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Johnson, Timothy C.; Zervos, Mihail

The solution to a second order linear ordinary differential equation with a non-homogeneous term that is a measure. (English) Zbl 1121.34007

Stochastics 79, No. 3-4, 363-382 (2007).
This paper is concerned with the solution of the ordinary differential equation \[ {1\over 2}\sigma^2 w'' + bw' - rw +h =0\;\; {\text{on}} \;(\alpha, \beta), \] where \(h\) is a locally finite measure and \(\sigma\), \(b\) and \(r\) are given functions. Both analytic and probabilistic expressions for the solution, with attendant properties, are derived, consideration being given to pertinent stochastic control models.
Reviewer: Andrew Dale (Durban)

Cited in 5 Documents

MSC:

34A30 Linear ordinary differential equations and systems
60J55 Local time and additive functionals
49J15 Existence theories for optimal control problems involving ordinary differential equations
34H05 Control problems involving ordinary differential equations
62L15 Optimal stopping in statistics

Keywords:

second-order linear ordinary differential equations; additive functionals; optimal stopping; singular control; measure-valued inhomogeneity
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References:

[1] DOI: 10.1007/s001860100161 · Zbl 1018.60045 · doi:10.1007/s001860100161
[2] DOI: 10.1137/S0363012900367825 · Zbl 0997.93102 · doi:10.1137/S0363012900367825
[3] DOI: 10.1214/aoap/1069786509 · Zbl 1072.60065 · doi:10.1214/aoap/1069786509
[4] DOI: 10.1007/s00245-004-0792-z · Zbl 1060.93105 · doi:10.1007/s00245-004-0792-z
[5] DOI: 10.1137/S0040585X9797852X · Zbl 0994.60046 · doi:10.1137/S0040585X9797852X
[6] Beneš V.E., Stochastics and Stochastics Reports 4 pp 39– (1980)
[7] Borodin A.N., Handbook of Brownian Motion–Facts and Formulae (2002) · Zbl 1012.60003
[8] Breiman L., Probability (1968)
[9] DOI: 10.1007/s002459900094 · Zbl 0928.93065 · doi:10.1007/s002459900094
[10] DOI: 10.1016/S0304-4149(03)00076-0 · Zbl 1075.60524 · doi:10.1016/S0304-4149(03)00076-0
[11] DOI: 10.1214/aoap/998926992 · Zbl 1083.91055 · doi:10.1214/aoap/998926992
[12] DOI: 10.2307/1969842 · Zbl 0057.09805 · doi:10.2307/1969842
[13] DOI: 10.1239/jap/1005091029 · Zbl 1026.91048 · doi:10.1239/jap/1005091029
[14] DOI: 10.1287/moor.8.3.439 · Zbl 0523.93068 · doi:10.1287/moor.8.3.439
[15] DOI: 10.1016/0304-4149(78)90059-5 · Zbl 0372.60116 · doi:10.1016/0304-4149(78)90059-5
[16] Itô K., Diffusion Processes and Their Sample Paths (1996) · Zbl 0837.60001
[17] Jacka S.D., Stochastics 10 pp 103– (1983) · Zbl 0495.93068 · doi:10.1080/17442508308833266
[18] DOI: 10.1214/aoap/1037125867 · Zbl 1027.93043 · doi:10.1214/aoap/1037125867
[19] DOI: 10.2307/1426435 · Zbl 0511.93076 · doi:10.2307/1426435
[20] DOI: 10.1016/S0304-4149(01)00157-0 · Zbl 1064.93049 · doi:10.1016/S0304-4149(01)00157-0
[21] Karatzas I., Stochastics and Stochastics Reports 71 pp 1– (2000) · Zbl 0979.93121 · doi:10.1080/17442500008834257
[22] Karatzas I., Brownian Motion and Stochastic Calculus (1988) · Zbl 0638.60065 · doi:10.1007/978-1-4684-0302-2
[23] DOI: 10.1214/aoap/1029962601 · Zbl 0938.93067 · doi:10.1214/aoap/1029962601
[24] Karlin S., A Second Course in Stochastic Processes (1981) · Zbl 0469.60001
[25] DOI: 10.1137/0330053 · Zbl 0759.93081 · doi:10.1137/0330053
[26] Mandl P., Analytical Treatment of One-dimensional Markov Processes (1968) · Zbl 0179.47802
[27] DOI: 10.1007/s007800050013 · Zbl 0957.60085 · doi:10.1007/s007800050013
[28] Øksendal, A., Mathematical models for investment under uncertainty, (PhD thesis, University of Oslo).
[29] Revuz D., Continuous Martingales and Brownian Motion, 2. ed. (1994) · Zbl 0804.60001
[30] Rogers L.C.G., Diffusions, Markov Processes and Martingales, Volume 2 (2000) · Zbl 0949.60003 · doi:10.1017/CBO9781107590120
[31] DOI: 10.1002/mana.19851240107 · Zbl 0594.60080 · doi:10.1002/mana.19851240107
[32] Shiryayev A.N., Optimal Stopping Rules (1978)
[33] Shiryaev A., Optimal Stopping and Free-Boundary Problems (2006) · Zbl 1115.60001
[34] DOI: 10.1137/0322005 · Zbl 0535.93071 · doi:10.1137/0322005
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