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121
AN EQUILIBRIUM CHARACTERIZATION OF THE TERM STRUCTURE
, 1977
"... The paper derives a general form of the term structure of interest rates. The following assumptions are made: (A.l) The instantaneous (spot) interest rate follows a diffusion process; (A.2) the price of a discount bond depends only on the spot rate over its term; and (A.3) the market is efficient. U ..."
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Cited by 619 (0 self)
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The paper derives a general form of the term structure of interest rates. The following assumptions are made: (A.l) The instantaneous (spot) interest rate follows a diffusion process; (A.2) the price of a discount bond depends only on the spot rate over its term; and (A.3) the market is efficient. Under these assumptions, it is shown by means of an arbitrage argument that the expected rate of return on any bond in excess of the spot rate is proportional to its standard deviation. This property is then used to derive a partial differential equation for bond prices. The solution to that equation is given in the form of a stochastic integral representation. An interpretation of the bond pricing formula is provided. The model is illustrated on a specific case.
Exponential Stability of Stochastic Differential Delay Equations
, 1994
"... : In this paper we study both pth moment and almost sure exponential stability of the stochastic differential delay equation dx(t)=f(t;x(t);x(t\Gammaø))dt+g(t;x(t);x(t\Gammaø))dw(t). Introduce the corresponding stochastic differential equation (without delay) dx(t)=f(t;x(t);x(t))dt+ g(t;x(t);x(t))dw ..."
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Cited by 55 (33 self)
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: In this paper we study both pth moment and almost sure exponential stability of the stochastic differential delay equation dx(t)=f(t;x(t);x(t\Gammaø))dt+g(t;x(t);x(t\Gammaø))dw(t). Introduce the corresponding stochastic differential equation (without delay) dx(t)=f(t;x(t);x(t))dt+ g(t;x(t);x(t))dw(t) and assume it is exponentially stable which is guaranteed by the existence of the Lyapunov function. We shall show that the original stochastic differential delay equation remains exponentially stable provided the time lag ø is sufficiently small, and a bound for such ø is obtained. Key Words: stochastic differential delay equations, Lyapunov function, Lyapunov exponent, BorelCantelli lemma. AMS 1991 Classifications: 60H10, 34K30 1. Introduction In many branches of science and industry stochastic differential delay equations have been used to model the evolution phenomena because the measurements of timeinvolving variables and their dynamics usually contain some delays (cf. Kolmanov...
On the optimal stopping problem for onedimensional diffusions, 2002. Working Paper (http://www.stat.columbia.edu/ ˜ik/DAYKAR.pdf
"... A new characterization of excessive functions for arbitrary one–dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessivity is equivalent to concavity in some suitable generalized sense. This permits a characterization of the value function of the ..."
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Cited by 36 (2 self)
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A new characterization of excessive functions for arbitrary one–dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessivity is equivalent to concavity in some suitable generalized sense. This permits a characterization of the value function of the optimal stopping problem as “the smallest nonnegative concave majorant of the reward function ” and allows us to generalize results of Dynkin and Yushkevich for standard Brownian motion. Moreover, we show how to reduce the discounted optimal stopping problems for an arbitrary diffusion process to an undiscounted optimal stopping problem for standard Brownian motion. The concavity of the value functions also leads to conclusions about their smoothness, thanks to the properties of concave functions. One is thus led to a new perspective and new facts about the principle of smooth–fit in the context of optimal stopping. The results are illustrated in detail on a number of non–trivial, concrete optimal stopping problems, both old and new.
Stochastic Differential Systems With Memory. Theory, Examples And Applications
 Ustunel, Progress in Probability, Birkhauser
, 1996
"... this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state. Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic timelag model with Gaussian noise, and the c ..."
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Cited by 22 (9 self)
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this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state. Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic timelag model with Gaussian noise, and the classical "heatbath" model of R. Kubo, modeling the motion of a "large" molecule in a viscous fluid. These examples are embedded in a general class of stochastic functional differential equations (sfde's). We then establish pathwise existence and uniqueness of solutions to these classes of sfde's under local Lipschitz and linear growth hypotheses on the coefficients. It is interesting to note that the above class of sfde's is not covered by classical results of Protter, Metivier and Pellaumail and DoleansDade. In Chapter II, we prove that the Markov (Feller) property holds for the trajectory random field of a sfde. The trajectory Markov semigroup is not strongly continuous for positive delays, and its domain of strong continuity does not contain tame (or cylinder) functions with evaluations away from 0. To overcome this difficulty, we introduce a class of quasitame functions. These belong to the domain of the weak infinitesimal generator, are weakly dense in the underlying space of continuous functions and generate the Borel
Inflation, asset prices and the term structure of interest rates in monetary economies
 Review of Financial Studies 9
, 1996
"... This article offers a tractable monetary asset pricing model. In monetary economies, the price level, inflation, asset prices, and the real and nominal interest rates have to be determined simultaneously and in relation to each other. This link allows us to relate in closed form each of the dependen ..."
