Splitting Methods for SPDEs: From Robustness to Financial Engineering, Optimal Control, and Nonlinear Filtering
In this survey chapter we give an overview of recent applications of the splitting method to stochastic (partial) differential equations, that is, differential equations that evolve under the influence of noise. We discuss weak and strong approximations s
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Splitting Methods for SPDEs: From Robustness to Financial Engineering, Optimal Control, and Nonlinear Filtering Christian Bayer and Harald Oberhauser
Abstract In this survey chapter we give an overview of recent applications of the splitting method to stochastic (partial) differential equations, that is, differential equations that evolve under the influence of noise. We discuss weak and strong approximations schemes. The applications range from the management of risk, financial engineering, optimal control, and nonlinear filtering to the viscosity theory of nonlinear SPDEs.
1 Introduction The theory of (ordinary/partial) differential equations has been very successful in modeling quantities that evolve over time. Many of these quantities can be profoundly affected by stochastic fluctuations, noise, and randomness. The theory of stochastic differential equations aims for a qualitative and quantitative understanding of the effects of such stochastic perturbations. This requires insights from pure mathematics and to deal with them in practice requires us to revisit and extend classic numerical techniques. Splitting methods turn out to be especially useful since they often allow to separate the problem into a deterministic and a stochastic part.
C. Bayer Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany e-mail: [email protected] H. Oberhauser () Department of Mathematics, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford OX2 6GG, UK e-mail: [email protected] © Springer International Publishing Switzerland 2016 R. Glowinski et al. (eds.), Splitting Methods in Communication, Imaging, Science, and Engineering, Scientific Computation, DOI 10.1007/978-3-319-41589-5_15
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C. Bayer and H. Oberhauser
White Noise and Brownian Motion The arguably simplest case of such a stochastic perturbation is an ODE driven by a vector field V that is affected by noise. Let us model this perturbation by a sequence of random variables N = (Nt )t≥0 which are picked up by a vector field W , dyt = V (yt ) +W (yt ) Nt . dt Noise
Often a reasonable assumption is that N = (Nt )t≥0 is white noise, that is 1. (independence) ∀s = t, Nt and Ns are independent, 2. (stationarity) ∀t1 ≤ · · · ≤ tn the law of (Nt1 +t , · · · , Ntn +t ) does not depend on t, 3. (centered) E [Nt ] = 0, ∀t ≥ 0. Above properties imply that the trajectory t → Nt cannot be continuous, and even worse if we assume that E[Nt2 ] = 1 then (ω ,t) → Nt (ω ) is not even measurable (see [60, 41]). Putting mathematical rigor aside, let us rewrite the above differential equation as an integral equation, i.e., we work with Bt = 0t Nr dr and since integration smoothes out we expect B = (Bt )t≥0 to have nicer trajectories than N. In this case the above becomes dyt = V (yt ) dt +W (yt ) dBt resp. yt =
t 0
V (yr ) dr +
t 0
W (yr ) dBr .
(15.1)
It turns out that B = (Bt )t≥0 can be rigorously defined as a stochastic process — i.e., a collection of (ω ,t)-measurable random variables carried on some p
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