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전형적인 확률미분방정식들의 해 📂확률미분방정식

전형적인 확률미분방정식들의 해

방정식 1

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

  • (L) Linear: {f(t,Xt)=at+btXtg(t,Xt)=ct+etXt\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} dXt=(at+btXt)dt+(ct+etXt)dWt 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: at=ct=0a_{t} = c_{t} = 0 dXt=btXtdt+etXtdWt d X_{t} = b_{t} X_{t} dt + e_{t} X_{t} d W_{t}
    • (LNS) in Narrow Sense: et=0e_{t} = 0 dXt=(at+btXt)dt+ctdWt d X_{t} = \left( a_{t} + b_{t} X_{t} \right) dt + c_{t} d W_{t}
  • (A) Autonomous: {f(t,Xt)=f(Xt)g(t,Xt)=g(Xt)\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} dXt=f(Xt)dt+g(Xt)dWt d X_{t} = f \left( X_{t} \right) dt + g \left( X_{t} \right) dW_{t}
    • (AL) Linear: {f(Xt)=a+bXtg(Xt)=c+eXt\begin{cases} f \left( X_{t} \right) = a + b X_{t} \\ g \left( X_{t} \right) = c + e X_{t} \end{cases} dXt=(a+b)dt+(c+eXt)dWt d X_{t} = \left( a + b \right) dt + \left( c + e X_{t} \right) d W_{t}
    • (ALH) Homogeneous: a=c=0a = c = 0 dXt=bXtdt+eXtdWt d X_{t} = b X_{t} dt + e X_{t} d W_{t}
    • (ALHS) in Narrow Sense: e=0e = 0 dXt=(a+bXt)Xtdt+cdWt 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: Xt=Φ(t)[X0+∫0t(as−escs)Φs−1ds+∫0tcsΦs−1dWs] 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 Φ\Phi 와 Φs−1\Phi_{s}^{-1} 는 다음과 같다. Φ(t)=exp⁡(∫0tesdWs+∫0t(bs−12es2)ds)Φs−1=exp⁡(−∫0seudWu−∫0s(bu−12eu2)du) \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: Xt=X0exp⁡(∫0t(bs−12es2)ds+∫0tesdWs) 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: Xt=e∫0tbsds[X0+∫0te∫0sbuduasds+∫0te−∫0sbuducsdWs] 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: Xt=exp⁡[e∫0tdWs+(b−12e2)∫0tds]⋅[X0+(a+ec)∫0tΦs−1ds+c∫0tΦs−1dWs] 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 Φs−1\Phi_{s}^{-1} 는 다음과 같다. Φs−1=exp⁡(−e∫0sdWu−(bu−12e2)∫0sdu) \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: Xt=X0exp⁡(∫0tbsds+∫0tesdWs−12∫0tes2ds) 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: Xt=X0ebt+a∫0teb(t−s)ds+c∫0teb(t−s)dWs 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. ↩︎