Convergence in multiscale financial models with non-Gaussian stochastic volatility
ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 2, pp. 500-518.

We consider stochastic control systems affected by a fast mean reverting volatility Y(t) driven by a pure jump Lévy process. Motivated by a large literature on financial models, we assume that Y(t) evolves at a faster time scale t/ϵ than the assets, and we study the asymptotics as ϵ0. This is a singular perturbation problem that we study mostly by PDE methods within the theory of viscosity solutions.

Received:
DOI: 10.1051/cocv/2015015
Classification: 93C70, 49L25, 35R09, 91B28
Keywords: Singular perturbations, stochastic volatility, jump processes, viscosity solutions, Hamilton–Jacobi–Bellman equations, portfolio optimization
Bardi, Martino 1; Cesaroni, Annalisa 2; Scotti, Andrea 3

1 Department of Mathematics, University of Padova, via Trieste 63, 35121 Padova, Italy
2 Department of Mathematics, currently at Department of Statistical Sciences, University of Padova, via C. Battisti 241, 35121 Padova, Italy
3 Department of Mathematics, current address Via Podgora 107, 30172, Mestre (VE), Italy
@article{COCV_2016__22_2_500_0,
     author = {Bardi, Martino and Cesaroni, Annalisa and Scotti, Andrea},
     title = {Convergence in multiscale financial models with {non-Gaussian} stochastic volatility},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {500--518},
     publisher = {EDP-Sciences},
     volume = {22},
     number = {2},
     year = {2016},
     doi = {10.1051/cocv/2015015},
     zbl = {1369.93713},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2015015/}
}
TY  - JOUR
AU  - Bardi, Martino
AU  - Cesaroni, Annalisa
AU  - Scotti, Andrea
TI  - Convergence in multiscale financial models with non-Gaussian stochastic volatility
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2016
SP  - 500
EP  - 518
VL  - 22
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2015015/
DO  - 10.1051/cocv/2015015
LA  - en
ID  - COCV_2016__22_2_500_0
ER  - 
%0 Journal Article
%A Bardi, Martino
%A Cesaroni, Annalisa
%A Scotti, Andrea
%T Convergence in multiscale financial models with non-Gaussian stochastic volatility
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2016
%P 500-518
%V 22
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2015015/
%R 10.1051/cocv/2015015
%G en
%F COCV_2016__22_2_500_0
Bardi, Martino; Cesaroni, Annalisa; Scotti, Andrea. Convergence in multiscale financial models with non-Gaussian stochastic volatility. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 2, pp. 500-518. doi : 10.1051/cocv/2015015. http://archive.numdam.org/articles/10.1051/cocv/2015015/

O. Alvarez and M. Bardi, Viscosity solutions methods for singular perturbations in deterministic and stochastic control. SIAM J. Control Optim. 40 (2001/02) 1159–1188. | DOI | MR | Zbl

O. Alvarez and M. Bardi, Singular perturbations of degenerate parabolic PDEs: a general convergence result. Arch. Ration. Mech. Anal. 170 (2003) 17–61. | DOI | MR | Zbl

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser, Boston (1997). | MR | Zbl

M. Bardi and A. Cesaroni, Optimal control with random parameters: a multiscale approach. Eur. J. Control. 17 (2011), 30–46. | DOI | MR | Zbl

M. Bardi and G. Terrone, Homogenization of some optimal control problems (to appear).

M. Bardi, A. Cesaroni and L. Manca, Convergence by Viscosity Methods in Multiscale Financial Models with Stochastic Volatility. SIAM J. Financial Math. 1 (2010) 230–265. | DOI | MR | Zbl

G. Barles, C. Imbert, Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25 (2008) 567–585. | DOI | Numdam | MR | Zbl

O.E. Barndorff–Nielsen and N. Shephard, Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. Royal Stat. Soc. B 63 (2001) 167–241. | DOI | MR | Zbl

