On explicit local solutions of Ito diffusions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:473

Article

Kouritzin, Michael A.¹  Remillard, Bruno² 

[1] Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada

[2] HEC Montreal, Serv Enseignement Methodes Quantitat Gest, Montreal, PQ H3T 2A7, Canada

关键词: Diffeomorphism;    Ito processes;    Explicit solutions;   

DOI  :  10.1016/j.jmaa.2018.12.067

来源: Elsevier

【 摘 要 】

Strong solutions of p-dimensional stochastic differential equations dX(t) = b(X-t, t)dt+sigma(X-t, t)dW(t), X-S = x that can be represented locally in explicit simulation form X-t = phi(x,s) (integral(t)(s) V(s,u)dW(u,t)) are considered. Here; W is a multidimensional Brownian motion; u -> V-s,V-u,V- phi(x,s) are continuous functions; and b, sigma, phi(x,s) are locally continuously differentiable. The following three-way equivalence is established: 1) There exists such a representation from all starting points (x, s), 2) V-s,V-u,V- phi(x,s) satisfies a set differential equations, and 3) b, a satisfy commutation relations. (For generality, the function V-s,V-t is allowed to depend upon phi(x,s) via V-s,V-t = U-s,U-t phi(x,s) for some operators U-s,U-t.) Moreover, construction theorems, based on a diffeomorphism between the solutions X and the strong solutions to a simpler Ito integral equation, with a possible deterministic component, are given. Finally, motivating examples are provided and its importance in simulation methods, including sequential Monte Carlo, financial risk assessment and path-dependent option pricing, is explained. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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