전형적인 확률미분방정식들의 해

전형적인 확률미분방정식들의 해

Solution to Typical SDEs

방정식 1

(G) General Form: $$ d X_{t} = f \left( t , X_{t} \right) dt + g \left( t , X_{t} \right) d W_{t} $$

  • (L) Linear: $\begin{cases} f \left( t , X_{t} \right) = a_{t} + b_{t} X_{t} \\ g \left( t , X_{t} \right) = c_{t} + e_{t} X_{t} \end{cases}$ $$ d X_{t} = \left( a_{t} + b_{t} X_{t} \right) dt + \left( c_{t} + e_{t} X_{t} \right) d W_{t} $$
    • (LH) Homogeneous: $a_{t} = c_{t} = 0$ $$ d X_{t} = b_{t} X_{t} dt + e_{t} X_{t} d W_{t} $$
    • (LNS) in Narrow Sense: $e_{t} = 0$ $$ d X_{t} = \left( a_{t} + b_{t} X_{t} \right) dt + c_{t} d W_{t} $$
  • (A) Autonomous: $\begin{cases} f \left( t , X_{t} \right) = f \left( X_{t} \right) \\ g \left( t , X_{t} \right) = g \left( X_{t} \right) \end{cases}$ $$ d X_{t} = f \left( X_{t} \right) dt + g \left( X_{t} \right) dW_{t} $$
    • (AL) Linear: $\begin{cases} f \left( X_{t} \right) = a + b X_{t} \\ g \left( X_{t} \right) = c + e X_{t} \end{cases}$ $$ d X_{t} = \left( a + b \right) dt + \left( c + e X_{t} \right) d W_{t} $$
    • (ALH) Homogeneous: $a = c = 0$ $$ d X_{t} = b X_{t} dt + e X_{t} d W_{t} $$
    • (ALHS) in Narrow Sense: $e = 0$ $$ d X_{t} = \left( a + b X_{t} \right) X_{t} dt + c d W_{t} $$

확률미분방정식 중에서도 선형, 동차, 자율 확률미분방정식들은 다음과 같이 해가 알려져있다.

솔루션

  • (L) Linear: $$ X_{t} = \Phi (t) \left[ X_{0} + \int_{0}^{t} \left( a_{s} - e_{s} c_{s} \right) \Phi_{s}^{-1} ds + \int_{0}^{t} c_{s} \Phi_{s}^{-1} d W_{s} \right] $$ 여기서 $\Phi$ 와 $\Phi_{s}^{-1}$ 는 다음과 같다. $$ \begin{align*} \Phi(t) =& \exp \left( \int_{0}^{t} e_{s} d W_{s} + \int_{0}^{t} \left( b_{s} - {{ 1 } \over { 2 }} e_{s}^{2} \right) ds \right) \\ \Phi_{s}^{-1} =& \exp \left( - \int_{0}^{s} e_{u} d W_{u} - \int_{0}^{s} \left( b_{u} - {{ 1 } \over { 2 }} e_{u}^{2} \right) du \right) \end{align*} $$
  • (LH) Homogeneous: $$ X_{t} = X_{0} \exp \left( \int_{0}^{t} \left( b_{s} - {{ 1 } \over { 2 }} e_{s}^{2} \right) ds + \int_{0}^{t} e_{s} d W_{s} \right) $$
  • (LNS) in Narrow Sense: $$ X_{t} = e^{\int_{0}^{t} b_{s} ds} \left[ X_{0} + \int_{0}^{t} e^{\int_{0}^{s} b_{u} du} as ds + \int_{0}^{t} e^{- \int_{0}^{s} b_{u} du} cs d W_{s} \right] $$
  • (AL) Linear: $$ X_{t} = \exp \left[ e \int_{0}^{t} d W_{s} + \left( b - {{ 1 } \over { 2 }} e^{2} \right) \int_{0}^{t} ds \right] \cdot \left[ X_{0} + (a + ec) \int_{0}^{t} \Phi_{s}^{-1} ds + c \int_{0}^{t} \Phi_{s}^{-1} d W_{s} \right] $$ 여기서 $\Phi_{s}^{-1}$ 는 다음과 같다. $$ \Phi_{s}^{-1} = \exp \left( - e \int_{0}^{s} d W_{u} - \left( b_{u} - {{ 1 } \over { 2 }} e^{2} \right) \int_{0}^{s} du \right) $$
  • (ALH) Homogeneous: $$ X_{t} = X_{0} \exp \left( \int_{0}^{t} b_{s} ds + \int_{0}^{t} e_{s} d W_{s} - {{ 1 } \over { 2 }} \int_{0}^{t} e_{s}^{2} ds \right) $$
  • (ALHS) in Narrow Sense: $$ X_{t} = X_{0} e^{bt} + a \int_{0}^{t} e^{b} (t-s) ds + c \int_{0}^{t} e^{b(t-s)} d W_{s} $$

  1. Panik. (2017). Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling: p146. ↩︎

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