F.E. Benth, K.H. Karlsen and K. Reikvam, Merton’s portfolio optimization problem in a Black-Scholes market with non-Gaussian stochastic volatility of Ornstein–Uhlenbeck Type. Math. Finance 13 (2003) 215–244. | DOI | MR | Zbl

A. Ciomaga, On the strong maximum principle for second order nonlinear parabolic integro-differential equations. Adv. Differ. Equ. 17 (2012) 635–671. | MR | Zbl

R. Cont and P. Tankov, Financial Modelling with Jump Processes. Chapman Hall/CRC, Boca Raton, Florida (2004). | MR | Zbl

F. Da Lio and O. Ley, Uniqueness results for second-order Bellman–Isaacs equations under quadratic growth assumptions and applications. SIAM J. Control Optim. 45 (2006) 74–106. | DOI | MR | Zbl

L.C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 359–375. | DOI | MR | Zbl

J.-P. Fouque, G. Papanicolau and R. Sircar, Derivatives in Financial Markets with Stochastic Volatility. Cambridge University Press, Cambridge, UK (2000). | MR | Zbl

J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Singular perturbations in option pricing. SIAM J. Appl. Math. 63 (2003a) 1648–1665. | DOI | MR | Zbl

J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale stochastic volatility asymptotics. Multiscale Model. Simul. 2 (2003b) 22–42. | DOI | MR | Zbl

J.-P. Fouque, G. Papanicolaou, R. Sircar and K. Solna, Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives. Cambridge University Press, Cambridge (2011). | MR | Zbl

W.H. Fleming and H.M. Soner, Controlled Markov Processes and Viscosity Solutions, 2nd edition. Springer-Verlag, New York (2006). | MR | Zbl

R.Z. Khasminskii, Stochastic Stability of Differential Equations, 2nd edition. Springer, Heidelberg (2012). | MR | Zbl

F. Hubalek and C. Sgarra, On the explicit valuation of geometric asian options in stochastic volatility models with jumps. J. Comput. Appl. Math. 235 (2011) 3355–3365. | DOI | MR | Zbl

F. Hubalek and C. Sgarra, On the Esscher transforms and other equivalent martingale measures for Barndorff–Nielsen and Shephard stochastic volatility models with jumps. Stoch. Process. Appl. 119 (2009) 2137–2157. | DOI | MR | Zbl

A.M. Kulik, Exponential ergodicity of the solutions to SDE’s with a jump noise. Stoch. Process. Appl. 119 (2009) 602–632. | DOI | MR | Zbl

M. Lorig and O. Lozano-Carbassé, Exponential Lévy-type models with stochastic volatility and jump intensity. Quant. Finance 15 (2015) 91–100. | DOI | MR | Zbl

E. Nicolato and E. Venardos, Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type. Math. Finance 13 (2003) 445–466. | DOI | MR | Zbl

H. Pham, Optimal stopping of controlled jump diffusion processes: a viscosity solution approach. J. Math. Systems Estim. Control 8 (1998) 1–27. | MR | Zbl

J. Picard, On the existence of smooth densities for jump processes. Probab. Theory Related Fields 105 (1996) 481–511. | DOI | MR | Zbl

E. Priola and J. Zabczyk, Densities for Ornstein–Uhlenbeck processes with jumps. Bull. Lond. Math. Soc. 41 (2009) 41–50. | DOI | MR | Zbl

K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999). | MR | Zbl

A. Sayah, Equations d’Hamilton–Jacobi du premier ordre avec termes intégro-différentiels. I. Unicité des solutions de viscosité. II. Existence de solutions de viscosité. Commun. Partial Differ. Equ. 16 (1991) 1057–1093. | DOI | Zbl

B. Simon, Functional Integration and Quantum Physics. Academic Press, New York (1979). | MR | Zbl

H.M. Soner, Optimal Control of Jump-Markov Processes and Viscosity Solutions, in Stochastic Differential Systems. Vol. 10 of Stochastic Control Theory and Applications. IMA Math. Appl. Springer, New York (1988) 501–511. | MR | Zbl

Cited by Sources: