = K@BKDO?YD <G=IQ ?5HXS = K y = H Q�= ?bI F B = !TZ = H QU= ?bI >Q¼/. �Vï »[ ¼/>206¼5W�V1¹#+E\ F 9]Z Q�< 9M?%K = BOZ>^�D@6 = 7C_ ="F ³B$ Ha` <GF DbB = HcB = _�? = BLD < 7CB Q ?RHTS =�d y = H QU=D F î =(  6 F5Q�= õ$ D�DOZ d HGBe?5Z�D�D <GF De` =fQ�F Hg_ = ¸GH =JFhQ�=�d D F 78H-9 =�F BOZ d@=fi ¹ ¼ º ?RHjD <>= B  GF5Q�=k Æ�t l j^V ­ Q�F 6:6 = _JD <>=Mmon�­qp�Tr ± �ts��XrS� �vu � ³ i ¹ ¼ º 7CBI_ = ¸GH = _J78HJD =�d 9�B'?5wxD <G=ý!> ï ì®5zyÀ>�¼45º¹ ?qwID <G=J{>d ? Q�= B@B 4 ³ G =|Q"F H-D <G= Hg68?T?&î F D�D <>= « ¾ "}B {GF5Q�= �`/7ND <d@= B {c=�Q DKD@? i ¹ ¼ º T?5wv~"� s ¯ ­�� ~ ° n ²$#�r ~ ° � ² ¯ u�n ² � ~�~ � ~ ³D$ D�D@Z d HcBK?RZ�DKD <GF D% TZGHG_ =�dBOZ>7ND F ^>6 =�Q ?RHG_�7:DO78?5HGB� �D <>= 7:HTD =��5d�F D@7:?RHP? {c="d@F DO? d 7CB FRQ DOZ F 686:� F H 0g+ ï »J¹@56>��B´ 7:w��7CB F B�789 { 6 = B�DO? Q�<GF B1D@7 QL{>d ? QU= B@B� XD <>= H����� Zj3¶t� � ­ �R4 � E ­����� ¾ X����L� ¾` <>="dO= D <>= H>? d 9�?5H�D <>= >20-ý�y?5�y�¼/.�� +20��%¹ 78BfD <>= « ¾ H>? d 9�?qwv« ¾T� k Æ

� l jËV ­ ´�� ���2� �` < 7:6 = D <GF Dv?5HLD <>= V1¹^îC5cyS¼4.��N+20��%¹ 7CB?=1ZcB1D�D <>= « ¾ H>? d 9�?5w?« ¾ � k ´v� ���2�� 7 = ³ D <>=Wx r u � r ± ² � ?5w ø 3¶ � � ­ �54 � E ­ F B FYd�F Hc_�?59�S Fqd 7 F ^>6 = � ³� < ZGB� v` =eQ�F H�ZGB = D <>= _ = HcB�7:D1��?5w�BO789 { 6 = B�DO? Q�<GF B1D@7 Q�{>d ? QU= BOB = BLD@? ¹O�&5R¹#.��D <G= _ = ¸GHG7ND@7:?RHo?5wD <>= 78HRD =��Rd@F 6cD@? F H�78BO?59 = D d �Z 3¶ � ù ­ �54 � E ­�� « ¾  � k Æ � l jËV ­ ´�� � � � sc¡ « � k ´v� � � �

® ="dO= �D <G= BOZ>^GB QUd 7 { D�'@¢ )M7:Hy'ó« ¾  ) dO= w =�d B£DO? ±�n{± r ± ° � ² �L¯ r�° � ± c�´ F D ="Q�< "H>7 Q�F 6 {>d ? {c="d D1�h?5w'?RZ d 7:HTD ="�5d�F HG_GBY` < 7 Q�< ` = `/786:62_>78B Q ZGB@BM9M? d@= 78H¤D <>= H = fXDB ="Q D@7:?RH ³$ HeD <>=IQ�F B = ` <G= H 4 78Bi½ d ?�`/H>7 F H � ?qD@7:?RH� TD <>= 9 ="F B�Z dO= » ¹ ¼ º DOZ d HGB£?5Z>D�D@?^ = H>?qD < 7:H � 9M? dO= D <cF HoD <>= ¯ u�n�w�¥ ²%°Ms � r ~ ¥u��» ¹ ¼ º X§¦©¨ ¿` <>="dO= ¦�78BxD <>=/{Gd ?R^ F ^>78687ND1�Y9 =�F BOZ d@= ?RH k F HG_ ¿ 7CBªD <G= « �T«K� ~ c ¥�� s � rX¬~ ¥vu�� ?RH � l j^V/­¯®h ? d�F]�5= H =�d�F 6 {Gd ? QU= BOB 4 � 78S = H F HX� <GF 6:w7"°? {c= He78HRD =�d S F 6 � ± j@² � F HG_ F HX��B = D³µ´ k �9 ="F BOZ d�F ^>6 = `/7:D <Jd@= B {�="Q D'DO?¶D <>= S Fqd 7 F ^>6 = 4 � ± � �D <>= 9 ="F B�Z dO= ?qwªD <G=d@="Q D F H � 6 = ³ ¨ � ± j�² � 7CBxD <>='·Rnc¸Dw �z¹"� nc¸ r ­�x r u � r ¸·R� � ?RH ³b� ?5w�D <>= � ¸�·qu�� s �X¸ ¹?qw 4 ?�S =�d D <G= DO789 = 78HTD ="d S F 6 � ± j@² � >` = 7 �5< D = _¶^X��D <>=b{>d ?5^ F ^>7:687:D1�e?qw » �

