Theory of stochastic process

Stochastic differential equations (SDEs) can model systems subject to fluctuations and have many applications in diverse research fields such as physics, chemistry, financial engineering, and biology. However, except for one-dimensional stationary cases, their solutions are rarely available even for one-dimensional systems. In our laboratory, we have developed a technique for obtaining time-dependent solutions of SDEs with the variational principle. The proposed method is called variational superposed Gaussian approximation (VSGA). Using VSGA, we can obtain accurate time-dependent solutions of systems driven by chaotic signals, which have often been solved by exhaustive Monte Carlo simulations.

確率微分方程式は物理，化学，電気工学，金融工学，生物などにおいて確率的なダイナミクスを記述するために用いられています．しかし，一次元の定常分布の場合を除いて，確率密度関数を計算することは容易ではありません．本研究室では，変分原理を用いた時間依存確率密度関数の計算方法VSGA（Variational Superposed Gaussian Approximation）を開発しました．VSGAを用いることで，今まではモンテカルロシミュレーションなどによって解かれていたカオス入力のある系の時間発展を，乱数を用いることなく正確に解くことが可能になりました．