## On the splitting-up method and stochastic partial differential equations.(English)Zbl 1028.60058

The authors consider stochastic partial differential equations essentially (generalisations are also given) of the form $du(t,x) = Lu(t,x)dt + M_ku(t,x) \circ dw_t^k, \quad x \in \mathbb{R}^d, \;t\in [0,T], \tag{1}$ and some initial data $$u_0 = u_0(x)$$. Here $$w_t^k$$ are $$k$$ independent one-dimensional Wiener processes and equation (1) is to be interpreted in the Stratonovich sense. The $$M_k$$’s are first-order operators and $$L$$ is a second-order elliptic operator, all acting on functions defined on $$R^d$$. The authors propose and analyse splitting-up methods for the numerical approximation of the solution of (1). By stretching out the time-scale in a certain way they can reformulate the splitting-up method as a stochastic equation and apply stochastic calculus for their proofs (previous proofs relied on semigroup theory). One of their theorems says that for each $$T>0$$ and $$p>0$$ there is a constant $$N$$ such that $$E \max_{t \in T_n} \|u_n(t) -u(t)\|_0^p \leq N/h^p$$ for all $$n>1$$, where $$u_n(t)$$ is the numerical approximation to (1), $$E$$ the expectation, $$T_n$$ a grid on $$[0,T]$$ and $$\|.\|_0$$ the usual $$L_2$$-norm on $$R^d$$. Auxiliary results, the proof of the above theorem, as well as generalisations of it are presented in Sections 2 to 5. The last section provides an application to nonlinear filtering.

### MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 65C30 Numerical solutions to stochastic differential and integral equations 35R60 PDEs with randomness, stochastic partial differential equations 93E25 Computational methods in stochastic control (MSC2010) 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 93E11 Filtering in stochastic control theory
Full Text:

### References:

  BENSOUSSAN, A., GLOWINSKI, R. and RASCANU, A. (1989). Approximation of Zakai · Zbl 1055.35002  equation by the splitting-up method. Stochastic Sy stems and Optimization (Warsaw 1988). Lecture Notes in Control Inform. Sci. 136 257-265. Springer, New York.  BENSOUSSAN, A., GLOWINSKI, R. and RASCANU, A. (1992). Approximation of some stochastic differential equations by the splitting up method. Appl. Math. Optim. 25 81- 106. · Zbl 0745.65089  FLORCHINGER, P. and LE GLAND, F. (1991). Time-discretization of the Zakai equation for diffusion processes observed in correlated noise. Stochastics Stochastics Rep. 35 233- 256. · Zbl 0729.60036  GERMANI, A. and PICCIONI, M. (1988). Semi-discretization of stochastic partial differential equations on Rd by a finite element technique. Stochastics 23 131-148. · Zbl 0641.60051  GRECKSCH, W. and KLOEDEN, P. E. (1996). Time-discretized Galerkin approximations of parabolic stochastic PDEs. Bull. Austral. Math. Soc. 54 79-85. · Zbl 0880.35143  GYÖNGY, I. and KRy LOV, N. (1982). On stochastic equations with respect to semimartingales II. Itô formula in Banach spaces. Stochastics 6 153-174. · Zbl 0481.60060  GYÖNGY, I. and KRy LOV, N. (1992). Stochastic partial differential equations with unbounded coefficients and applications III. Stochastics Stochastics Rep. 40 75-115. · Zbl 0791.60045  GYÖNGY, I. and NUALART, D. (1997). Implicit scheme for stochastic parabolic partial differential equations driven by space-time white noise. Potential Anal. 7 725-757. · Zbl 0893.60033  ITO, K. and ROZOVSKII, B. (2000). Approximation of the Kushner equation for nonlinear filtering. SIAM J. Control Optim. 38 893-915. · Zbl 0952.93126  KRy LOV, N. (1995). Introduction to the Theory of Diffusion Processes. Amer. Math. Soc., Providence, RI. · Zbl 0844.60050  KRy LOV, N. and ROZOVSKII, B. (1981). Stochastic evolution equations. J. Soviet Math. 16 1233-1277. · Zbl 0462.60060  KRy LOV, N. and ROZOVSKII, B. (1986). On the characteristics of degenerate second order parabolic Itô equations. J. Soviet Math. 32 336-348. · Zbl 0584.60068  LOTOTSKY, S. V. (1996). Problems in statistics of stochastic differential equations. Ph.D. dissertation, Univ. Southern California.  NAGASE, N. (1995). Remarks on nonlinear stochastic partial differential equations: An application of the splitting-up method. SIAM J. Control Optim. 33 1716-1730. · Zbl 0845.60062  ROZOVSKII, B. (1987). On the kinematic dy namo problem in random flow. In Probability Theory and Mathematical Statistics 2 509-516. VNU, Utrecht.  ROZOVSKII, B. (1990). Stochastic Evolution Sy stems: Linear Theory and Applications to Nonlinear Filtering. Kluwer, Dordrecht.  YOO, H. (1998). An analytic approach to stochastic differential equations and its applications. Ph.D. dissertation, Univ. Minnesota.  MINNEAPOLIS, MINNESOTA 55455 E-MAIL: kry lov@math.umn.edu
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.