Page 134: Stochastic Processes and Stochastic Integration - Marcus Pivato

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X » ¼¾½8¿ ÀLÇ � 4 � ± ¡Å² � � ¾4 � ± � ð ³ÉÈ Ä"¦�Ê Ë Ì � ³ ­$ D�ö?�Í B�9 = D < ?�_j`/7:686/`�? d î�wz? d�F ^ d ? F _ Q 6 F BOB�?5w'B1D@? Q�<GF B�DO7 Qe{>d ? QU= BOB = B"ÎK78H{GF5d D@7 Q Z>6 F5d �7:D2`/786:6�`£? d îe` <>= H = S =�d 4 78B F s�r u ¹�� ¸ ccr ­Ï� `/7ND < « ¾ 78H Q�dO= 9 = HTD@B"®� <>= ^ = B1D = f { ?RBO7:DO78?5H�?5w]D <>= $ D/ö?Ð7:HTD =��5d�F 66 E78HÑD <>=hQ ?RHRD = fXD�?qwB« ¾ "¯^c?RZ>HG_ = _9 Fqd DO78H �RF 6 = B@ X7CB � Ã$­°®Ò n�¸ r ¸ ¹�� · �zÓ)�z¹"� ¸ c ± ¸ ¹ � c u r ¸Dw ~ r ¸JwÑÔ r u ¹"� ¸ c�r ­�� ¦ u�n Ó �Tu ¹�� � ~G =b="Fqd 687 ="d 9M?5DO78S F D = _¶D <>=�Q ?RHGB�D d Z Q D@7:?RHP?5wvD <G= ~�¹ n�· #r ~"¹"� · � ¸ ¹ � c u r ­Z�\ �R4F B F ` F �¶?qw �5= H =�d�F D@7:H �fÕ�Ö � 3 V:×45¯Ø��,. Ö 0g+UØ Î�` = D F î = D <>= B1D@? Q�<cF B�DO7 QI{>d ? Q�= B@B4 � ?5Z d '�H>?R78B = B�?RZ d�QU= ) � F HG_ ×/565¯Ø@. 3 ×45¯Ø 7NDE^X�¶D <>=�{Gd ? QU= BOBÏ\JDO? �5= D/D <>= îX78HG_?qw)H>?R78B = ` = ` F HTD"®$ w2D < 78BYH>?R78B = 7CBYDO?|^ = D d Z>6:� >@×/.�� Ö�Õ < ?�` = S =�d D <>= H¤` = _�?5HvÍ DM` F HTD]D <G=wzZ>H Q DO78?5H³\aD@?o^ =eF ^G6 = DO?fBO��B1D = 9 F D@7 Q"F 686:�y'O^>7 F B^)eD <>= 9M?�S = 9 = HTDI?qwKD <>= H>?R78B =DO?�` Fqd _>B2BO?59 = _�7 d@="Q DO78?5HGB F HG_ F ` F �¶w d ?R9Ù?5D <>="d B"® h ? d'= f F 9 { 6 = �BOZ {>{ ?RB = D <G=wzZ>H Q DO78?5Hí\f` F B Q ?RHGB�D d Z Q D = _eZcB�78H � 7:H>wz? d 9 F DO78?5H F ^�?5Z�D�D <G= wzZ>DOZ d@= 9�?�S = 9 = HTD�B?qw 4 ® � <>= Ho` =LQ ?5Z>6C_¶^GZ>7:6C_Ä\JBO? F BKD@?M^ = '�6 Fqd@�5= )�` <>= H = S =�d 4 9M?�S = _³'OZ { )1 F Hc_Ð'OBO9 F 686`)|` <>= H = S ="d 4 9M?�S = _Ð'O_�?�`/H1)G® $ HjD < 78B�` F �! x` =�Q ?5ZG68_jZGB = D <G=9M?�_�Z>6 F DO78H � wzZ>H Q DO78?5H�\MD@?bB���B�D = 9 F DO7 Q�F 6868� +�ÚRØ@ì D <>= H>?57CB = D@?'w F SR?5Z d 9M?�S = 9 = HTD78HoBO?59 = _�7 d@="Q DO78?5HGB F HG_ F ` F �]w d ?59Û?qD <G=�d B�®)Ü26:DO789 F D = 6:�& >^X� Q�F5dO= wzZ>6:68��^>Z>786C_�7:H �\D £` = 9M7 �R< D = S = H¤^ =JF ^>6 = DO?h9 F î = D <>= 78HRD =��Rd@F 6 {>d ? Q�= B@B ø \ �R4 SX7 d D@Z F 686:��5Ø@5}Ø#> Õ 07.10g+2560�� 78H�7ND�Bx9M?�S = 9 = HTD@B"® � < 78BK`�?RZ>68_ Q 6 =�Fqd 6:��_ = w =�F DK?RZ dx= HTD@7 d@= 78HRD = HTD"®� ? = 6:789M7:H F D = D < 78B { ?RB@B�78^>786:7:D1�! �` = 9]ZcB1De_ = 9 F HG_jD <GF D�D <>=f{>d ? QU= B@Bb\g^ =¸�nc¸ r ¸ ¹"� · ��Ó r ¹�� ¸ c G7:H�D <>= B = HGB = D <GF D% F D F HX� � 78S = H¶D@7:9 = E >D <>= S F 68Z = ?5w®\ � E �Q�F H�?RH>68��_ ="{c= HG_a?5Ha^T��7:H>wz? d 9 F DO78?5H F ^�?5Z�DbD <>=�Ý ×�+25 9M?�S = 9 = HTD@BL?qw 4 F HG_9]ZcB1D2^ = DO?5D F 6868� 0-ý4. Ö >O×4.15 ?5wvwzZ�D@Z d@= 9�?�S = 9 = HTD�B�®� <>="dO=|F5dO= 9 F HX�g_�7NÞ =�d@= HTDo` F ��B�D@?jwz? d 9 F 6:68�-_ =Uß H =�¸�nc¸ r ¸ ¹"� · ��Ó r ¹�� ¸ c ` < 7 Q�< 6 ="F _�DO?�_�7:Þ ="dO= HRD�D =�Q�< H>7 Q�F 6�_ = S = 6:? { 9 = HTD@B� ª^>Z�D F 686 {>d ?X_>Z QU= ^ F BO7 Q"F 686:�D <G= B F 9 =PQ ?5H Q 6:ZGBO78?5H�® � <G= 789 { ? d D F HTD]78_ ="F 7CBbD <GF D�\-_ =�{�= Hc_>B�?5H>68��?RHhD <G=Ý ×�+25 ?qw 4 F Hc_P7CB�78HG_ =�{�= Hc_ = HTDE?5w7:D@BxàC3 5 3 >OØ 9M?�S = 9 = HTD@B"®$ w(\�7CB ¸�nc¸ r ¸ ¹�� · ��Ó r ¹"� ¸ c F HG_¤D <G=J{>d ? Q�= B@B 4 78B F s�r u ¹"� ¸ c�r ­�� �D <>= HD <G= 7:HTD =��5d�F 6 {Gd ? QU= BOB ø \ �54 7CB ×�V<+ Ö 9 Fqd DO78H �RF 6 = ® � < 78Be78B¶?5H = ?qwID <>= 9M?RB�D789 { ? d D F HTD {Gd ? {c="d D@7 = B�?qwvD <>= $ D�ö?Ù78HRD =��Rd@F 6Ï®

Page 135: Stochastic Processes and Stochastic Integration - Marcus Pivato

º à 8 % LNMPO'&)( Q+*SR-, &/. % LNM , & T�%§T10 &)( UHQ M2& T . 0� »'á �Tu�· ��~ � ��â ?5HcB1D d Z Q D F Hh78HTDOZ>7:DO78S =¶F5dO� Z>9 = HTDb` < � ø \ �54 B < ?5Z>6C_a^ =PF9 Fqd DO78H �RF 6 = 7:w®\|78B�HG?5H F HRD@7 Q 7 {cF DO78H � ® �ã ¹ n�· #�r ~�¹�� ·Jä r ­�·R¥J­Ï¥ ~� D@? Q�<cF B�DO7 Q 78HTD ="�5d�F 6CBvwz? d 9åD <>= ^ F B�7CB�wz? d _ =�ß H>78H �bF H = HTDO7 d@= ��×qV:� 3 V 3 + ?qw�B1D@? Q�<cF BË"DO7 Q�{>d ? QU= B@B = B"® � <>= a ¥�¸Dw r>s �T¸ ¹ r ­�æ # �XnGu�� s ncçom � ° �Tu��T¸ ¹�� r ­�· r ­�·R¥J­Ï¥ ~D = 6:6CB/ZGB£D <GF D% �7Nw�D <>= wzZGH Q D@7:?RH�è©7CB/_ =Uß H = _o^X� �è � V � � X Z�3é \ � E � � E jD <G= H� �wz? d2F HX��V( � è� E � V � X¸\ � V � [$ H F BO7:9M786 F5d w F B < 78?5Hu �7:w F B�DO? Q�<GF B�DO7 Q){>d ? QU= B@Bvè-7CB �RØ�êA.�Ø�� SX7 F D <>= B1D@? Q�<GF B�DO7 Q78HRD =��Rd@F 6 è � V � X Z 3é \ � E � �R4 � E ­ � º �D <G= H]` = 9M7 �5< Dvwz? d 9 F 6:68�bB F �'D <GF D®\ �R4 7CBD <>= 'O_ ="d 78S F DO78S = )/?qw�èb F HG_b` d 7:D = �� èX \ �R4 �Ïë �$ DED@Z d HGBE?RZ�DED <GF D'D < 7CBL78BOH�Í DA=1ZGB�D FeQ Z�D =�F ^>ZGB = ?qwxHG?qD F DO78?5H�Î�D < 78BLH>?5D F DO78?5HF5Q DOZ F 6:68�oDOZ d HGB'?5Z�D'DO?P^ = !TZ>7ND =YF5{>{>d ? {>d 7 F D = G^ ="Q�F ZGB = ` =YQ�F H|_ ="d 78S =]F H F 68? � BDO? F 686ª?qwKD <>= ZGBOZ F 6 d Z>6 = BI?qw£_�7NÞ =�d@= HTD@7 F DO78?5H� �BOZ Q�<�F B'D <>= «K� �T«�¸ ��ì�í ¥D­���F HG_D <G=Yä #�r � ¸ í ¥D­�� ®Ü26:DO789 F D = 68�! G7:D'^ ="Q ?59 = B29�? dO=�Q ?5HXS = H>7 = HTD F HG_�9M? dO=b{ ?�` ="d wzZG6�DO?e_ = B Q�d 78^ =B�DO? Q�<GF B�DO7 QI{>d ? QU= B@B = B/78HfD <G= wz? d 9Û?5w = !TZ F DO78?5H �Ïë � d@F D <>="d D <cF H = !TZ F DO78?5H � º � ®$ HG_ ="= _¯ �` =bQ�F H = S = H¶wz? d 9]Z>6 F D = ~"¹ nv· #�r ~"¹"� ·_w � ° �Ru��X¸ ¹"� r ­K�Xî�¥ r ¹"� nc¸ ~I78H¶D < 78B` F �! F HG_LD <>= H]BO?568S = D <>= 9Ä $D@? Q ?RHGB1D d Z Q D .�Ø#ì {>d ? QU= B@B = B�`/7:D < BOZ>^�D@6 =�{>d ? {�=�d DO7 = B"®

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Page 136: Stochastic Processes and Stochastic Integration - Marcus Pivato

*SR �*RbQ M 0 ¨M.��J(�M , © Q ( , M , &�L'(í¨'( QHT�� M2& T�� ( , . , &/. % LNM , & T�% O Q . % ( ,G, ( ,�º à ÂG =Y<GF S = B =�= H|7:H � 8 ® ë ® ë ­ < ?�` 7:DI78B { ?TBOBO78^>6 = DO?P^>Z>7868_ F B�DO? Q�<GF B1D@7 Q]{>d ? Q�= B@B?5H F S ="Q DO? d B {cF5QU=�� ^X�oB {c=�Q 7:wz�X7:H � D <>=Yd@F HG_�?59Ù� ¸�·qu�� s �X¸ ¹"~b?5wªD <>=]{>d ? Q�= B@B?�S ="d/F 686�DO789 = "°78HTD ="d S F 6CB�?qw)H>?RH�� ="d ?�6 = H � D < ®h ? d2= S =�d ����� E 78H�� ��� ���� G` = _ =�ß H = _ FMd@F HG_�?59 S Fqd 7 F ^>6 =�� �"! # cBO?MD <cF D2D < 78BQ ?R6:6 ="Q DO78?5Ho?qw)S Fqd 7 F ^>6 = B�B F DO7CB ßG= B£D <>=�Q ?5HG_>7ND@7:?RHGB �

