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A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents
The multi-dimensional stochastic Stefan financial model for a portfolio of assets
1. | Department of Mathematics and Applied Mathematics, University of Crete, GR–714 09 Heraklion, Greece |
2. | Institute of Applied and Computational Mathematics, FORTH, GR–711 10 Heraklion, Greece |
3. | Computer Science Department, University of Crete, Voutes University Campus, HERAKLION, Crete, GR-70013, Greece |
4. | Department of Mathematical and Physical Sciences, University of Chester, Thornton Science Park, CH2 4NU, UK |
5. | Institute of Applied and Computational Mathematics, FORTH, GR–711 10 Heraklion, Greece |
The financial model proposed involves the liquidation process of a portfolio through sell / buy orders placed at a price $ x\in\mathbb{R}^n $, with volatility. Its rigorous mathematical formulation results to an $ n $-dimensional outer parabolic Stefan problem with noise. The moving boundary encloses the areas of zero trading. We will focus on a case of financial interest when one or more markets are considered. We estimate the areas of zero trading with diameter approximating the minimum of the $ n $ spreads for orders from the limit order books. In dimensions $ n = 3 $, for zero volatility, this problem stands as a mean field model for Ostwald ripening, and has been proposed and analyzed by Niethammer in [
References:
[1] |
N. D. Alikakos, P. W. Bates and X. Chen,
Convergence of the Cahn-Hilliard Equation to the Hele-Shaw Model, Arch. Rational Mech. Anal., 128 (1994), 165-205.
doi: 10.1007/BF00375025. |
[2] |
N. D. Alikakos and G. Fusco,
Ostwald ripening for dilute systems under quasistationary dynamics, Comm. Math. Phys., 238 (2003), 429-479.
doi: 10.1007/s00220-003-0833-5. |
[3] |
N. D. Alikakos, G. Fusco and G. Karali,
The effect of the geometry of the particle distribution in Ostwald ripening, Comm. Math. Phys., 238 (2003), 481-488.
doi: 10.1007/s00220-003-0834-4. |
[4] |
N. D. Alikakos, G. Fusco and G. Karali,
Ostwald ripening in two dimensions- The rigorous derivation of the equations from Mullins-Sekerka dynamics, J. Differential Equations, 205 (2004), 1-49.
doi: 10.1016/j.jde.2004.05.008. |
[5] |
A. Altarovici, J. Muhle-Karbe and H. M. Soner,
Asymptotics for fixed transaction costs, Finance Stoch., 19 (2015), 363-414.
doi: 10.1007/s00780-015-0261-3. |
[6] |
D. C. Antonopoulou, D. Blömker and G. D. Karali,
The sharp interface limit for the stochastic Cahn-Hilliard equation, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), 280-298.
doi: 10.1214/16-AIHP804. |
[7] |
D. C. Antonopoulou, G. D. Karali and A. N. K. Yip,
On the parabolic Stefan problem for Ostwald ripening with kinetic undercooling and inhomogeneous driving force, J. Differential Equations, 252 (2012), 4679-4718.
doi: 10.1016/j.jde.2012.01.016. |
[8] |
British Pound v US Dollar Data, https://www.poundsterlinglive.com.,, |
[9] |
X. Chen,
The Hele-Shaw problem and area-preserving curve shortening motions, Arch. Rational Mech. Anal., 123 (1993), 117-151.
doi: 10.1007/BF00695274. |
[10] |
X. Chen,
Global asymptotic limit of solutions of the Cahn-Hilliard equation, Journal of Differential Geometry, 44 (1996), 262-311.
|
[11] |
X. Chen, X. Hong and F. Yi,
Existence, uniqueness and regularity of classical solutions of Mullins-Sekerka problem, Comm. Partial Differential Equations, 21 (1996), 1705-1727.
doi: 10.1080/03605309608821243. |
[12] |
X. Chen and M. Dai,
Characterization of optimal strategy for multiasset investment and consumption with transaction costs, SIAM J. Financial Math., 4 (2013), 857-883.
doi: 10.1137/120898991. |
[13] |
X. Chen and F. Reitich,
Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling, J. Math. Anal. Appl., 164 (1992), 350-362.
doi: 10.1016/0022-247X(92)90119-X. |
[14] |
R. Cont and A. de Larrard,
Price dynamics in a Markovian limit order market, SIAM J. Financial. Math., 4 (2013), 1-25.
