The name is absent



Taking the total derivative we get

lɪ+y,   [σ^ φ ( χσ—c ) - 2σ2φ, ( χσc ) [Bkr+2xcr+r2+дст2] h(χ°^c}dc

{£ [r [ɪ - *    ■ i)] + ⅛φ    ■  c) [Bkr + . Xcr + 2 + ∆σ2∣]    "«.} d = 0

and hence

√=       ʃ ∞ [r [ɪ - Φ (—)] + ɪφ (—) [Bkr + 2Xcr + r2 +∆σ2] dh(xc) dXc

dc     _    J-∞ LL        ' σ,- ∕J    2σv ∖ σv ) I 1 c ' 1 "lJ χc c

χc  -  ɪ + - ʃɪ [rφ (χσ∏ - ⅛φ' (χσ-) [Bkr+2χr+r2+^2] ⅛(χc)dχc

=  ʃɪ [r [ɪ - * (χσ-c)] + 2⅛φ (χσ-c) [Bkr+2χcr+r2+1] 1 c

ɪ + - ʃɪ Φ ( xσ-c ) [r + x2σ-c [Bkr + 2χcr + r2 + ^21] ⅛(χc )dχc

where the second line uses the fact that φ'(x) =


ɪ— e
√2πσχc


[χc-μχc ]2
2
σ2
χc


we get


[χc-μχc ]2

dh(Χc)      ɪ - ~~2σζ

χ      χc

χc     √2π


(x). Assuming a normal density h(Xc) =


[χc-μχc ]2      ɪ

σ4       σ2

χc        χc


and hence


- [Xc-μXc ]2

d-c    σXc ʃɪ [r - * (χσ-c)] + 2⅛φ (χσ-c) [Bkr + 2χcr+ r2 + ^2]][ɪ-  χ -X x     ^∏σχ-e   2σχc  c

χc                                                                                                   - [Xc-μXc]2

ɪ + ɪ ʃ φ ( '   ) [r + ¾-c [Bkr + cr + r2 + ∆σ21] ɔ-i---e   27XC  c

1 σv .1 ∞        σ∙. ; L        2σJ; l ' c 1       1      2jJ √2πσXc                          c

33



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