$&% �'�(� E ð)� �*� ����j �,+-! # X �.+-! � µ �,�"! # [ � º �G = D <>= He_ =Uß H = _�BO?59 =/d@F HG_�?R9+S Fqd 7 F ^>6 =� é F B)D <>= 'OB�D Fqd DO78H �I{ ?578HRDC)I?qw�D <G={>d ? QU= B@B� F HG_o6 =�� 78BO6 F D = D <cF D � �"! # 78Bk=1ZGB1DED <>= 'O7:H Q�dO= 9 = HTDó)1 � � �b¡ E � >D <G= H�D <G={>d ? QU= B@B � H F DOZ d�F 6868�Mw F 6868B/78HTDO? { 6 F5Q�= ®$ H�B�?R9 = B = HGB = v7:Hh_ =�ß H>78H � D <>=�d�F HG_>?59ó78H QUd@= 9 = HTD�S Fqd 7 F ^>6 =�� �"! # v` =�Fqd@=d@="F 6868��BO? d D]?qw F BOBO7 � H>78H �JFfd�F HG_�?R9 s � r ~ ¥u$�0/ D@?fD <G= 78HRD =�d S F 6 � �&j E �e´ � �*� ���$®

1 = H QU= ªD <G=PQ ?R6:6 ="Q DO78?5H�?5w d�F HG_�?R9 78H Q�dO= 9 = HTD@B32 � �"! # Î4�&j E ð)� �*� ���65P7CB d@="F 6:68� FB {c=�Q 7 F 6 Q�F B = ?5w�` <GF D/78B Q"F 686 = _ F�u r ¸Jw�n sÉs � r ~ ¥vu�� ®798;:98=<

mo�St�¸ �z¹�� n�¸?> ×/.�� ÖqÕ@ Ø�×�+ 3 >OØe = D � 4 jBA � ^ =]F 9 ="F B�Z d@F ^>6 = B {GF5Q�= ® ò u r ¸Jw�n s�s � r ~ ¥u$� 78B FMd@F HG_�?59S Fqd 7 F ^>6 = D F îX78H � 7ND�B�S F 6:Z = B/78H¶D <>= B {GF5Q�= ?5w�9 =�F BOZ dO= B£?5H � 4 jBA � ®h ? d 9 F 6:68�! �` = 6 = D � k j-C jO¦ � _ = H>?qD = B�?R9 =o{Gd ?R^ F ^>78687ND1�jB {GFRQU= F HG_ � F½ F H FRQ�< B {GF5Q�=*D �D <>= H F�u r ¸Jw�n s�s � r ~ ¥vu�� ?5H 4 XD F îX78H � S F 6:Z = B�78H � >7CB FwzZ>H Q DO78?5H � » � k ¨)A�sc¡ �BO?]D <GF D �º ® h ? d2F HX� ß f = _]ï£ð k XD <>= wzZ>H Q DO78?5H»FE � A�sc¡ �78B FG� s/S F 68Z = _P9 ="F B�Z d@= ®

HBI  %Ì9Æ � � � Æ6Å�~r�k� �BJ �|Æ®Ì �6ÆJÑ �C� Å��|Æ®Ì9�LKiÖ�{/}2~�Ì9�#�6Å%Ì �6Ì Ñ#��Æ6Ì �H��� Ì9Ê|Ì � � �4Ì �Ï� Ø2Ì9���MK � � Ó Ó � �r ��|��r �Ì9ÆJÌ9Æ�NFÅ�Ær� � Æ6}2~r�u}BO*ÆrÌ9Ø2���|� � � � Æ6Å�~r��Ì9�N�r ��kÊ �<� Æ6ÆrÌ9Ê �C� Æ6�|��Ær�#Ö I  ��®�QP�Å � �6Ì9}2� �=R^� Ì9Æ��r ��|�NFÅ�Ær��r ��kÆr� � ��� � ~6�TSVUXWZY-[ â]\X\ WZYBW_^0WZY ê Ê|}2���%Ì �6Ì9}2�Þèu �Ì9Ê6  � � �P� � � ÆrÅ�~6� � Å�Ær��Æ � �6Ì9Æ=O � Ö` I  %� Çba¯� � � Ê� ÞÆ �/� Ê|�|ÉHÊË}2�%��Ì �6Ì9}2�PÌ9Æ � ��Ì9ÆF�6~ � Ê|�rÌ9��ØÕ�r�|Ê� ���Ì9Ê �C� Ì � � ÖdcB��è � �@� � Ñ#�|Ê¡�6}2~uÆ ��� ÊË�Æ6}Hè¯�AÊ � � � ���W�6 �Ì9�%Ø2ÆD�6}2Ø2�¡�r ��|~^� � ���;è¯��è � �@� �ea¯� � � Ê6 (�r} � } � }2Ø � Æ6}H�r  � �®�r �� � Ê|}2�@Ñ#�|~6Ø2�2Ö×2Å�Ær���r �Ì9� J }BO&K � ÆAÓ Ó

Page 137: Stochastic Processes and Stochastic Integration - Marcus Pivato

º Ã�f % LNMPO'&)( Q+*SR-, &/. % LNM , & T�%§T10 &)( UHQ M2& T . 0ë ® h ? d2F HX� ß f = _P9 ="F BOZ d�F ^>6 = B�ZG^GB = D ³ ð�A)g>D <>= wzZ>H Q DO78?5H �» � ³ ­ � kih ï?j¡ »FE � ³ ­ ð �78B Õ Ø�×�kml?>O×RWUV:Ø ® � � <cF D27CB*g�7:D Q ?RHGB1D@7ND@Z�D = B F >O×on�� ÖqÕ [ ×/>mp�×RWUV8Ørq �

798;:98;:»'á r>s Ó ­��ts p Õ2Ý V:Ø�u Ö p;kBk Ö n @ Ø�×okBlXv@Ømk ¦ u��Tu��Tî�¥ ��~��z¹ � ~�� � w ®  ® ë ­e = D�¢ � k ¨ � l jBx � s�¡zy {�^ =eF)| ?R78B@B�?RH {>d ? QU= BOBL`/7:D < 78HRD = HGBO7:D1� �~} l ®� < ZGBrgRwz? dKß f = _vïegTD <G='{GF D < ¢ E � � � jBx � sc¡�y {�7CB F�{�= 7 QU= `/7CB =LQ ?5HGB�D F HRD {GF D <` < 7 Q�< 7:H Q�dO=�F B = B£^X�eZ>H>7:D����1ZG9 { B-� F D27 d@dO="� Z>6 Fqd 7:HTD =�d S F 68B"®G =KQ�F HID < 78H*îI?qw>¢ F Bv^ = 7:H � D <G=xdO= B�ZG6NDv?5wG� ¸ ¹ � c u r ¹"� ¸ c F/d�F HG_�?R9�9 ="F B�Z dO= gF B£wz?R6:68?�`2B �e = D 4 · j 4 ¾ j 4 / j [@[�[ ^ = 78HG_ ="{c= HG_ = HRD d�F HG_�?R9 S F5d 7 F ^G6 = B�?5H k g F 686 � á�Ó n ¬¸��T¸ ¹�� r ­r­I²lw �Ï~�¹ u � «)¥ ¹ �Sw `/7:D < {GF5d@F 9 = D ="d�� ®�� =�ß H = D <>=fd@F HG_�?R9ÅS F5d 7 F ^G6 = B� · j � ¾ j � / j [�[@[ ^X� �

� · X 4 ·� ¾ X 4 · µ 4 ¾�/X 4 · µ 4 ¾ µ 4 /®®®� � < ZGB�D <>= S F5d 7 F ^>6 = B � � � � ���U� ��� dO="{>d@= B = HTDKD <>= ¬ ¥ s Ój¹�� s � ~E?qw�D <>=�| ?57CBOBO?5H{>d ? QU= B@B�¢h® �h ? dxF HX� ½£? d@= 6cB�Z>^cB = D ³ ´ � l jBx � gT_ =Uß H = » � ³ ­>DO?b^ = D <>=Ed@F HG_�?59+S F5d 7 F ^G6 = �» � ³ ­ � X��=�¾Ê�� � 2��,ð�y {�Î � � ð ³ 5�­D <cF DM78BrgxD <>= HXZ>9Y^ ="d ?qw4�1ZG9 { DO789 = BYD <cF D�6:7 = `/7ND < 7:H ³ ® � <G= H » � k ¨� � l jBx � s�¡�y {g7CB F¶u r ¸Dwn s s � r ~ ¥u$� ?RH � l jBx � g�_ =Uß H = _|SX7 F D <>=Md@F HG_�?59{ 6 F5Q�= 9 = HTD¶?5w FjQ ?5ZGHRD F ^>6 =|Q ?5686 =�Q DO78?5H ?5w�� F D@?59�B"��g ="F5Q�< ?5wbZ>H>7:DP9 F B@B*g F HG_

Page 138: Stochastic Processes and Stochastic Integration - Marcus Pivato

*SR �*RbQ M 0 ¨M.��J(�M , © Q ( , M , &�L'(í¨'( QHT�� M2& T�� ( , . , &/. % LNM , & T�% O Q . % ( ,G, ( ,�º Ã��B {GF5Q�= _ F5{GFqd D�`/7:D <P= f { ?RH = HTD@7 F 6:68��_�78B�D d 78^>Z�D = _¶_�7CB�D F H QU= B�^ = D1` =�= H¶D <>= 9�® � »'á ¬�Tu�· ��~ � � � =�d 7Nwz�¶D < 7CB�® �� H F B = HGB = gK` =JQ�F HjD < 7:H1îj?qw » F BM^ = 78H � D <>= �RØ�vmp [ ×o��p [ Ø ?qwLD <G=�| ?57CBOBO?5H{>d ? QU= B@BI¢�g�^ ="Q"F ZGB = ` =�Q�F H �5= H =�d�F D =�F S ="d BO78?5H|?5w�¢ ^T�|^T�J� ¸ ¹ � c u r ¹"� ¸ c »DO? {>d ?�_�Z Q�= D <>= B�DO? Q�<GF B1D@7 Q'{>d ? Q�= B@Be� � ke��� � l jmx � s�¡�y {�gG_ =Uß H = _ �� ��� � ��� » � l j � ­b � »'á �Tu�· ��~ � �Kâ <>="Q î�D <GF D¡�ü7CB F BO789 { 6 =(| ?R78B@B�?RH {>d ? QU= BOB£?qw)78HRD = HGBO7:D1� � � ®� <>= 9 ="F B�Z d@= » 7CB Q�F 6:6 = _ F ~�� s Ó ­Ï� ¦ n �Ï~"~ nc¸ s � r ~ ¥u$� ®

� < 7CB = f F 9 { 6 = 7CBYS =�d �aBOZ �R�5= B�DO78S = ®£¢ nc� d@F HG_�?59 � s/S F 6:Z = _h9 ="F B�Z dO= ?RH� ��� ��� Q"F HP^ = ZGB = _¶DO?�_ =�ß H =bF B�DO? Q�<GF B1D@7 Q'{>d ? QU= B@B"® 1 = H Q�= g�` =IQ"F H¶D < 78H*îe?5wvD <G=d�F HG_�?R9µ9 ="F B�Z d@= B F B�^ = 7:H � D <>= �5Ø*vmp [ ×o��p [ Ø�k ?5w)B�DO? Q�<GF B1D@7 QL{>d ? QU= BOB = B"®798;:98¥¤

æ # �XnGu�� s � nd�}Ø�ý]vO×]�¦p§ndý¶× > ×]n�� ÖqÕÛÕ Ø�×�kmlXvOب = D � ^ =LF S =�Q D@? d B {GFRQU= gT6 = D l ð)� ��� ��� ´ � l�© x � g F HG_�6 = D » ^ =IF�� s/S F 6:Z = _u r ¸Dwn s s � r ~ ¥u�� ?RH�� ��� ����® ¨ = D � é ^ =YF�d@F HG_�?R9 S Fqd 7 F ^>6 = D F îT78H � S F 6:Z = B278H� g�78HG_ ="{c= HG_ = HRD2?5w » ® � <G= HPD <>="dO=I=�ª 7CB1D�B F B1D@? Q�<cF B�DO7 QE{>d ? QU= BOB � � � ��� ���Ps�¡ �BO?YD <GF D �º ® � � l � �Z� � é ®ë ®¬«>? d2F 6:6­�(� � ð)� ��� ��� © � �=� � s � � � � � »x� � © � ­°®

«>Z d D <>=�d 9M? d@= g�D <>=�{>d ? QU= BOB ��<GF B�B F 9 { 6 =£{GF D < B�?qw F 6:9M?TB1DªB�Z dO= 6:� «Kn�¥�¸Jw��dwx r u � r ¹"� n�¸ 8¦ u�n�n�ç �� <>= B�DO? Q�<GF B1D@7 Q�{>d ? QU= B@B � 7CB Q ?RHGB�D d Z Q D = _a^X�|B {�="Q 7Nwz�X78H � 7:D@B�� ¸·5u$� s �X¸ ¹�~?�S ="d2<GF 6Nw§®¯? {�= HfD@7:9 = 7:HTD ="d S F 68Brg F BE7CBE_ = B QUd 7:^ = _f78H � w ® ë ® ë ­¯®T«>? d'= S ="d �)��� � 78H� ��� ����g>` = _ =Uß H =

� �"! # �Z� »x� � © � ­� <>= H�D <>=PdO= B�Z>6:D@B]?qw � w ® ë ® ë ­ = HGBOZ dO= D <GF DYD <>="dO= 78B F B�DO? Q�<GF B�DO7 QM{>d ? QU= B@B �B F DO7CB�wz�T78H � º F HG_ ë ®

Page 139: Stochastic Processes and Stochastic Integration - Marcus Pivato

º Ã�¯ % °T±¡² &)(4³ *�´-, &/. % °T±L, &,µ % µ 0 &)(T¶¬³ ± &,µ�. 0� D dO= 9 F 7:HcB2D@?PB < ?�` D <GF D �¤<cF BLB F 9 { 6 =�{GF D < B'?qw F 6:9M?RB�DbB�Z d@= 68��^c?RZ>HG_ = _S Fqd 7 F DO78?5H�®�½£Z�DKwz? dKßdªX= _bïh7:H k g5D <>= wzZ>H Q D@7:?RH � E � � ��� ���Ps�¡ � 78B�?5^�D F 78H = _M^X�BO7:9 { 68�¶78HTD =��Rd@F DO78H � D <>= 9 ="F B�Z dO= » E ÎXD <GF D278B �

� E �=� � ��� » E � ld© � ­»FE 7CB FG� s/S F 6:Z = _,½£? dO= 6�9 =�F BOZ dO= g�B�? F HX��wzZGH Q D@7:?RHo_ =�ß H = _P78HoD < 7CB/9 F H�®H ="d 9YZGB1D <GF S = ^�?5Z>HG_ = _JS F5d 7 F D@7:?RH9g�^T��^ F B�7 Q�d@= BOZ>6:D@BL?qw d@="F 6 F H F 68��B�7CB � B =�= � f�­� <>= ? d@= 9ÈÃG® ë ª � ®·â F Ho` =bdO= S =�d B = D < 7CB {>d ? Q�= B@B õ¹¸ 7:S = H F B�DO? Q�<GF B�DO7 QL{>d ? Q�= B@B*g Q"F H�` = _ =Uß H =7:D@BG�@_ =�d 7:S F DO78S = � F B F BOZ>7:D F ^G6 =Id�F HG_�?R9µ9 ="F B�Z d@= õ

798;:98»ºæ # �XnGu�� s½¼�yGØ¿¾�Ø*vmp [ ×]�¦p [ Ø Ö à × s � Ö ��y>×okB��p��Àu�v Ö �@Ø�kBk Ö à¡Á Ö lXn��5Ø���Âc×]vmp�×o��p Ö n¨ = D � � k ¨Ã� � � ��� s�¡ � ^ =�F B�DO? Q�<GF B�DO7 Q�{>d ? Q�= B@B*g F HG_aB�Z {G{ ?TB = D <cF DbD <G=B F 9 { 6 =L{cF D < B/?qw �h<GF S =bF 6:9M?RB�D2B�Z dO= 6:� «�nc¥�¸Jw��dw£x r u � r ¹�� n�¸ ®«>? dE=�F5Q�<fß�ª�= _Äï�78H k g�_ =Uß H = D <>= 9 =�F BOZ dO= » E ?RH <GF 6Nw§®¯? {�= HJ78HTD ="d S F 6CB'?qw� ��� ��� F B£wz?56868?�`2B �$ �À� � ðt� �r� ��� © »FE)� � © � ­ ��� � �=� � s � � � �F Hc_Fg5wz? dK=�F5Q�< ïeg =�ª D = HG_MDO?]D <>= ½£? d@= 6cBO7 � 9 F ® F 6 �5= ^ d@F 78H�D <>= B1D F HG_ Fqd _M` F �R®� <>= H¶D <>=IdO= B�Z>6:DO78H � wzZ>H Q DO78?5H » � k ¨ � � � �*� ��� � s�¡ � 78B F�� s/S F 68Z = _ u r ¸Dwn ss � r ~ ¥vu�� ®¦ u�n�n�ç �«�7 ª ï2® � <>=�{cF D <?� E � � ��� ���,s�¡ � 7CB F wzZ>H Q DO78?5Hh?5w£^�?5Z>Hc_ = _�S Fqd 7 F DO78?5Hv®� <>= 9 ="F BOZ d@= »FE _ =Uß H = _ F ^�?�S = 7CBED <>= H��1ZGB�DID <>=MQ ? dOd@= B { ?RHG_�78H � « �T«K� ~6Ä ¥�� ¬ã ¹"� �d­ ¹ ¬ � ~e� ¸ ¹ � Ä u r ­ wz? d D < 78B {GF D < ® � «>? d _ = D F 7:6CB2?RHfD < 78B Q ?RHGB1D d Z Q DO78?5H­g�B =�= � f�­B ="Q D@7:?RH_Ãd   g {>{ ® ª  � ®·

