[...]... for the Reader (ii) Engineers, quantitative analysts and others with a more technical background in mathematical and quantitative methods who are interested in applying stochastic differential equations with jumps, and in implementing efficient simulation methods or developing new schemes could use the book according to the following suggested flowchart Without too much emphasis on proofs the selected material... larger class of SDEs driven by fairly general semimartingales The class of SDEs driven by Wiener processes and Poisson jump measures with finite intensity appears to be large enough for realistic modeling of the dynamics of quantities in finance Here continuous trading noise and a few single events model the typical sources of uncertainty Furthermore, stochastic jump sizes and stochastic intensities,... weighted index ODE ordinary differential equation SDE stochastic differential equation PDE partial differential equation PIDE partial integro differential equation Iν (·) modified Bessel function of the first kind with index ν modified Bessel function of the third kind with index λ time step size of a time discretization Kλ (·) Δ i l = i! l!(i−l)! combinatorial coefficient C k (Rd , R) set of k times continuously... determinant of a matrix A A−1 inverse of a matrix A (x, y) inner product of vectors x and y N = {1, 2, } set of natural numbers 1 2 d d with ith component xi XX Basic Notation ∞ in nity (a, b) open interval a < x < b in [a, b] closed interval a ≤ x ≤ b in = (−∞, ∞) + = [0, ∞) d set of real numbers set of nonnegative real numbers d-dimensional Euclidean space Ω sample space ∅ empty set A∪B the union of. .. uniqueness of solutions of SDEs These tools and results provide the basis for the application and numerical solution of stochastic differential equations with jumps 1.1 Stochastic Processes Stochastic Process If not otherwise stated, throughout the book we shall assume that there exists a common underlying probability space (Ω, A, P ) consisting of the sample space Ω, the sigma-algebra or collection of events... Stochastic Differential Equations with Jumps in Finance, Stochastic Modelling and Applied Probability 64, DOI 10.1007/978-3-642-13694-8 1, © Springer-Verlag Berlin Heidelberg 2010 1 2 1 SDEs with Jumps We set the time set to the interval T = [0, ∞) if not otherwise stated On some occasions the time set may become the bounded interval [0, T ] for T ∈ (0, ∞) or a set of discrete time points {t0 , t1 , t2... fundamental building blocks in stochastic modeling and, thus, in financial modeling The random increments Xtj+1 − Xtj , j ∈ {0, 1, , n−1}, of these processes are independent for any sequence of time instants t0 < t1 < < tn in [0, ∞) for all n ∈ N If t0 = 0 is the smallest time instant, then the initial value X0 and the random increment Xtj − X0 for any other tj ∈ [0, ∞) are also required to be independent... features of advanced models in many areas of application with uncertainties are often event-driven In finance and insurance one has to deal with events such as corporate defaults, operational failures or insured accidents By analyzing time series of historical data, such as prices and other financial quantities, many authors have argued in the area of finance for the presence of jumps, see Jorion (1988)... dynamics of financial quantities specified by stochastic differential equations (SDEs) with jumps have become increasingly popular Models of this kind can be found, for instance, in Merton (1976), Bj¨rk, Kabanov & Runggaldier o (1997), Duffie, Pan & Singleton (2000), Kou (2002), Sch¨nbucher (2003), o Glasserman & Kou (2003), Cont & Tankov (2004) and Geman & Roncoroni (2006) The areas of application of SDEs with. .. implementations of weak schemes For the approximation of the expected value of a function g of the solution X T at a final time T , there exist alternative numerical methods Under suitable conditions, the pricing function u(x, t) = E(g(X T )|X t = x) can be expressed as a solution of a partial integro differential equation (PIDE) Therefore, an approximation of the pricing function u(x, t) can be obtained by solving . Perkins For other titles in this series, go to http://www.springer.com/series/602 Eckhard Platen r Nicola Bruti-Liberati Numerical Solution of Stochastic Differential Equations with Jumps in Finance Eckhard. 34 1.6 StochasticDifferentialEquations 38 1.7 LinearSDEs 45 1.8 SDEswithJumps 53 1.9 ExistenceandUniquenessofSolutionsofSDEs 57 1.10 Exercises . . . . . 59 2 Exact Simulation of Solutions of SDEs. approximation of SDEs with jumps represents the focus of the monograph. The reader learns about powerful numerical methods for the solution of SDEs with jumps. These need to be implemented with care.