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Cited by 21 (2 self)
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This article offers a tractable monetary asset pricing model. In monetary economies, the price level, inflation, asset prices, and the real and nominal interest rates have to be determined simultaneously and in relation to each other. This link allows us to relate in closed form each of the dependent entities to the underlying real and monetary variables. Among other features of such economies, inflation can be partially nonmonetary and the real and nominal term structures can depend on fundamentally different risk factors. In one extreme, the process followed by the real term structure is independent of that followed by its nominal counterpart. This article studies the endogenous and simultaneous determination of the price level, inflation, asset prices, and the term structure of interest rates, both real and nominal. The setup is an intertemporal monetary economy where there is fiat money for transaction purposes and where investors find it useful to hold cash balances. The objective is to relate, in the spirit of Cox, Ingersoll, and Ross (1985b) (henceforth, The authors would like to thank Franklin Allen (the editor) and the anonymous referee for helpful comments and suggestions. Any remaining errors
A stochastic particle method for the McKeanVlasov and the Burgers equation
 Math. Comp
, 1997
"... Abstract. In this paper we introduce and analyze a stochastic particle method for the McKeanVlasov and the Burgers equation; the construction and error analysis are based upon the theory of the propagation of chaos for interacting particle systems. Our objective is threefold. First, we consider a ..."
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Cited by 20 (3 self)
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Abstract. In this paper we introduce and analyze a stochastic particle method for the McKeanVlasov and the Burgers equation; the construction and error analysis are based upon the theory of the propagation of chaos for interacting particle systems. Our objective is threefold. First, we consider a McKeanVlasov equation in [0,T]×R with sufficiently smooth kernels, and the PDEs giving the distribution function and the density of the measure µt, the solution to the McKeanVlasov equation. The simulation of the stochastic system with N particles provides a discrete measure which approximates µk∆t for each time k∆t (where ∆t is a discretization step of the time interval [0,T]). An integration (resp. smoothing) of this discrete measure provides approximations of the distribution function (resp. density) of µk∆t. We show that the convergence rate is O 1 / √ N + √) ∆t for the approximation in L1 (Ω × R) ofthecumulative distribution function at time T, and of order O ε2 + 1
Weighted Stochastic Sobolev Spaces and Bilinear SPDE's Driven by Spacetime White Noise
"... In this paper we develop basic elements of Malliavin calculus on a weighted L 2(\Omega\Gamma2 This class of generalized Wiener functionals is a Hilbert space. It turns out to be substantially smaller than the space of Hida distributions while large enough to accommodate solutions of bilinear stoch ..."
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Cited by 17 (5 self)
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In this paper we develop basic elements of Malliavin calculus on a weighted L 2(\Omega\Gamma2 This class of generalized Wiener functionals is a Hilbert space. It turns out to be substantially smaller than the space of Hida distributions while large enough to accommodate solutions of bilinear stochastic PDE's. As an example, we consider a stochastic advectiondiffusion equation driven by spacetime white noise in IR d . It is known that for d ? 1, this equation has no solutions in L 2(\Omega\Gamma4 In contrast, it is shown in the paper that in an appropriately weighted L 2(\Omega\Gamma there is a unique solution to the stochastic advectiondiffusion equation for any d 1. In addition we present explicit formulas for the HermiteFourier coefficients in the Wiener chaos expansion of the solution.
Dynamics: a probabilistic and geometric perspective
 In Proceedings of the International Congress of Mathematicians
, 1998
"... ..."
An Evaluation of MultiFactor CIR Models Using LIBOR, Swap Rates, and Cap and Swaption Prices
, 2001
"... ..."
Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator and applications
 SIAM J. APPL. MATH
, 1986
"... We construct asymptotic expansions for the exponential growth rate (Lyapunov exponent) and rotation number of the random oscillator when the noise is large, small, rapidly varying or slowly varying. We then apply our results to problems in the stability of the random oscillator, the spectrum of the ..."
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Cited by 14 (2 self)
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We construct asymptotic expansions for the exponential growth rate (Lyapunov exponent) and rotation number of the random oscillator when the noise is large, small, rapidly varying or slowly varying. We then apply our results to problems in the stability of the random oscillator, the spectrum of the onedimensional random Schrödinger operator and wave propagation in a onedimensional random medium.