ENDOGENOUS DOWNWARD JUMP DIFFUSION AND BLOW UP PHENOMENA BEFORE CRASH

• Kwon, Young-Mee (DEPARTMENTS OF MULTIMEDIA HANSUNG UNIVERSITY) ;
• Jeon, In-Tae (DEPARTMENTS OF MATHEMATICS CATHOLIC UNIVERSITY OF KOREA) ;
• Kang, Hye-Jeong (DEPARTMENTS OF MATHEMATICS SEOUL NATIONAL UNIVERSITY)
• Published : 2010.11.30

Abstract

We consider jump processes which has only downward jumps with size a fixed fraction of the current process. The jumps of the pro cesses are interpreted as crashes and we assume that the jump intensity is a nondecreasing function of the current process say $\lambda$(X) (X = X(t) process). For the case of $\lambda$(X) = $X^{\alpha}$, $\alpha$ > 0, we show that the process X shold explode in finite time, say $t_e$, conditional on no crash For the case of $\lambda$(X) = (lnX)$^{\alpha}$, we show that $\alpha$ = 1 is the borderline of two different classes of processes. We generalize the model by adding a Brownian noise and examine the blow up properties of the sample paths.

References

1. M. C. Adam and A. Szafarz, Speculative bubbles and financial markets, Oxford Econmic Papers 44 (1992), no. 4, 626-640. https://doi.org/10.1093/oxfordjournals.oep.a042068
2. J. V. Andersen and D. Sornette, Fearless versus fearful speculative financial bubbles, Phys. A 337 (2004), no. 3-4, 565-585. https://doi.org/10.1016/j.physa.2004.01.054
3. O. J. Blanchard, Speculative bubbles, crashes and rational expectations, Economic Letters 3 (1979), 387-389. https://doi.org/10.1016/0165-1765(79)90017-X
4. O. J. Blanchard and M. W. Watson, Bubbles, Rational Expectations and Speculative Markets, in: P. Wachtel, eds., Crisis in Economic and Financial Structure: Bubbles, Bursts, and Shocks. Lexington Books: Lexington, 1982.
5. C. Camerer, Bubbles and fads in asset prices, Journal of Economic Surveys 3 (1989), no. 1, 3-41. https://doi.org/10.1111/j.1467-6419.1989.tb00056.x
6. K. L. Chung and R. J. Williams, Introduction to Stochastic Integration, Second edition. Probability and its Applications. Birkhauser Boston, Inc., Boston, MA, 1990.
7. G. W. Evans, Pitfalls in testing for explosive bubbles in asset prices, American Economic Review 81 (1991), 922-930.
8. Y. Fukuta, A simple discrete-time approximation of continuous-time bubbles, J. Econom. Dynam. Control 22 (1998), no. 6, 937-954. https://doi.org/10.1016/S0165-1889(97)00086-9
9. I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Springer-Verlag, New York-Heidelberg, 1972.
10. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.
11. I. Jeon, Stochastic fragmentation and some sufficient conditions for shattering transition, J. Korean Math. Soc. 39 (2002), no. 4, 543-558. https://doi.org/10.4134/JKMS.2002.39.4.543
12. A. Johansen, O. Ledoit, and D. Sornette, Crashes as critical points, Int. J. Theo. & Appl. Finance 3 (2000), no. 2, 219{255. https://doi.org/10.1142/S0219024900000115
13. A. Johansen and D. Sornette, Critical crashes, Risk 12 (1999), no. 1, 91-95.
14. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1988.
15. T. Lux and D. Sornette, On rational bubbles and fat tails, Journal of Money, Credit, and Banking 34 (2002), no. 3, 589-610. https://doi.org/10.1353/mcb.2002.0004
16. Y. Malevergne and D. Sornette, Multi-dimensional rational bubbles and fat tails, Quantitative Finance 1 (2001), 533-541. https://doi.org/10.1080/713665876
17. D. Porter and V. Smith, Stock market bubbles in the laboratory, Applied Mathematical Finance (1994), 111-127.
18. A. V. Skorohod, Studies in The Theory of Random Processes, Inc. Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965.
19. V. Smith, G. Suchanek, and A. Williams, Bubbles crashes, and endogenous expectations in experimental stock asset markets, Econometrica 56 (1988), 1119-1151. https://doi.org/10.2307/1911361
20. D. Sornette and J. V. Andersen, A nonlinear super-exponential rational model of speculative financial bubbles, Int. J. Mod. Phys. C 13 (2002), no. 2, 171-188. https://doi.org/10.1142/S0129183102003085