Errata corrige optimal portfolio and consumption with habit formation in a jump diffusion market

Applied Mathematics and Computation 232 (2014) 235–236

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Errata corrige optimal portfolio and consumption with habit formation in a jump diffusion market Xinfeng Ruan a,⇑, Wenli Zhu a, Jin Hu b, Jiexiang Huang a a b

School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, PR China School of Science, Chongqing Jiaotong University, Chongqing 400047, PR China

a r t i c l e

i n f o

Keywords: Portfolio and consumption Habit formation Stochastic control Jump diffusions market The maximum principle Equity premium puzzle

a b s t r a c t We correct (17), (19), (20), (21), (46) and (48) in [1]. Published by Elsevier Inc.

According to errata corrige of [2] in [3], we find (17), (19), (20), (21), (46) and (48) in [1] are incorrect. Now we present corrected formulas in (1)–(7) as follows, respectively. 

   1 f_ ðtÞX  ðtÞc1 þ ðc  1Þf ðtÞX  ðtÞc2 rt XðtÞ þ ðlt  rt Þu ðtÞ  Ay ðtÞ þ ðc  1Þðc  2Þf ðtÞX  ðtÞc3 r2t u ðtÞ2 2  Z c1 c1 c2      f ðtÞ½ðX ðtÞ þ hðt; zÞu ðtÞÞ  X ðtÞ  ðc  1ÞX ðtÞ hðt; zÞu ðtÞmðdzÞ dt þ R0 Z h i c2 c1 e þðc  1Þf ðtÞðX  ðtÞÞ rt u ðtÞdBðtÞ þ dzÞ; f ðtÞ ðX  ðt Þ þ hðt; zÞu ðt  ÞÞ  X  ðt Þc1 Nðdt;

dp ðtÞ ¼

ð1Þ

R0

h i c1  k ðt; zÞ ¼ f ðtÞX  ðtÞc1 ð1 þ hðt; zÞu ðtÞX  ðtÞ1 Þ  1 ; (

1 c f_ ðtÞ þ Wt f ðtÞ þ ð1  cÞðAc edt Þc1 f ðtÞc1 ¼ 0;

f ðTÞ ¼ 0; where

1 wt ¼ crt þ ðc  1Þðlt  r t Þu ðtÞX  ðtÞ1 þ ðc  1Þðc  2ÞX  ðtÞ2 r2t u ðtÞ2 2 Z h i 1 c1   þ ð1 þ hðt; zÞu ðtÞX ðtÞ Þ  1 þ ðc  1Þhðt; zÞu ðtÞX  ðtÞ1 mðdzÞ; R0

⇑ Corresponding author. E-mail address: [email protected] (X. Ruan). 0096-3003/$ - see front matter Published by Elsevier Inc. http://dx.doi.org/10.1016/j.amc.2014.01.060

ð2Þ

ð3Þ

236

X. Ruan et al. / Applied Mathematics and Computation 232 (2014) 235–236

Fðm ðtÞÞ ¼ lt  r t þ ðc  1Þr2t m ðtÞ þ Fðm Þ ¼ l  r þ ðc  1Þr2 m þ k

Z

Z

hðt; zÞ½ð1 þ hðt; zÞm ðtÞÞc1  1mðdzÞ ¼ 0;

ð4Þ

R0

z½ð1 þ zm Þc1  1 f ðzÞdz ¼ 0;

ð5Þ

R0

1 2

W ¼ cr þ ðc  1Þðl  rÞm þ ðc  1Þðc  2Þr2t m2 þ k

Z

½ð1 þ zm Þc1  1  ðc  1Þzm  f ðzÞdz;

ð6Þ

R0

References [1] X. Ruan, W. Zhu, J. Hu, J. Huang, Optimal portfolio and consumption with habit formation in a jump diffusion market, Appl. Math. Comput. 222 (2013) 391–401. [2] N.C. Framstad, B. Øksendal, A. Sulem, Sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance, J. Optim. Theory Appl. 121 (1) (2004) 77–98. [3] N.C. Framstad, B. Øksendal, A. Sulem, Errata corrige sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance, J. Optim. Theory Appl. 124 (2) (2005) 511–512.