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## LONG-TERM RETURNS IN STOCHASTIC INTEREST RATE MODELS: APPLICATIONS B¥

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Citations: | 9 - 3 self |

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3281 |
An introduction to probability theory and its applications. Vol
- Feller
- 1971
(Show Context)
Citation Context ...check that: (√ ∫ ( ) √ −2β3 nt1 ∫ ( ) ) −2β3 ntk du, ···, du δn 0 Xu + δu 2β δn ( L −→ B ′ t1 , ···,B′ tk 0 ) . Xu + δu 2β We recall the following well known theorem of probability theory (See Feller =-=[9]-=- p. 247): Theorem A Suppose that a sequence Zn :Ω−→ IR k is given. If Zn = Un +Vn where then • Un • Vn L −→ U with U ∼ µ a probability measure IP −→ 0 Zn L −→ U. From the stochastic differential equat... |

1978 | A theory of the term structure of interest rates
- Cox, Ingersoll, et al.
- 1985
(Show Context)
Citation Context ...terest rate models: Convergence in law. G. Deelstra F. Delbaen Vrije Universiteit Brussel Departement Wiskunde Pleinlaan 2 B-1050 Brussel Belgium Abstract Using an extension of the Cox-Ingersoll-Ross =-=[1]-=- stochastic model of the short interest rate r, we study the convergence in law of the longterm return in order to make some approximations. We use the theory (√ of Bessel processes and observe the co... |

1869 |
Continuous Martingales and Brownian Motion
- Revuz, Yor
- 1999
(Show Context)
Citation Context ...odels. If we define X by a transformation of the CIR square root process r, namely X = 4 σ2 r, then X is a Besselsquare process with drift −κ/2 and dimension 4κγ σ2 . The many results obtained by Yor =-=[6,7]-=-, convinced us that these processes are very tractable. Therefore, we consider a family of stochastic processes X, which contains the Besselsquare processes with drift. More precisely, we study proces... |

1745 |
Brownian Motion and Stochastic Calculus
- Karatzas, Shreve
- 1991
(Show Context)
Citation Context ...nd vector. We will show that each component 12 converges in probability to zero. This result follows from stochastic integration theory. We cite the theorem that we will use, see e.g. Karatzas-Shreve =-=[10]-=- p. 147, proposition 2.26 which remains valid for T ≡ ∞: Theorem B Let us denote L2,0 = { H| H predictable with ∫∞ 0 H2udu <∞ } and let us define L2,0 −→ L0(Ω,F∞, IP ) H 7−→ (H ·B)∞ If ∫∞ 0 (Hnu ) 2du... |

1086 |
Limit Theorems for Stochastic Processes
- Jacod, Shiryayev
- 1980
(Show Context)
Citation Context ... 0 δu ) ) 2β du converge in law to (B ′ t1 , ···,B′ tk ). We will now prove that the laws of (Y n ) form a weakly relatively compact sequence. We have to check two conditions (see e.g. Jacod–Shiryaev =-=[11]-=-, p. 320): For all N ∈ IN ∗ ,ε>0, there exists n0 ∈ IN ∗ and K ∈ IR + such that for all n ≥ n0 : and Aldous’ criterion, namely For all N ∈ IN ∗ ,ε>0: lim lim sup θ→0 n ( IP sup |Y t≤N n ) t | >K ≤ ε. ... |

484 | An Empirical Comparison of Alternative Models of the Short-Term Interest Rate - Chan, Karolyi, et al. - 1992 |

294 |
Pricing interest-rate-derivative securities.
- Hull, White
- 1990
(Show Context)
Citation Context ...ersion level. However, this is a constant level, namely the long-term value γ. It is more reasonable to conjecture that the market will influence this level. Schaefer and Schwartz [3], Hull and White =-=[4]-=- and Longstaff and Schwartz [5], proposed time-dependent parameters. We extend the CIR model by assuming a stochastic reversion level. In this way, we can treat more factor models. If we define X by a... |

228 |
Interest rate volatility and the term structure: a twofactor general equilibrium model.
- Longstaff, Schwartz
- 1992
(Show Context)
Citation Context ...a constant level, namely the long-term value γ. It is more reasonable to conjecture that the market will influence this level. Schaefer and Schwartz [3], Hull and White [4] and Longstaff and Schwartz =-=[5]-=-, proposed time-dependent parameters. We extend the CIR model by assuming a stochastic reversion level. In this way, we can treat more factor models. If we define X by a transformation of the CIR squa... |

199 | On Some Exponential Functionals of Brownian Motion.” Adv. - Yor - 1992 |

174 | Exploiting the Conditional Density in Estimating the Term Structure: An Application to the Cox, Ingersoll, and Ross Model,” - Pearson, Sun - 1994 |

127 | The distribution of a perpetuity, with applications to risk theory and pension funding - Dufresne - 1990 |

90 | Mr. Keynes and the "Classics"; A suggested interpretation. - Hicks - 1937 |

68 | Long forward and zero-coupon rates can never fall. - Dybvig, Ingersoll, et al. - 1996 |

63 | Pricing Interest Rate Options in a Two-Factor Cox-Ingersoll-Ross Model of the Term Structure,” Review of Financial Studies, - Chen, Scott - 1992 |

44 | A Two-Factor Model of the Term Structure: An Appropriate Analytical Solution - Schaefer, Schwartz - 1984 |

17 | Stochastic analysis of the interaction between investment and insurance risks”. - PARKER - 1997 |

16 |
Adecomposition of Bessel bridges, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete
- Pitman, Yor
- 1982
(Show Context)
Citation Context ...odels. If we define X by a transformation of the CIR square root process r, namely X = 4 σ2 r, then X is a Besselsquare process with drift −κ/2 and dimension 4κγ σ2 . The many results obtained by Yor =-=[6,7]-=-, convinced us that these processes are very tractable. Therefore, we consider a family of stochastic processes X, which contains the Besselsquare processes with drift. More precisely, we study proces... |

10 | Problems in certain two-factor term structure models, - Hogan - 1993 |

10 | Distribution of the present value of future cash flows - Parker - 1993 |

9 | The term structure of real interest rates and the Cox, Ingersoll, and Ross model. - Brown, Schaefer - 1994 |

9 | Squared Bessel Processes and Their Applications to the Square Root Interest Rate Model - Shirakawa - 2002 |

9 | A Note on the Behavior of Long Zero Coupon Rates in a No Arbitrage Framework - Karoui, Frachot, et al. - 1997 |

8 | Moments of the present value of a portfolio of policies - Parker - 1994 |

6 | Stochastic modelling of interest rates: Actuarial vs. equilibrium approach - Giacotto - 1986 |

6 | The Distribution of Annuities - Vanneste, Goovaerts, et al. - 1994 |

6 | A non-linear model of the term-structure of interest rates - Tice, Webber - 1997 |

4 | Term Structure Modeling and Asymptotic Long Rate - Yao - 1999 |

2 | Do interest rates converge?, working paper - Dybvig, Ingersoll, et al. - 1985 |

2 | An Application of Stochastic Interest Rates Models in Life Assurance - PARKER - 1992 |

2 |
Existence of Solutions of Stochastic Differential Equations related to the Bessel process, forthcoming
- Deelstra, Delbaen
(Show Context)
Citation Context ...ion: dXs =(2βXs + δs)ds +2 √ XsdBs ∀s ∈ IR + with δ a non-negative adapted stochastic process and β < 0. In ”Existence of Solutions of Stochastic Differential Equations related to the Bessel process” =-=[8]-=-, we have shown that this stochastic differential equation has a unique (nonnegative) strong solution as soon as ∫ t 0 δu du < ∞ a.e. for all t ∈ IR +. In this paper, we will assume that 1 ∫ t a.e. t ... |

1 | Extra randomness in certain annuity modes, hzsurance - J, FUELLING - 1991 |

1 | Actuarial Mathematics. Society of Actt,aries - BOWERS, GERBER, et al. - 1986 |

1 | Long-term returns in stochastic interest rate models - STRA - 1995 |

1 | Yield Options in the Generalised Cox-lngersoli-Ross Model - DEELSTRA - 2000 |

1 | Evaluation of interest randomness for pension valuation - GOOVAERTS, TEUNEN - 1995 |

1 | Solution of the extended CI R term structure and bond option valuation - MAGHSOODI - 1996 |

1 | The present value of a stochastic perpetuity and the Gamma distribution - MILEVSKV - 1997 |

1 |
Long-term interest rate models, forthcoming
- Deelstra, Delbaen
(Show Context)
Citation Context ...derivative securities and to construct a hedging strategy. They are also a necessary tool in managing long-term life insurance contracts. As in “Long-term returns in stochastic interest rate models.” =-=[2]-=-, we study the long-term return using an extension of the Cox, Ingersoll and Ross [1] stochastic model of the short interest rate r. Cox, Ingersoll and Ross express the short interest rate dynamics as... |

1 |
An Empirical Comparison of Altermative Models of the Short-Term Interest Rate
- Chan, Karolyi, et al.
- 1992
(Show Context)
Citation Context ...otion is too slow. In practice however, the period of interest usually is shorter than 40 or 50 years. Using the parameters estimated within the empirical work of Chan, Karolyi, Longstaff and Sanders =-=[12]-=-, namely κ = 0.2339, γ∗ = 0.0808 and σ =0.0854, we have calculated the exact bond prices and the proposed approximation. As the approximation does not depend on the present interest rate r0, we do not... |