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[]\TU/lmr/m/it/10.95 A strong solution to the stochastic differential equation ([][]1.2.8[][]) on probability space $\OT1/cmr/m/n/10.95 (
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\TU/lmr/m/n/10.95 points $\OML/cmm/m/it/10.95 t[]; t[]; [] ; t[] \OMS/cmsy/m/n/10.95 2 []$ \TU/lmr/m/n/10.95 it is the case that the vectors $[][]$ and $[][]$
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\OML/cmm/m/it/10.95 C[] \U/msb/m/n/10.95 x \OT1/cmr/m/n/10.95 0$\TU/lmr/m/n/10.95 , $\U/msb/m/n/10.95 F x F[]$\TU/lmr/m/n/10.95 , $\OML/cmm/m/it/10.95 ^^X \U/msb/m/n/10.95 x \OML/cmm/m/it/10.95 x; ^^V \U/msb/m/n/10.95 x \OML/cmm/m/it/10.95 a[]; ^^[ \U/msb/m/n/10.95 x \OML/cmm/m/it/10.95 b[]; W \U/msb/m/n/10.95 x \OML/cmm/m/it/10.95 W[]; X \U/msb/m/n/10.95 x []$
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[]\TU/lmr/m/n/10.95 However note also that since $\OML/cmm/m/it/10.95 G[]$ \TU/lmr/m/n/10.95 is upper semi-continuous, further the fact that, $\OML/cmm/m/it/10.95 ^^^[] \OMS/cmsy/m/n/10.95 2 []$\TU/lmr/m/n/10.95 ,
 []


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\TU/lmr/m/n/10.95 and then $\OT1/cmr/m/n/10.95 ([][]3\OML/cmm/m/it/10.95 :\OT1/cmr/m/n/10.95 2\OML/cmm/m/it/10.95 :\OT1/cmr/m/n/10.95 27[][])$\TU/lmr/m/n/10.95 , and $\OT1/cmr/m/n/10.95 ([][]3\OML/cmm/m/it/10.95 :\OT1/cmr/m/n/10.95 2\OML/cmm/m/it/10.95 :\OT1/cmr/m/n/10.95 16[][])$\TU/lmr/m/n/10.95 , imply for all $\OML/cmm/m/it/10.95 " \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 (0\OML/cmm/m/it/10.95 ; \OMS/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$ \TU/lmr/m/n/10.95 we have that: $[][] [] \OT1/cmr/m/n/10.95 =
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\TU/lmr/m/it/10.95 ery $\OML/cmm/m/it/10.95 r \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 (0\OML/cmm/m/it/10.95 ; \OMS/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$ \TU/lmr/m/it/10.95 satisfy the condition that $\OML/cmm/m/it/10.95 O[] \OMS/cmsy/m/n/10.95 ^^R O$\TU/lmr/m/it/10.95 , where $\OML/cmm/m/it/10.95 O[] \OT1/cmr/m/n/10.95 = \OMS/cmsy/m/n/10.95 f\OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 2 O \OT1/cmr/m/n/10.95 : []\OMS/cmsy/m/n/10.95 g$
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[]\TU/lmr/bx/n/10.95 Corollary 3.3.1.1. []\TU/lmr/m/it/10.95 Let $\OML/cmm/m/it/10.95 T \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 (0\OML/cmm/m/it/10.95 ; \OMS/cmsy/m/n/10.95 1\OT1/cmr/m/n/10.95 )$\TU/lmr/m/it/10.95 , let $[]$ be a probability space, let $\OML/cmm/m/it/10.95 u[] \OMS/cmsy/m/n/10.95 2 \OML/cmm/m/it/10.95 C[] []$\TU/lmr/m/it/10.95 ,
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\TU/lmr/m/it/10.95 following functions (referred to as auxiliary functions) satisfying for all $\OML/cmm/m/it/10.95 ^^W \OMS/cmsy/m/n/10.95 2 []$\TU/lmr/m/it/10.95 : 
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[] \OMS/cmsy/m/n/10.95 2 []$\TU/lmr/m/it/10.95 , and $\OML/cmm/m/it/10.95 ^^W[] \OT1/cmr/m/n/10.95 = [] \OMS/cmsy/m/n/10.95 2
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[]\TU/lmr/m/n/10.95 Since it is the case that for all $\OML/cmm/m/it/10.95 j \OMS/cmsy/m/n/10.95 2 f\OT1/cmr/m/n/10.95 1\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; :::; n\OMS/cmsy/m/n/10.95 g$ \TU/lmr/m/n/10.95 that: $[] [] \OT1/cmr/m/n/10.95 = []$\TU/lmr/m/n/10.95 , where $\OML/cmm/m/it/10.95 l[]; l[]; :::; l[]; L \OMS/cmsy/m/n/10.95 2
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\TU/lmr/m/n/10.95 See Remark 5.3.2. Let $\OML/cmm/m/it/10.95 x[]; x[]; :::; x[] \OMS/cmsy/m/n/10.95 2 []$ \TU/lmr/m/n/10.95 satisfy for all $\OML/cmm/m/it/10.95 i \OMS/cmsy/m/n/10.95 2 f\OT1/cmr/m/n/10.95 1\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; :::; n\OMS/cmsy/m/n/10.95 g$
