670 research outputs found
Smoothness of the density for solutions to Gaussian rough differential equations
We consider stochastic differential equations of the form
driven by a multi-dimensional Gaussian
process. Under the assumption that the vector fields and
satisfy H\"{o}rmander's bracket condition, we demonstrate
that admits a smooth density for any , provided the driving
noise satisfies certain nondegeneracy assumptions. Our analysis relies on
relies on an interplay of rough path theory, Malliavin calculus and the theory
of Gaussian processes. Our result applies to a broad range of examples
including fractional Brownian motion with Hurst parameter , the
Ornstein-Uhlenbeck process and the Brownian bridge returning after time .Comment: Published in at http://dx.doi.org/10.1214/13-AOP896 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The Fractional OU Process: Term Structure Theory and Application
The paper revisits dynamic term structure models (DTSMs) and proposes a new way in dealing with the limitation of the classical affine models. In particular, this paper expands the flexibility of the DTSMs by applying a fractional Brownian motion as the governing force of the state variable instead of the standard Brownian motion. This is a new direction in pricing non defaultable bonds with offspring in the arbitrage free pricing of weather derivatives based on fractional Brownian motions. By applying fractional Ito calculus and a fractional version of the Girsanov transform, a no arbitrage price of the bond is recovered by solving a fractional version of the fundamental bond pricing equation. Besides this theoretical contribution, the paper proposes an estimation methodology based on the Kalman filter approach, which is applied to the US term structure of interest ratesFractional bond pricing equation, fractional Brownian motion, fractional Ornstein-Uhlenbeck process, long memory, Kalman Filter
The Fractional Ornstein-Uhlenbeck Process: Term Structure Theory and Application
The paper revisits dynamic term structure models (DTSMs) and proposes a new way in dealing with the limitation of the classical affine models. In particular, this paper expands the flexibility of the DTSMs by applying a fractional Brownian motion as the governing force of the state variable instead of the standard Brownian motion. This is a new direction in pricing non defaultable bonds with offspring in the arbitrage free pricing of weather derivatives based on fractional Brownian motions. By applying fractional Ito calculus and a fractional version of the Girsanov transform, a no arbitrage price of the bond is recovered by solving a fractional version of the fundamental bond pricing equation. Besides this theoretical contribution, the paper proposes an estimation methodology based on the Kalman filter approach, which is applied to the US term structure of interest rates.Fractional bond pricing equation; fractional Brownian motion; fractional Ornstein-Uhlenbeck process; long memory; Kalman filter
Montecarlo simulation of long-term dependent processes: a primer
As a natural extension to LeĆ³n and Vivas (2010) and LeĆ³n and Reveiz (2010) this paper briefly describes the Cholesky method for simulating Geometric Brownian Motion processes with long-term dependence, also referred as Fractional Geometric Brownian Motion (FBM). Results show that this method generates random numbers capable of replicating independent, persistent or antipersistent time-series depending on the value of the chosen Hurst exponent. Simulating FBM via the Cholesky method is (i) convenient since it grants the ability to replicate intense and enduring returns, which allows for reproducing well-documented financial returnsā slow convergence in distribution to a Gaussian law, and (ii) straightforward since it takes advantage of the Gaussian distribution ability to express a broad type of stochastic processes by changing how volatility behaves with respect to the time horizon. However, Cholesky method is computationally demanding, which may be its main drawback. Potential applications of FBM simulation include market, credit and liquidity risk models, option valuation techniques, portfolio optimization models and payments systems dynamics. All can benefit from the availability of a stochastic process that provides the ability to explicitly model how volatility behaves with respect to the time horizon in order to simulate severe and sustained price and quantity changes. These applications are more pertinent than ever because of the consensus regarding the limitations of customary models for valuation, risk and asset allocation after the most recent episode of global financial crisis.Montecarlo simulation, Fractional Brownian Motion, Hurst exponent, Long-term Dependence, Biased Random Walk. Classification JEL: C15, C53, C63, G17, G14.
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