Numerical Regularization for SDEs: Construction of Nonnegative Solutions

In the numerical solution of stochastic diierential equations (SDEs) such appearances as sudden, large uctuations (explosions), negative paths or unbounded solutions are sometimes observed in contrast to the qualitative behaviour of the exact solution. To overcome this dilemma we construct regular (bounded) numerical solutions through implicit techniques without discretizing the state space. For discussion and classiication, the notation of life time of numerical solutions is introduced. Thereby the task consists in construction of numerical solutions with lengthened life time up to eternal one. During the exposition we outline the role of implicitness for this`process of numerical regularization'. Boundedness (Nonneg-ativity) of some implicit numerical solutions can be proved at least for a class of linearly bounded models. Balanced implicit methods (BIMs) turn out to be very eecient for this purpose. Furthermore, the local property of conditional positivity of numerical solutions is shown constructively (by special BIMs). The suggested approach also gives some motivation to use BIMs for the construction of numerical solutions for SDEs on bounded manifolds with`natural conditions' on their boundaries. Finally we suggest to apply these methods to population dynamics in Biology, innovation diiusion in Marketing and to mean reverting processes in Finance, such as stochastic interest rates. 1 This contribution exhibits a rst trial to numerical analysis of Stochastic Diierential Algebraic Equations (SDAEs) with nonanticipating algebraic constraints. It is an improved version of discussion paper no. 40 refereed and published within Sonderforschungsbereich 373 at Humboldt University in Berlin (1994).

[1]  S. Jaschke.  Exploratory Data Analysis of Short-Term Interest Rates , 1994 .

[2]  J. Hull.  Options, futures, and other derivative securities , 1989 .

[3]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[4]  Oldrich A. Vasicek.  An equilibrium characterization of the term structure , 1977 .

[5]  F. Bass.  A new product growth model for consumer durables , 1976 .

[6]  Kazimierz Sobczyk,et al.  Stochastic Differential Equations , 1991 .

[7]  Eckhard Platen,et al.  Approximation of itô integral equations , 1980 .

[8]  Georges Courtadon.  The Pricing of Options on Default-Free Bonds , 1982, Journal of Financial and Quantitative Analysis.

[9]  D. Duffie.  Dynamic Asset Pricing Theory , 1992 .

[10]  Denis Talay,et al.  Simulation and numerical analysis of stochastic differential systems : a review , 1990 .

[11]  S. Ross,et al.  A theory of the term structure of interest rates'', Econometrica 53, 385-407 , 1985 .

[12]  J. M. Clark,et al.  The maximum rate of convergence of discrete approximations , 1980 .

[13]  Sang Bin Lee,et al.  Term Structure Movements and Pricing Interest Rate Contingent Claims , 1986 .

[14]  John B. Donaldson,et al.  On the term structure of interest rates , 1990 .

[15]  R. Tschernig,et al.  Nonlinear Interest Rate Dynamics and Implications for the Term Structure , 1996 .

[16]  D. Talay,et al.  Discretization and simulation of stochastic differential equations , 1985 .

[17]  E. Wong,et al.  ON THE RELATION BETWEEN ORDINARY AND STOCHASTIC DIFFERENTIAL EQUATIONS , 1965 .

[18]  Campbell R. Harvey,et al.  An Empirical Comparison of Alternative Models of the Short-Term Interest Rate , 1992 .

[19]  J. Quadrat.  Numerical methods for stochastic control problems in continuous time , 1994 .

[20]  H. Schurz.  Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise , 1996 .

[21]  Nigel J. Newton.  Asymptotically efficient Runge-Kutta methods for a class of ITOˆ and Stratonovich equations , 1991 .

[22]  S. Zacks,et al.  Introduction to stochastic differential equations , 1988 .

[23]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .