📂Stochastic Differential Equations

Solutions to Typical Stochastic Differential Equations

Equations 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: $e = 0$ $$ d X_{t} = \left( a + b X_{t} \right) X_{t} dt + c d W_{t} $$

Among the stochastic differential equations, solutions to linear, homogeneous, autonomous stochastic differential equations are known as follows.

Solution

• (L) Linear: $$ X_{t} = \Phi (t) \left[ X_{0} + \int_{0}^{t} \left( a_{s} - e_{s} c_{s} \right) \Phi_{s}^{-1} ds + \int_{0}^{t} c_{s} \Phi_{s}^{-1} d W_{s} \right] $$ Here, $\Phi$ and $\Phi_{s}^{-1}$ are as follows. $$ \begin{align*} \Phi (t) =& \exp \left( \int_{0}^{t} e_{s} d W_{s} + \int_{0}^{t} \left( b_{s} - {{ 1 } \over { 2 }} e_{s}^{2} \right) ds \right) \\ \Phi_{s}^{-1} =& \exp \left( - \int_{0}^{s} e_{u} d W_{u} - \int_{0}^{s} \left( b_{u} - {{ 1 } \over { 2 }} e_{u}^{2} \right) du \right) \end{align*} $$
• (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: $$ X_{t} = \exp \left[ e \int_{0}^{t} d W_{s} + \left( b - {{ 1 } \over { 2 }} e^{2} \right) \int_{0}^{t} ds \right] \cdot \left[ X_{0} + (a + ec) \int_{0}^{t} \Phi_{s}^{-1} ds + c \int_{0}^{t} \Phi_{s}^{-1} d W_{s} \right] $$ Here, $\Phi_{s}^{-1}$ is as follows. $$ \Phi_{s}^{-1} = \exp \left( - e \int_{0}^{s} d W_{u} - \left( b_{u} - {{ 1 } \over { 2 }} e^{2} \right) \int_{0}^{s} du \right) $$
• (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: $$ X_{t} = X_{0} e^{bt} + a \int_{0}^{t} e^{b} (t-s) ds + c \int_{0}^{t} e^{b(t-s)} d W_{s} $$