798;:98;7í � sjr u�Å� <>= B = D <G= ? d@= 9µB < ?�`�D <GF D d�F Hc_�?59µ9 =�F BOZ dO= B Fqd@=LF H =*ª�Q�= 686 = HTD�` F ��?qw)_ = ®B QUd 7:^>78H � B�DO? Q�<GF B1D@7 QI{>d ? QU= BOB = B <GF SX78H � ×�V ÕMÖ kB�kmlXvOØ�V¾�¶W Ö lXn��5Ø�� [ ×]vmp�×o��p Ö n ® � < 78B7CB F H-7:9 { ? d D F HRDe68789�7:D F DO78?5H9g�^ ="Q"F ZGB = 9 F HX� {>d ? QU= B@B = Brg�BOZ Q�<-F B�Æ u$n0Ç�¸ � r ¸s n ¹"� nc¸ g5_�? n Ö � <GF S =2F 6:9M?RB�DKBOZ dO= 6:��^�?5Z>HG_ = _�S Fqd 7 F DO78?5H­g F HG_�D < ZGBrg ��×onVn Ö � ^ =d@=�{>d@= B = HTD = _ F B�7:HTD ="�5d�F 6CB£?qw d@F HG_�?59Û9 =�F BOZ d@= B"®

Page 140: Stochastic Processes and Stochastic Integration - Marcus Pivato

*�´�ÈX´ µ 0 &)(T¶¬³ ± &,µ 0 ¶ ± ¶ ± µ 0', & ² ³b. % ( ,>, ( , . tÉb.�© 0TÊ ( ÊË�0± ³¬µ ± &,µ�. 0gº à ªÌMïBÍ Î9Ï�Ð ÷eÑ]ù ò¿ÐÓÒ*Ï Ñ ò Ñ ò(ÒrÏ�ø9Ð �©ùªô��/÷ ø�ø ÷ ø ô]ÿÕÔÛô�Ö Ï)× ÷ ×

Ø ò ù Ò�ò¿Ð Ò ô ÏäYnGu��Xî�¥ ��~��z¹ � ~�� � Ù ® º ­� <>=ª=�F BO7 = B1DvBO? d D�?qw�B1D@? Q�<cF B�DO7 Q 7:HTD ="�5d�F D@7:?RH'7CB�D <cF Dv`/7:D <bd@= B {c=�Q D�D@? F B1D@? Q�<cF BÚ®DO7 Qb{>d ? QU= BOB/` < ?RB =b{GF D < B Fqd@=�F 689M?RB�DEB�Z dO= 6:�P?qw «Kn�¥¸,Û�0Û'Ü r u � r ¹"� nc¸ 8 � H�D < 78BQ�F B = gX` =LQ�F HL�1ZGB�D F5{>{ 68��D <>=bQ 6 F B@BO7 Q"F 6ÞÝ �T«K� ~rÄ ¥�� ¬ ã ¹�� �0ß ¹bà � ~b� ¸ ¹ � Ä u r ß D@? =�F5Q�<{GF D < B ="{c="d@F D = 68��g�7:H-? d _ ="d D@?a?R^�D F 7:H F B = HGB�78^>6 = _ =Uß HG7ND@7:?RH¤?5wLB�DO? Q�<GF B�DO7 Q 7:HX®D ="�5d�F D@7:?RH�® � <>= ?RH>6:� d@= !TZ>7 dO= 9 = HTDe78B¶D <GF D¶` = 9]ZGB�DP789 { ?TB =�Q ?5Hc_�7ND@7:?RHGBe?RH?5Z d p§nd�}Øbá]v@×on�� DO? � Z Fqd�F HTD =�= D <cF D]D <>= ¨ = ^ = B � Z =tâ DO7 = 6:D=� = B]78HTD =��Rd@F 68B Q ?5HXS =�d@�5={>d ? {�=�d 6:�¶?5H F 689�?TB1D = S =�d � {GF D < ®ãä¦å¬äÚæ ç'èFéeêÚèFë½ì4í�î(è­ï�è­ð�ñ4òeèÞó�ôMõ�êQè.ö÷õ�ø�è­ð�ù�úMõXè9ñüûdý ö

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mo����¸ �z¹�� n�¸�¸ p;km� Ö v��"¹4p VZ�¦v@×o��p Ö n� <>= # ��~"¹ ncu\¢ª� ß ¹ u r ¹"� nc¸ ?qw�D <>= B1D@? Q�<cF B�DO7 QI{>d ? Q�= B@B � � k ¨Õ� ��� ��� � ¡ � 78BD <G=Iß 6ND d@F DO78?5H�?qwªBO7 � 9 FMF 6 �R= ^ d�F B�º�» # � À �d¼¼ � ���5½ gG` <>=�d@= g�wz? d2F HX��þ~ÿÕ� ��� ��� © » 3 78BD <G= B�7 � 9 F ® F 6 �5= ^ d�F ?5H k �R= H ="d@F D = _¶^X��D <>=bd@F HG_�?R9µS Fqd 7 F ^>6 = Bb� � # � À �\¾ ¿�À Ág � ®798¥¤­8=<�:í � sjr u�ź ® � < ZcB*g$D <>= B�7 � 9 F ® F 6 �5= ^ d�F » 3 Q ?5HTD F 7:HcB F 6:6X78H�wz? d 9 F DO78?5H F ^�?5Z�D)D <>= ^ =�<cF ST78?5Z d?qw�D <>=b{>d ? QU= BOB � Z { D@?YD@7:9 = þLgT^GZ�DEH>?RH = D <>="dO=�F w D ="d ®ÃBI�Z\f W [^] ��ÄÅ`Æ ÄÅ INÇ R u

Page 146: Stochastic Processes and Stochastic Integration - Marcus Pivato

*�´�ÈX´ µ 0 &)(T¶¬³ ± &,µ 0 ¶ ± ¶ ± µ 0', & ² ³b. % ( ,>, ( , . tÉb.87 0TÊ ( ÊË�0± ³¬µ ± &,µ�. 0gº w Ùë ® � <>=b{>d ? Q�= B@B � 78B F sjr u ¹"� ¸ Ä r ß�� p à ×]n�� Ö nGVN�Gp à¡gXwz? dEF HX� � � þ�78H)� � � ��� g» ¼¾½8¿ ÀÈ À � � � þ � ­ � � ��� � ©` < 7 Q�< g>7:HPDOZ d H9gG78B�D d Z = p à ×on�� Ö ncV¾�Gp à» ¼ ½8¿ ÀÈ À � � ��� ¡ þ � ­ � l  798¥¤­8=<X¤mo����¸ �z¹�� n�¸��]Ö n�×onV�¦p��*p Ý ×]�¦p§n0ᨠ= D"# � k ¨¹� ��� ��� � ¡ � � ^ = B1D@? Q�<cF B�DO7 QM{>d ? Q�= B@B�®%#j7CB Q�F 6:6 = _ ¸�nc¸ r ¸ ¹"� · � ¬Ó r ¹"� ¸ ÄP?qw � 7:wQgRwz? d£F 6:6 � � þÑ78H3� � � ��� gTD <>=Ld�F HG_�?R9©S F5d 7 F ^>6 = # �=� � 7CB Õ Ø�×�kmlXvO×5W�V:Ø`/7:D <�dO= B {�="Q D�DO?O» # ®� < 7CBM^ F BO7 Q�F 6868��9 ="F HcBMD <GF D�D <>=f{Gd ? QU= BOBn# Q�F H>H>?5D£� {>d@= _�7 Q DB�|D <>= wzZ�D@Z d@=78H QUd@= 9 = HTD�B�?qw � ®� <>= Hf` = B�DO? Q�<GF B1D@7 Q"F 686:�P7:HTD =��5d�F D = #�`/7ND <JdO= B {�="Q D2DO? � g�` =YFqd@= ^ F BO7 Q"F 686:�ZGBO7:H � # F B F�Q ?5HTD d ?56XwzZ>H Q DO78?5HMD@? Õ�Ö �]l�V:×o�}Ø D <>= 78H QUd@= 9 = HRD�Bª?qw � ® � <>=E¸�nc¸ r ¸ ¬¹"� · ��Ó r ¹�� ¸ Ä {>d ? {�=�d D1�YB F ��B)D <cF DKD < 7CBK9M?�_�Z>6 F D@7:?RH�9]ZGB�D F 68` F �XBx^ = ��Z>HX^>7 F B = _��GÎ` =YQ�F H�Í D'ZGB = #�D@? �OB ,5= `e��D <>=�= SR?568Z�DO78?5Hf?5wªD <>= 78HRD =��Rd@F 6Ï® � <G= H =*ª DLD <G= ? d@= 99 F�,R= B£D < 78B {>d@="Q 78B = ®798¥¤­8=<0ºæ # �XnGu�� s¨ = D � ^ =IF B�DO? Q�<GF B1D@7 QE{>d ? Q�= B@BK`/7:D <o{GF D < B£?qw F 6:9M?TB1D/B�Z dO= 6:� � � � g F HG_¶6 = D#f^ = � ¸ ¹ � Ä u r « ß�� `/7ND <od@= B {�="Q D�D@? � g â Z {>{ ?RB = D <GF D �º ® � 78B F sjr u ¹"� ¸ Ä r ß�� ®ë ®`#f7CB ¸n�¸ r ¸ ¹"� · �zÓ r ¹�� ¸ Äf?qw � ®/G® � <>= 7:HTD =��5d�F 6?¡�# � � Ä u�n0Ç ~�~ ß�n0ÇGßi¢ 8� <>= H¶D <>=b{>d ? Q�= B@B`¡�# � � 78B F 6CB�? F s�r u ¹"� ¸ Ä r ßÏ� ®¦ u�n�n�ç �¨ = D2è©_ = H>?qD = D <>= B�DO? Q�<GF B1D@7 QL{>d ? QU= B@B �

Page 147: Stochastic Processes and Stochastic Integration - Marcus Pivato

º w f % °T±¡² &)(4³ *�´-, &/. % °T±L, &,µ % µ 0 &)(T¶¬³ ± &,µ�. 0è �Z� ' # � �  

«>? d�F HX�j¢ ÿ y {Mg£6 = D¶è   � k ¨£� �r� ��� � ¡ � ^ = D <>= B�DO? Q�<GF B�DO7 QP{>d ? Q�= B@B_ =�ß H = _ F BKwz?56868?�`2B�®Ó«>? d/F HX� ß�ª�= _�þ ÿ)� ��� ����g�6 = DAÉþ   ^ = D <>= V:×ov�áRØ�km� Õ l�VZ�¦p Ý V:Ø Ö à�±UÊ V:Ø�kBk�� yG×on þ]® � <cF D/78Brg Éþ   �Z� Ë ë   Ä�þ`Ì� <>= Ho_ =�ß H = �è   � þ � ��� Í3 Ê���� � # P ² � ºë   Q Ä �ÏÎ ² � ºë   ¡ ²ë  nÐ

H # P Éþ  ë   Q Ä �ÑÎ Éþ  ë   ¡ þ Ð   � º   º �  � Ho?qD <G=�d `£? d _>B*gXwz? d2F HX��;'ÿ k © F HG_tþ ÿ)� ��� ��� ©

è   � ; © þ � �Z� Í3 Ê��(� � # P ; © ² � ºë   Q Ä Î��ÓÒ ; © ²ë  �Ô � � P ; © ² � ºë   Q ÐH # P ; © Éþ  ë   Q Ä Î � � ; © þ � � � P ; © Éþ  ë   Q Ð   � º   ë �� < ZGBrg>è   7CB F BO? d D£?5w¿�@_�78B QUd@= D = DO789 = � Fq{>{>d ? ª 7:9 F DO78?5H¶?5w�èM®ä�ß r � s < �«>? d2F 689�?TB1D F 6:6<;eg F HG_ F 6869þ(g6:789  ; � è   � ; © þ � � è � ; © þ �  

¦ u�n�n�ç ��«�7 ª ;2® � <>= H�D <>=P=�ª�{>d@= B@B�78?5H � º   ë � wz? d è   � ; © þ � 78Be�1ZGB1D F H Fq{ ®{>d ? ª 7:9 F D@7:?RHewz? d D <>= Ý �R«�� ~6Ä ¥� ¬ ã ¹"� �0ß ¹�à � ~Õ$ ¸ ¹ � Ä u r ß' 3é # E ��� � � � E � � ­. B/¢ ¡ xig�D < 78B2`/7:686 Q ?RHXS =�d@�5= DO?�D <G=�Q ? d@d@="Q D�S F 68Z = ®· � º ­

Page 148: Stochastic Processes and Stochastic Integration - Marcus Pivato

*�´�ÈX´ µ 0 &)(T¶¬³ ± &,µ 0 ¶ ± ¶ ± µ 0', & ² ³b. % ( ,>, ( , . tÉb.87 0TÊ ( ÊË�0± ³¬µ ± &,µ�. 0gº w �ä�ß r � s : �«>? d2F HX�Gþ ÿ)� �*� ����g

Ý � � 687:9  ; � è   � þ � � è � þ � ©` <>="dO= D <G= 6:789M7ND/D F�,5= B { 6 F5QU= 78HoD <>= Ý � H>? d 9µ?5H d�F HG_�?R9µS Fqd 7 F ^>6 = B"®¦ u�n�n�ç � � < 78BKwz?56868?�`2Bª7:9M9 = _>7 F D = 68��w d ?R9 â 6 F 789 º F HG_�D <>= ¨ = ^ = B � Z = �E?R9�®78H F D = _ â ?5HXS ="dO�R= H Q�= � <>= ? d@= 9�®· � ë ­ä�ß r � s ¤ �� < 78H , ?5w)è   F B/_ =Uß H>78H ��FGÛ ��~ ·qu�� ¹ � ¹"� s � Ó u�n�·R� ~"~ �Öè   � k ¨¹y { � ¡ �` <>="dO= Öè   � ² � �Z� è  G× �±+ÊNØ ®� <>= H9g�wz? dIF HX� ß�ª�= _�¢�ÿ y {�g�D <G=Y{>d ? QU= BOB Öè   7CB F�Û ��~ ·qu�� ¹ � ¹"� s � s�r u ¬¹"� ¸ Ä r ßÏ� ®¦ u�n�n�ç �¨ = D º » # � À �d¼¼ � ���5½ ^ = D <>= # �Ï~�¹ nGu\¢ª�,ß ¹ u r ¹"� nc¸ ?5w � ®«>? d£F HX�O²¹ÿ�y {Mg�6 = D_» ® �\Ù ��� » �Ú Ê ® � < ZGBrg � » ® �\Ù � �q���� � 78B£D <>= _�78B QUd@= D = ®°DO789 =# ��~"¹ ncuJ¢ª�,ß ¹ u r ¹�� n�¸ ?qw�D <>= _�7CB QUd@= D = ®ÏD@7:9 = 9 Fqd DO78H �RF 6 =

2 � é © � ´Ú Ê © � ÚÚ Ê © �ÜÛÚ Ê ©  * r _5â ?VgXwz? d2F HX�eH F DOZ d@F 6�HTZG9]^ =�d Be� é ��Ý�g

» ¼ ½8¿ ÀÈ ¾ Þ ¿  Π# Ò Ýë  *Ô Ä �ÑÎ Ýë   ¡ Ý H ºë   Ð�Ð� » ¼¾½8¿ ÀÈ ¾ Þ ¿  Π» ¼¾½8¿ ÀÈ ¾ �  Π# Ò Ýë  �Ô Ä � Î Ýë   ¡ Ý H ºë   Ð�Ð�Ð� » ¼¾½8¿ ÀÈ ¾ Þ ¿  Π# Ò Ýë   Ô Ä"» ¼ ½C¿ ÀÈ ¾ �  Χ�ÏÎ Ýë   ¡ Ý H ºë   Ð�Ð�Ð � B�78H QU= #J78B ¸�n�¸ r ¸ ¹�� · �zÓ r ¹"� ¸ Ä �� » ¼¾½8¿ ÀÈ ¾ Þ ¿  Á # Ò Ýë  %Ô Ä l Æ � BO78H QU=�� 7CB F s�r u ¹"� ¸ Ä r ßÏ� �� l� < ZGBrgXwz? d2F HX��� é � � � g

Page 149: Stochastic Processes and Stochastic Integration - Marcus Pivato

º w ¯ % °T±¡² &)(4³ *�´-, &/. % °T±L, &,µ % µ 0 &)(T¶¬³ ± &,µ�. 0» ¼ ½8¿ ÀÈ ¾ Þ ¿  Á Öè   � � � � Æ � » ¼¾½8¿ ÀÈ ¾ Þ ¿ Â Ç Öè   � � é � H ß ´�à&� ß ¿