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\TU/lmr/m/n/10.95 This and the assumption that $\OT1/cmr/m/n/10.95 ^^H \OMS/cmsy/m/n/10.95 2 \OML/cmm/m/it/10.95 C []$ \TU/lmr/m/n/10.95 along with the assumption that $[][] \OMS/cmsy/m/n/10.95 j [] [] ^^@
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\TU/lmr/m/n/10.95 The assumption that for all $[][] \OMS/cmsy/m/n/10.95 j [] [] ^^@ [] [] j \OT1/cmr/m/n/10.95 = 0$ \TU/lmr/m/n/10.95 and the assumption that $[][] \OMS/cmsy/m/n/10.95 j\OML/cmm/m/it/10.95 x[] \OMS/cmsy/m/n/10.95 ^^@
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\TU/lmr/m/n/10.95 By symmetry note also that $[] [] \OT1/cmr/m/n/10.95 = [] []$ \TU/lmr/m/n/10.95 and also that $[] [] \OT1/cmr/m/n/10.95 =
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\TU/lmr/m/n/10.95 Note that since $\U/euf/m/n/10.95 p[] \OMS/cmsy/m/n/10.95 2 O []$ \TU/lmr/m/n/10.95 for $\OML/cmm/m/it/10.95 n \U/msa/m/n/10.95 > \OT1/cmr/m/n/10.95 2$\TU/lmr/m/n/10.95 , it is the case for all $\OML/cmm/m/it/10.95 x \OMS/cmsy/m/n/10.95 2 \U/msb/m/n/10.95 R$ \TU/lmr/m/n/10.95 then that $[] \OMS/cmsy/m/n/10.95 2
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[]\TU/lmr/m/it/10.95 that for all $\OML/cmm/m/it/10.95 ^^R \OMS/cmsy/m/n/10.95 2 \OT1/cmr/m/n/10.95 ^^B$\TU/lmr/m/it/10.95 , $\OML/cmm/m/it/10.95 n \OMS/cmsy/m/n/10.95 2 \U/msb/m/n/10.95 N[]$\TU/lmr/m/it/10.95 , $\OML/cmm/m/it/10.95 t \OMS/cmsy/m/n/10.95 2 []$\TU/lmr/m/it/10.95 , that $[][] \U/msa/m/n/10.95 6 [][] [] []$ 
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[]\TU/lmr/bx/n/10.95 Definition 8.1.1 \TU/lmr/m/n/10.95 (Activation ANN)\TU/lmr/bx/n/10.95 . []\TU/lmr/m/it/10.95 Let $\OML/cmm/m/it/10.95 n \OMS/cmsy/m/n/10.95 2 \U/msb/m/n/10.95 N$\TU/lmr/m/it/10.95 . We denote by $\U/euf/m/n/10.95 i[] \OMS/cmsy/m/n/10.95 2 [] ^^R
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[]\TU/lmr/bx/n/10.95 Lemma 8.2.5. []\TU/lmr/m/it/10.95 Let $\OML/cmm/m/it/10.95 ^^W \OMS/cmsy/m/n/10.95 2 []$ \TU/lmr/m/it/10.95 with end-widths $\OML/cmm/m/it/10.95 d$\TU/lmr/m/it/10.95 . It is then the case that $[][] [] [] \OT1/cmr/m/n/10.95 = [][] [] =
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\TU/lmr/m/n/10.95 then tells us that for all $\OML/cmm/m/it/10.95 i \OMS/cmsy/m/n/10.95 2 f\OT1/cmr/m/n/10.95 0\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 1\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; :::; N\OMS/cmsy/m/n/10.95 g$\TU/lmr/m/n/10.95 , $[] [] \OT1/cmr/m/n/10.95 = []$\TU/lmr/m/n/10.95 , $[][] [] \OMS/cmsy/m/n/10.95 2
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\OMS/cmsy/m/n/10.95 f\OT1/cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 3\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 4\OML/cmm/m/it/10.95 ; :::\OMS/cmsy/m/n/10.95 g$\TU/lmr/m/n/10.95 , $\OML/cmm/m/it/10.95 x \OT1/cmr/m/n/10.95 = \OMS/cmsy/m/n/10.95 f\OML/cmm/m/it/10.95 x[]; x[]; :::; x[]\OMS/cmsy/m/n/10.95 g 2 \U/msb/m/n/10.95 R[]$ \TU/lmr/m/n/10.95 it holds that $[] [] \OMS/cmsy/m/n/10.95 2 \OML/cmm/m/it/10.95 C []$\TU/lmr/m/n/10.95 . and $[] [] \OT1/cmr/m/n/10.95 =
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[]\TU/lmr/m/it/10.95 it holds that $[] [] \U/msa/m/n/10.95 6 [] [] \OT1/cmr/m/n/10.95 + 3 [] \OMS/cmsy/m/n/10.95 ^^A [] \OT1/cmr/m/n/10.95 + 7 + 3\OML/cmm/m/it/10.95 N \OT1/cmr/m/n/10.95 + \OML/cmm/m/it/10.95 d[] [] \OT1/cmr/m/n/10.95 + \OML/cmm/m/it/10.95 d []$ 
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[]\TU/lmr/m/n/10.95 Note that for all $\OML/cmm/m/it/10.95 i \OMS/cmsy/m/n/10.95 2 f\OT1/cmr/m/n/10.95 1\OML/cmm/m/it/10.95 ; \OT1/cmr/m/n/10.95 2\OML/cmm/m/it/10.95 ; [] ; \U/euf/m/n/10.95 n\OMS/cmsy/m/n/10.95 g$\TU/lmr/m/n/10.95 , Lemma [][]9.0.3[][] tells us that $[][] [] \OMS/cmsy/m/n/10.95 2 \OML/cmm/m/it/10.95 C[] []$\TU/lmr/m/n/10.95 .
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