Symmetry in nonlinear stochastic dynamical systems and its applications
Project/Area Number  08640300 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Nagoya City University 
Principal Investigator 
MISAWA Tetsuya Nagoya City Univ., Fact.of Economics, Associate Prof., 経済学部, 助教授 (10190620)

CoInvestigator(Kenkyūbuntansha) 
HASHIMOTO Yoshiaki Nagoya City Univ., Inst.of Natural Sci., Prof., 自然科学研究教育センター, 教授 (50106259)
MIYAHARA Yoshio Nagoya City Univ., Fact.of Economics, Prof., 経済学部, 教授 (20106256)

Project Fiscal Year 
1995 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1996 : ¥1,100,000 (Direct Cost : ¥1,100,000)

Keywords  nonlinear / stochastic systems / symmetry & conserved quantities / similarity method / mathematical finance / martingale measure / relative entropy / wavelet analysis / 非線形確率系 / 対称性 / 保存量 / 簡約化 / 数理ファイナンス / マルチンゲル測度 / 相対エントロピー / ウェーブレット解析 
Research Abstract 
The present study focuses on the theory of symmetry for stochastic nonlinear dynamical systems described by stochastic differential equations. Here symmetry means an oneparameter continuous transformation which leaves the stochastic system invariant. Within the framework, the following results are obtained. 1) A method for deriving conserved quantities from symmetry is developed, and thereby the new conserved quantities are obtained for the nonlinear stochastic systems. 2) The similarity method is formulated to stochastic systems ; that is, if a stochastic dynamical system admits symmetry, it follows that the order of stochastic equations describing the system can be reduced. It is examined that the method is useful to analyze stochastic nonlinear systems. As the related topics, numerical simulations of a stochastic Kaldor business cycle model, which is a typical example of stochastic nonlinear system, and stochastic analysis of the pricing problem of contingent claims are treated. In the first topic, the numerical results indicate that noise in the model may not only obscure the underlying dynamical structures, but also reveal the hidden structures, for example, the chaotic attractors near a window of chaos or the periodic attractors near a small chaotic parameter region. In the second topic, an equivalent martingale mesure for the probability measure assigned to the price process of stocks, which may be regarded as a genaral stodhastic dynamical system, plays an important role to determine the price of contingent claims. If the market is incomplete, there are many equivalent martingale measures. Hence the minimization principle of relative entropy is adopted for a criterion of reasonable martingale measure ; the obtained measure is called the canonical martingale measure (CMM). The existence of CMM and the relations between CMM and the minimal martingale measure are investgated.

Report
(3results)
Research Output
(12results)