# Ò Ýë  %Ô Ä �ÏÎ Ýë   ¡ Ý H ºë   Ð È� Öè   � � é � H ß ´�à&� ß ¿

» ¼¾½8¿ ÀÈ ¾ Þ ¿  Π# Ò Ýë  �Ô Ä �ÏÎ Ýë   ¡ Ý H ºë   Ð�Ð� Öè   � � é � H ß ´�à&� ß ¿

l  · � /q­ä�ß r � s º �¨ = D2¢�gV� é g F HG_Õ� � ^ = 7:H£y {�g F HG_oBOZ {>{ ?RB = þ é � ß ¿± Ê ® � <>= H �» ¼ ½8¿ ÀÈ Á ¿ Î è Î � éë   ¡ � �ë  zÐ�Ð � l¦ u�n�n�ç �� = `/7:686�_�?MD < 78B/SX7 F D <>= Ý � H>? d 9�Î�` = `/786:6vB < ?�`�D <GF D ����� » ¼ ½C¿ ÀÈ Á ¿ Î è Î � éë   ¡ � �ë   Ð�Ð ���� � � l«�7 ª*áe} l ® � � â 6 F 7:9 ë g ß Hc_*â } ¢ý6 F5dO�R='= H>?5Z �R< BO?�D <GF Dè P � éë   Q ã ä èæå P � éë   Q ± ² � è P � �ë   Q ã ä èæå P � �ë   Q  ` <>="dO= _�7CB1D F H Q�= 78B�9 ="F B�Z dO= _¶78HoD <>= Ý � H>? d 9�® � < ZcB �è Î � éë   ¡ � �ë   Ð ãÚwç ä è å Î � éë   ¡ � �ë  KÐ� < ZGBrg » ¼¾½8¿ ÀÈ Á ¿ Î è Î � éë   ¡ � �ë  zÐèÐ ãÚwç ä » ¼¾½8¿ ÀÈ Á ¿ � èL­ å Î � éë   ¡ � �ë  zÐ� Z�D/^X� â 6 F 789é/dg ÖèRå+78B F 9 F5d D@7:H �TF 6 = g�BO?» ¼¾½8¿ ÀÈ Á ¿ Î è å Î � éë   ¡ � �ë  zÐ�Ð � l  

Page 150: Stochastic Processes and Stochastic Integration - Marcus Pivato

*�´�ÈX´ µ 0 &)(T¶¬³ ± &,µ 0 ¶ ± ¶ ± µ 0', & ² ³b. % ( ,>, ( , . tÉb.87 0TÊ ( ÊË�0± ³¬µ ± &,µ�. 0gº w�ê� »'á �Tu�· ��~ � ® �� < ZGBrg » ¼ ½C¿ ÀÈ Á ¿ Î è Î � éë   ¡ � �ë  KÐ�Ð ãÚ�ç ä l  � < 7CB�78B/D d Z = wz? d2F HX� áT} l gGB�?M` = Í dO= _�?RH = ®· � w ­ä�ß r � s 7 �èå7CB FM·5n�¸ ¹"� ¸�¥n�¥ ~M¹"� s � s�r u ¹�� ¸ Ä r ß�� 8¦ u�n�n�ç �¨ = D¬þ é � þ � 7:HÕ� ��� ���$®-� = ` F HRD�D@?eB < ?�` �» ¼ ½C¿ ÀÈ Á ¿ � è � þ é ¡ þ � ­8­ � l  � = `/7:686�_�?MD < 78B/SX7 F D <>= Ý � H>? d 9�Î�` = `/786:6vB < ?�`�D <GF D ���� » ¼¾½8¿ ÀÈ Á ¿ � è � þ é ¡ þ � ­:­ ��� � � l  � ?�`�g�^X� < � { ?qD <>= BO7CB*g�D <G= B�DO? Q�<GF B1D@7 Q 7:HTD ="�5d�F 6 Ä u�n0Ç ~o~ ßÏn0Ç�ßi¢ ® � < ZcB*g�wz? dF HX� áT} l gXD <>="dO= 7CB F ¢ ÿ£y {�gGBO?]D <GF D6g>7Nw�ë�� �± Ê gXD <>= H����� ' 3¥¤?ì3 ¿ # � � ����� � � áF Hc_ ����� ' 3 ´3 ´ � ì # � �

����� � � á  ` <>="dO= ` =/F5dO= D F(, 78H � D <>= Ý � ®¯H>? d 9�B�?qwGD <>= B = 7:HTD =��5d�F 6CB F B d@F HG_�?R9 S F5d 7 F ^G6 = B?5HPD <G= 9 =�F BOZ d@= B {GFRQU= � k © C © ¦ � ®� < ZGBrg�7Nw)` =Lß Hc_PHXZ>9Y^ ="d Be� é F HG_)� � BO?YD <GF D� éë   } þ é ± ² � � éë   ã í þ é ©F Hc_� �ë   � þ � ± ² � � �ë   ã í þ � ©

Page 151: Stochastic Processes and Stochastic Integration - Marcus Pivato

º6Ù l % °T±¡² &)(4³ *�´-, &/. % °T±L, &,µ % µ 0 &)(T¶¬³ ± &,µ�. 0D <G= H ���� è Î þ é ¡ � éë  zÐ ���� � ± ² � ���� è Î � �ë   ¡ þ � Ð ���� � � �� ?�`�g�HG?qD = D <GF D �» ¼ ½8¿ ÀÈ Á ¿ � è � þ é ¡ þ � ­8­ � » ¼¾½8¿ ÀÈ Á ¿ Î è Î þ é ¡ � éë  zÐ H è Î � éë   ¡ � �ë  zÐ H è Î � �ë   ¡ þ � Ð�Ð

� » ¼¾½8¿ ÀÈ Á ¿ Î è Î þ é ¡ � éë  zÐ�Ð H » ¼ ½C¿ ÀÈ Á ¿ Î è Î � éë   ¡ � �ë  KÐ�ÐH » ¼¾½8¿ ÀÈ Á ¿ Î è Î � �ë   ¡ þ � Ð�Ð  

� » ¼¾½8¿ ÀÈ Á ¿ Î è Î þ é ¡ � éë  zÐ�Ð H l H » ¼ ½8¿ ÀÈ Á ¿ Î è Î � �ë   ¡ þ � Ð�Ð  ^X� â 6 F 789 w ® � < ZGBrg��� » ¼ ½C¿ ÀÈ Á ¿ � è � þ é ¡ þ � ­8­ ��� �ïî ���� » ¼¾½8¿ ÀÈ Á ¿ Î è Î þ é ¡ � éë  KÐ�Ð ���� � H ���� » ¼ ½C¿ ÀÈ Á ¿ Î è Î � �ë   ¡ þ � Ð�Ð ���� �

����� » ¼¾½8¿ ÀÈ Á ¿ Î è Î þ é ¡ � éë   Ð�Ð ���� � H ���� » ¼ ½C¿ ÀÈ Á ¿ Î è Î � �ë   ¡ þ � Ð�Ð ���� � � � � â

î ���� è Î þ é ¡ � éë  zÐ ���� � H ���� è Î � �ë   ¡ þ � Ð ���� � � � � � d 7 F H � 6 = � H = !TZ F 6î á H á � ²Uð � �O�� ë Ä á  � =�Q�F H­g�?qw Q ?5Z d B = g�9 F�,R=|áEFqd ^>7ND d@F5d 7868�eB�9 F 686°® 1 = H Q�= g>` =bQ ?RH Q 68ZG_ = ���� » ¼¾½8¿ ÀÈ Á ¿ � è � þ é ¡ þ � ­:­ ��� � � l  

· � w ­·

ÌMï�ñ Î9Ï�Ð ÷eÑ]ù ò¿ÐÓÒ*Ï Ñ ò Ñ ò(ÒrÏ�ø9Ðóò¤ò õ ÷õôj÷ þnö �©ùªô��/÷ ø�ø ÷ ø¦ u��Tu$�Xî�¥ ��~��z¹ � ~�� � Ù ® /]­4g � w ® Ù ® w ­ � w ® fG® Ù ­

Page 152: Stochastic Processes and Stochastic Integration - Marcus Pivato

*�´^÷X´ µ 0 &)(¡¶¬³ ± &,µ 0 ¶ ± ¶ ± µ 0L, & & ± �|(ùøv( �`ú ² ³b. % ( ,G, ( , º6Ù>ºäYnGu��Xî�¥ ��~��z¹ � ~�� � Ù ® º ­â ?�w Fqd gx` =P<GF S = D <>d@=�=�Fq{>{>d ? FRQ�<>= B�DO?hB�DO? Q�<GF B�DO7 Q 7:HTD ="�5d�F D@7:?RH�® � w � 7CB F

