No abstract

A class of (possibly) degenerate integro-differential equations of parabolic type is considered, which includes the Kolmogorov equations for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces and in Sobolev-Slobodeckij spaces. Generalisations to stochastic integro-differential equations, arising in filtering theory of jump diffusions, will be given in a forthcoming paper.

A class of (possibly) degenerate integro-differential equations of parabolic type is considered, which includes the Kolmogorov equations for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces and in Sobolev-Slobodeckij spaces. Generalisations to stochastic integro-differential equations, arising in filtering theory of jump diffusions, will be given in a forthcoming paper.

A class of (possibly) degenerate stochastic integro-differential equations of parabolic type is considered, which includes the Zakai equation in nonlinear filtering for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces.2010 Mathematics Subject Classification. Primary 45K05, 60H15, 60H20, 60J75; Secondary 35B65.Its derivatives in x ∈ R d up to order max{ m , 3} exist and are continuous in x ∈ R d such that |D k ξ| ≤ξ k = 0, 1, 2, ..., max{ m , 3} :=m for all (ω, t, x, z) ∈ Ω × H T × Z. Moreover, K −1 ≤ | det(I + θDξ t,z (x))| for all (ω, t, x, z, θ) ∈ Ω × H T × Z × [0, 1], where I is the d × d identity matrix, and Dξ denotes the Jacobian matrix of ξ in x ∈ R d . Assumption 2.3. The function η = (η i ) maps Ω × [0, T ] × R d × Z into R d such that Assumption 2.2 holds with η andη in place of ξ andξ, respectively.

In this paper we introduce a multilevel Picard approximation algorithm for semilinear parabolic partial integro-differential equations (PIDEs). We prove that the numerical approximation scheme converges to the unique viscosity solution of the PIDE under consideration. To that end, we derive a Feynman-Kac representation for the unique viscosity solution of the semilinear PIDE, extending the classical Feynman-Kac representation for linear PIDEs. Furthermore, we show that the algorithm does not suffer from the curse of dimensionality, i.e. the computational complexity of the algorithm is bounded polynomially in the dimension d and the prescribed reciprocal of the accuracy ε.Key words and phrases. Multilevel Picard approximation, Nonlinear PIDE, overcoming the curse of dimensionality, Monte Carlo methods, Feynman-Kac representation.Financial support by the Nanyang Assistant Professorship Grant (NAP Grant) Machine Learning based Algorithms in Finance and Insurance is gratefully acknowledged.