doi: 10.1137/110856605. |
[15] |
R. Cont, S. Stoikov and R. Talreja,
A stochastic model for order book dynamics, Oper. Res., 58 (2010), 549-563.
doi: 10.1287/opre.1090.0780. |
[16] |
E. Ekström, Selected Problems in Financial Mathematics, PhD Thesis, Uppsala Universitet, Sweden, 2004. |
[17] |
L. C. Evans, H. M. Soner and P. E. Souganidis,
Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.
doi: 10.1002/cpa.3160450903. |
[18] |
T. Funaki,
Singular limit for stochastic reaction-diffusion equation and generation of random interfaces, Acta Math. Sin. (Engl. Ser.), 15 (1999), 407-438.
doi: 10.1007/BF02650735. |
[19] |
M. D. Gould, M. A. Porter, S. Williams, M. McDonald, D. J. Fenn and S. D. Howison,
Limit order books, Quant. Finance, 13 (2013), 1709-1742.
doi: 10.1080/14697688.2013.803148. |
[20] |
V. Henderson,
Prospect theory, liquidation, and the disposition effect, Management Science, 58 (2012), 445-460.
|
[21] |
T. Lybek and A. Sarr, Measuring Liquidity in Financial Markets, International Monetary Fund, work-in-progress, No. 02/232, 2002. |
[22] |
H. M. Markowitz, Portfolio selection: Efficient diversification of investments, John Wiley and Sons, Inc., New York, 1959. |
[23] |
R. C. Merton,
Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics, 51 (1969), 247-257.
doi: 10.2307/1926560. |
[24] |
M. Müller,
Stochastic Stefan-type problem under first-order boundary conditions, Ann. Appl. Probab., 28 (2018), 2335-2369.
doi: 10.1214/17-AAP1359. |
[25] |
B. Niethammer,
Derivation of the LSW-theory for Ostwald ripening by homogenization methods, Arch. Rational Mech. Anal., 147 (1999), 119-178.
doi: 10.1007/s002050050147. |
[26] |
B. Niethammer,
The LSW model for Ostwald ripening with kinetic undercooling, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 1337-1361.
doi: 10.1017/S0308210500000718. |
[27] |
W. Ostwald, Blocking of Ostwald ripening allowing long-term stabilization, Z. Phys. Chem., 37 (1901), 385 pp. |
[28] |
C. Parlour and D. Seppi, Handbook of Financial Intermediation & Banking, North-Holland (imprint of Elsevier), Amsterdam, eds. A. Boot and A. Thakor, 2008. |
[29] |
Z. Zheng, Stochastic Stefan problems: Existence, uniqueness, and modeling of market limit orders, PhD Thesis, University of Illinois at Urbana-Champaign, 2012. |
[30] |
G. Zimmerman, 2 Portfolio Protection Strategies That Don't Work - and 2 That Do, Advisors Voices, 2016. https://www.nerdwallet.com/blog/investing/2-portfolio-protection-strategies-dont-work/ |
show all references
References:
[1] |
N. D. Alikakos, P. W. Bates and X. Chen,
Convergence of the Cahn-Hilliard Equation to the Hele-Shaw Model, Arch. Rational Mech. Anal., 128 (1994), 165-205.
doi: 10.1007/BF00375025. |
[2] |
N. D. Alikakos and G. Fusco,
Ostwald ripening for dilute systems under quasistationary dynamics, Comm. Math. Phys., 238 (2003), 429-479.
doi: 10.1007/s00220-003-0833-5. |
[3] |
N. D. Alikakos, G. Fusco and G. Karali,
The effect of the geometry of the particle distribution in Ostwald ripening, Comm. Math. Phys., 238 (2003), 481-488.
doi: 10.1007/s00220-003-0834-4. |
[4] |
N. D. Alikakos, G. Fusco and G. Karali,
Ostwald ripening in two dimensions- The rigorous derivation of the equations from Mullins-Sekerka dynamics, J. Differential Equations, 205 (2004), 1-49.
doi: 10.1016/j.jde.2004.05.008. |
[5] |
A. Altarovici, J. Muhle-Karbe and H. M. Soner,
Asymptotics for fixed transaction costs, Finance Stoch., 19 (2015), 363-414.
doi: 10.1007/s00780-015-0261-3. |
[6] |
D. C. Antonopoulou, D. Blömker and G. D. Karali,
The sharp interface limit for the stochastic Cahn-Hilliard equation, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), 280-298.
doi: 10.1214/16-AIHP804. |
[7] |
D. C. Antonopoulou, G. D. Karali and A. N. K. Yip,
On the parabolic Stefan problem for Ostwald ripening with kinetic undercooling and inhomogeneous driving force, J. Differential Equations, 252 (2012), 4679-4718.
doi: 10.1016/j.jde.2012.01.016. |
[8] |
British Pound v US Dollar Data, https://www.poundsterlinglive.com.,, |
[9] |
X. Chen,
The Hele-Shaw problem and area-preserving curve shortening motions, Arch. Rational Mech. Anal., 123 (1993), 117-151.
doi: 10.1007/BF00695274. |
[10] |
X. Chen,
Global asymptotic limit of solutions of the Cahn-Hilliard equation, Journal of Differential Geometry, 44 (1996), 262-311.
|
[11] |
X. Chen, X. Hong and F. Yi,
Existence, uniqueness and regularity of classical solutions of Mullins-Sekerka problem, Comm. Partial Differential Equations, 21 (1996), 1705-1727.
doi: 10.1080/03605309608821243. |
[12] |
X. Chen and M. Dai,
Characterization of optimal strategy for multiasset investment and consumption with transaction costs, SIAM J. Financial Math., 4 (2013), 857-883.
doi: 10.1137/120898991. |
[13] |
X. Chen and F. Reitich,
Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling, J. Math. Anal. Appl., 164 (1992), 350-362.
doi: 10.1016/0022-247X(92)90119-X. |
[14] |
R. Cont and A. de Larrard,
Price dynamics in a Markovian limit order market, SIAM J. Financial. Math., 4 (2013), 1-25.
doi: 10.1137/110856605. |
[15] |
R. Cont, S. Stoikov and R. Talreja,
A stochastic model for order book dynamics, Oper. Res., 58 (2010), 549-563.
doi: 10.1287/opre.1090.0780. |
[16] |
E. Ekström, Selected Problems in Financial Mathematics, PhD Thesis, Uppsala Universitet, Sweden, 2004. |
[17] |
L. C. Evans, H. M. Soner and P. E. Souganidis,
Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123.
doi: 10.1002/cpa.3160450903. |
[18] |
T. Funaki,
Singular limit for stochastic reaction-diffusion equation and generation of random interfaces, Acta Math. Sin. (Engl. Ser.), 15 (1999), 407-438.
doi: 10.1007/BF02650735. |
[19] |
M. D. Gould, M. A. Porter, S. Williams, M. McDonald, D. J. Fenn and S. D. Howison,
Limit order books, Quant. Finance, 13 (2013), 1709-1742.
doi: 10.1080/14697688.2013.803148. |
[20] |
V. Henderson,
Prospect theory, liquidation, and the disposition effect, Management Science, 58 (2012), 445-460.
|
[21] |
T. Lybek and A. Sarr, Measuring Liquidity in Financial Markets, International Monetary Fund, work-in-progress, No. 02/232, 2002. |
[22] |
H. M. Markowitz, Portfolio selection: Efficient diversification of investments, John Wiley and Sons, Inc., New York, 1959. |
[23] |
R. C. Merton,
Lifetime portfolio selection under uncertainty: The continuous-time case, Review of Economics and Statistics, 51 (1969), 247-257.
doi: 10.2307/1926560. |
[24] |
M. Müller,
Stochastic Stefan-type problem under first-order boundary conditions, Ann. Appl. Probab., 28 (2018), 2335-2369.
doi: 10.1214/17-AAP1359. |
[25] |
B. Niethammer,
Derivation of the LSW-theory for Ostwald ripening by homogenization methods, Arch. Rational Mech. Anal., 147 (1999), 119-178.
doi: 10.1007/s002050050147. |
[26] |
B. Niethammer,
The LSW model for Ostwald ripening with kinetic undercooling, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 1337-1361.