Û�� ¹ �Tu s � ¸ ��~"¹"� ·�{>d ? QU= B@BM?qwI^�?5Z>HG_ = _-S Fqd 7 F DO78?5H9g�D <>= Hg?qw Q ?RZ d B = ` =JQ"F H _ = ®ß H = B�DO? Q�<GF B1D@7 Q 7:HTD ="�5d�F 6CB]`/7:D <-d@= B {c=�Q D�DO? � �1ZGB1D�^X�h9 ="F HGB�?qwED <>=�Q 6 F B@B�7 Q�F 6¨ = ^ = B � Z = ® â D@7 = 6ND¦� = B � HTD =��5d�F 6°® � w_ûg7CB F s�r u ¹�� ¸ Ä r ß�� ` < ?TB = 78H Q�dO= 9 = HTD@B <GF S =�)¸ ��¹ � Ü r u � r ¸�·5� g)D <>= H¤` =oQ�F H�ZGB = D <G= 9 F5Q�< 78H ="d �a?qw/D <>= � D Ö?É78HTD =��Rd@F 6xD@?78HRD =��Rd@F D = `/7:D <�dO= B {�="Q D'D@?%ûK®L«�7:H F 6:68��g�7:w�üh7CB F¶{>d ? Q�= B@BE` < ?RB = B F 9 { 6 =]{GF D < B<GF S = ±  �� «Kn�¥¸,Û���Û Ü r u � r ¹�� n�¸ g�D <>= H�` =eQ�F H Fq{G{ 6:�|D <>= ¨ = ^ = B � Z = ® â DO7 = 6:D=� = BD =�Q�< H>7C!TZ = B ="{c="d@F D = 6:�MD@? ="F5Q�<o{GF D < g F B/_�78B Q ZcBOB = _P7:H � Ù ® /Y­K®� � Q ?R9]^>78H>78H � D <>= B = D <>d@=�=bF5{>{>d ? F5Q�<G= BrgR` =LQ"F H�_ =Uß H = B1D@? Q�<cF B�DO7 Q 78HTD =��Rd@F ®DO78?5Howz? d2F ^ d ? F _ Q 6 F BOB£?5w Ý ��Ü5¢ Ó u$nv·5� ~"~ � ~�® � ?�_�?�D < 78Brg>` = `/78686 = 9 { 68?���D <G=· r ¸n�¸ � · r ߣu�� Ó u�� ~ �T¸ ¹ r ¹"� nc¸ _ = S = 68? {�= _P7:H � w ® f>® Ù ­K®798»º.8=<mo����¸ �z¹�� n�¸ ¼G× Õ Ø �¨ = DÕý � k ¨�� �*� ��� � ¡ � ^ =oF ¨|þ= SX� {>d ? QU= BOBrg F HG_�BOZ {>{ ?RB = D <cF D"ý <GF B· r ¸n�¸ � · r ߣu�� Ó u�� ~ �T¸ ¹ r ¹"� nc¸ � ý � � H û H üº ® � �=� � ��� � Ä�ÿ± wz? d B�?R9 =Lß�ª�= _ � H>?5H d@F HG_�?59 � S =�Q D@? d ÿ± ÿ � Îë ®&û ��� � �Z��� � ��� � Ä�ÿ� � H � ± ��� � Ä�ÿ� ± H  r *  H � � �=� � Ä�ÿ� � g F Hc_b` <>=�d@= � � ©  * r  � � Fqd@=� 7:HG_ =�{�= HG_ = HTDrg�?5H = ®¯_�789 = HGBO78?5H F 6­Æ u�n0Ç�¸ � r ¸ s n ¹�� n�¸ ~�gc` <>=�d@= ÿ� � ©  * r gÿ� �Fqd@='ßdªX= _ � H>?5H d@F HG_�?59 � S =�Q D@? d B�78H � Î/G®Eü Ff·5n s Ó n�¥¸,Û ¦ n ��~�~ n�¸ Ó u�n�·5� ~�~�?5H � `/7:D < � ¸ ¹ �X¸ ~��z¹ ¢ s � r ~ ¥vu�� �Óg7:HG_ =�{�= HG_ = HTD2?qw � � ©  * r  © � � Îâ�F �LD <cF D�ý�7CBv¹ r>s � 7:w>D <>=| ?57CB@B�?RH {>d ? Q�= B@B?üM78B�?5w F 689M?RB�D�BOZ dO= 6:� «Kn�¥�¸.Û��0Û

Ü r u � r ¹"� n�¸ ®� !TZ>78SX7:6 F HTD@6:��gEýÛ78B¶¹ r>s � 7:wLD <>= 9 =�F BOZ dO= � <GF B �)¸ ��¹ �G��u ~"¹ s n s �X¸ ¹� w ® Ù ® w ­ª®798»º.8;:í � sjr u�Å� HTDOZ>7:DO78S = 6:��gTD < 7CB�B F ��BKD <>=L{>d ? Q�= B@BK7CBOH�Í DG��DO?X?���_>78B Q ?RHRD@7:HXZ>?RZGB"® � <G=N�£d ?�`/HX®7 F H Q ?R9 { ?5H = HTD <GF B ·5n�¸ ¹"� ¸�¥n�¥ ~YB F 9 { 6 =b{cF D < B*g F HG_oD <>= _ = D =�d 9M7:HG78B�DO7 Q�Q ?R9�®{ ?5H = HRDY78B F5Q DOZ F 6:68� ß � ¸�� r u ® � <>=�| ?57CBOBO?5H {>d ? QU= B@Bb78HTD d ?X_>Z QU= B Ff{ ?qD = HTDO7 F 686:�

Page 153: Stochastic Processes and Stochastic Integration - Marcus Pivato

º6Ù ë % °T±¡² &)(4³ *�´-, &/. % °T±L, &,µ % µ 0 &)(T¶¬³ ± &,µ�. 078H ß H>7:D =IQ ?R6:6 ="Q DO78?5He?5wv_�78B Q ?RHTDO78HTZG7ND@7 = B*gT^>Z�D�` ='dO= !TZ>7 d@= D <GF D���9M?RB�D"�b?5w�D <>= B = ^ =d@= 6 F DO78S = 6:���@B�9 F 6:6;��7:H�B�7¥� = ®798»º.8¥¤

mo����¸ �z¹�� n�¸�� nd�}Øbá]v@×o��p Ö n 9Óp§�zy)vOØ�k Ý Ø��*�M� Ö ¼c× Õ Ø �¨ = DRý � k ¨¬� �m� ��� � ¡ � ^ =£F ¹ r>s � Ý p��Ü5¢ Ó u�n�·5� ~�~Tg F HG_b6 = Dæý � � H û H ü g` <>="dO=�� © ûüg F HG_%ü Fqd@=bF B�^ = wz? d@= ®¨ = D%# � k ¨?� � � ����� ¡ � � ^ = BO?59 =fdO=�F 6»®°S F 6:Z = _jB�DO? Q�<GF B1D@7 Q�{>d ? QU= BOBrg F HG_BOZ {>{ ?RB = �º ®`#f7CB ¸n�¸ r ¸ ¹"� · �zÓ r ¹�� ¸ Äf?qwæûx®ë ®`#�ÿ)Ý ± × k ¨t� �*� ���$Î_� ® � Ù Ø gX` <>="dO= � ® � Ù 78BKD <G='monVßqp� r ¸ ~ s � r ~ ¥vu��]F BOBO? Q 7 F D = _DO?�ûx®/G®`#f7CB2� ¸ ¹ � Ä u r « ß�� `/7:D <�d@= B {c=�Q D�DO?�ü)®� =Uß H = D <G= ~"¹ n�· #�r ~"¹"� · � ¸ ¹ � Ä u r ß ?5w¥#�`/7ND <Yd@= B {�="Q DDO?Ný|D@?E^ = D <>= B1D@? Q�<GF BÚ®DO7 QI{>d ? QU= B@B � ' # � ý � k ¨t� �*� ��� � ¡ �_ =�ß H = _9g�wz? d2F 6:6Fþ ÿt� �r� ���og�^T� �' 3é # � ý �Z� � ' 3é # � � Ä�ÿ± H ' 3é # � û H ' 3é # � ü ©` <>="dO= D <>= ê4vBkm� 7:HTD ="�5d�F 6I?5H D <G=ad 7 �5< D <GF HG_�BO78_ = 7CB��1ZGB�DfD <>= B�D F HG_ Fqd _Ý �T«K� ~6Ä ¥�� � ¸ ¹ � Ä u r ß ?5H�� ��� ����g£D <>= k�Ø�� Ö n�� 7:HTD =��5d�F 6/?RH-D <>=fd 7 �R< D <GF HG_ BO78_ =7CB F Hq$�¹�n � ¸ ¹ � Ä u r ß g F HG_jD <>= �zy0p§vO� 78HTD ="�5d�F 6�?5H¤D <>=fd 7 �5< D <cF HG_-B�7C_ = 7CB FÓ r ¹ # Ç ��~ � ~"¹ n�· #�r ~"¹"� · � ¸ ¹ � Ä u r ß 8

Page 154: Stochastic Processes and Stochastic Integration - Marcus Pivato

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