分野別セミナー

確率微分方程式の解の大偏差原理の問題は，数理ファイナンス分野への応用の観点から重要な問題である．本講演では，確率積分が駆動する一次元確率微分方程式の大偏差原理について得られた結果を紹介する．当結果は，確率積分の大偏差原理と（通常の）ラフパス理論を組み合わせることで証明できる．まず初めに．α-Uniformly
Exponentially Tightness
と呼ばれる新しい概念に着目することで，確率積分の大偏差原理が証明できることを述べる．その後，ラフパス理論を用いることで確率積分が駆動する一次元確率微分方程式の大偏差原理が証明できることを述べる．

The large deviation principle for stochastic differential equations is
important from the viewpoint of applications in mathematical finance. In
this talk, I will present results on the large deviation principle for
one-dimensional stochastic differential equations driven by stochastic
integrals. Our results can be proved by combining the large deviation
principle of stochastic integrals and rough path theory. First of all,
we will show that the large deviation principle for stochastic integrals
can be proved by focusing on a new concept called $\alpha$-Uniformly
Exponentially Tightness. Then, we will show that rough path theory can
be applicable to prove the large deviation principle for one-dimensional
stochastic differential equations driven by stochastic integrals.