doi: 10.1017/S0308210500000718. |
[27] |
W. Ostwald, Blocking of Ostwald ripening allowing long-term stabilization, Z. Phys. Chem., 37 (1901), 385 pp. |
[28] |
C. Parlour and D. Seppi, Handbook of Financial Intermediation & Banking, North-Holland (imprint of Elsevier), Amsterdam, eds. A. Boot and A. Thakor, 2008. |
[29] |
Z. Zheng, Stochastic Stefan problems: Existence, uniqueness, and modeling of market limit orders, PhD Thesis, University of Illinois at Urbana-Champaign, 2012. |
[30] |
G. Zimmerman, 2 Portfolio Protection Strategies That Don't Work - and 2 That Do, Advisors Voices, 2016. https://www.nerdwallet.com/blog/investing/2-portfolio-protection-strategies-dont-work/ |




Time |
||||
9:00 | 30.25 | 29.75 | 0.5 | 30 |
9:02 | 30.75 | 29.50 | 1.25 | 30.125 |
9:04 | 31.00 | 29.25 | 1.75 | 30.125 |
9:06 | 31.50 | 29.00 | 2.50 | 30.25 |
9:08 | 35.00 | 28.75 | 6.25 | 31.875 |
Sum | 158.5 | 146.25 | 12.25 | 152.375 |
Time |
||||
9:00 | 30.25 | 29.75 | 0.5 | 30 |
9:02 | 30.75 | 29.50 | 1.25 | 30.125 |
9:04 | 31.00 | 29.25 | 1.75 | 30.125 |
9:06 | 31.50 | 29.00 | 2.50 | 30.25 |
9:08 | 35.00 | 28.75 | 6.25 | 31.875 |
Sum | 158.5 | 146.25 | 12.25 | 152.375 |
Time |
||||
9:00 | 15.00 | 14.25 | 0.75 | 14.625 |
9:02 | 15.25 | 14.25 | 1.00 | 14.75 |
9:04 | 15.25 | 15.00 | 0.25 | 15.125 |
9:06 | 15.50 | 15.25 | 0.25 | 15.375 |
9:08 | 15.75 | 15.50 | 0.25 | 15.625 |
Sum | 76.75 | 74.25 | 2.50 | 75.50 |
Time |
||||
9:00 | 15.00 | 14.25 | 0.75 | 14.625 |
9:02 | 15.25 | 14.25 | 1.00 | 14.75 |
9:04 | 15.25 | 15.00 | 0.25 | 15.125 |
9:06 | 15.50 | 15.25 | 0.25 | 15.375 |
9:08 | 15.75 | 15.50 | 0.25 | 15.625 |
Sum | 76.75 | 74.25 | 2.50 | 75.50 |
Time |
||||
9:00 | 20.75 | 19.50 | 1.25 | 20.125 |
9:02 | 21.00 | 19.50 | 1.50 | 20.25 |
9:04 | 21.25 | 19.25 | 2.00 | 20.25 |
9:06 | 22.00 | 18.25 | 3.75 | 20.125 |
9:08 | 25.50 | 18.50 | 7.00 | 22.00 |
Sum | 110.5 | 95 | 15.50 | 102.75 |
Time |
||||
9:00 | 20.75 | 19.50 | 1.25 | 20.125 |
9:02 | 21.00 | 19.50 | 1.50 | 20.25 |
9:04 | 21.25 | 19.25 | 2.00 | 20.25 |
9:06 | 22.00 | 18.25 | 3.75 | 20.125 |
9:08 | 25.50 | 18.50 | 7.00 | 22.00 |
Sum | 110.5 | 95 | 15.50 | 102.75 |
Asset | ||||
1 | 550 | 550/2.45=224.49 | 550/1600=0.34375 | 77.168 |
2 | 750 | 750/0.5=1500 | 750/1600=0.46875 | 703.125 |
3 | 300 | 300/3.1=96.774 | 300/1600=0.1875 | 18.145 |
Sum | 1600 |
Asset | ||||
1 | 550 | 550/2.45=224.49 | 550/1600=0.34375 | 77.168 |
2 | 750 | 750/0.5=1500 | 750/1600=0.46875 | 703.125 |
3 | 300 | 300/3.1=96.774 | 300/1600=0.1875 | 18.145 |
Sum | 1600 |
Asset | ||||
1 | 550 | 550/0.080437=6837.64 | 550/1600=0.34375 | 2350.438 |
2 | 750 | 750/0.03312=22644.92 | 750/1600=0.46875 | 10614.806 |
3 | 300 | 300/0.15114=1984.91 | 300/1600=0.1875 | 372.170 |
Sum | 1600 |
Asset | ||||
1 | 550 | 550/0.080437=6837.64 | 550/1600=0.34375 | 2350.438 |
2 | 750 | 750/0.03312=22644.92 | 750/1600=0.46875 | 10614.806 |
3 | 300 | 300/0.15114=1984.91 | 300/1600=0.1875 | 372.170 |
Sum | 1600 |
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