Weak solution of Stochastic Differential Equation

• Concept of weak solution

Fokker-Plank 방정식을 만족하는 probablity density 를 가진 diffusion process가 어떤 stochastic differential equation의 해 인 경우를 생각해보자. 이는 stochastic differential equation의 weak solution과 깊이 연관되어 있다.

Concept of weak solution

Let the stochastic differential equation as follows:

\[dX_t = a(t, X_t)dt + b(t, X_t)dW_t \label{eq01:concept} \tag{1}\]

with the initial value $X_{t_0} = Y_{t_0}$ for any Wiener process $W = { W_t, t \geq 0 }$. 이때, coefficients는 strong solution을 만족한다고 가정하자 (Measurability, Lipschitz continuous, linear growth, Strong solution theorem). 그러면, for each Wiener process, there will be a solution $X$.

이러한 Solution $X​$는 the given diffusion process $Y​$에 대해 동등한 stochastic process이다. 그리고 $X​$와 $Y​$의 probablity law는 같다. 그러나 일반적으로는 **$X​$와 $Y​$의 Sample Path는 다르다. **

따라서, 위의 특성을 만족할 수 있는 Wiener process를 만드는 것이 목표가 된다.

Let $Y$ be a given difussion process on $[0, T]$ with drift $a(t, y)$ and strictly positive diffusion coefficient $b(t, y)​$

We define functions $g$ and $\bar{a}​$ by

\[g(t,y) = \int_0^y \frac{1}{b(t,x)} dx \label{eq02:concept} \tag{2}\]

and

\[\bar{a}(t,x) = \left( \frac{\partial g}{\partial t} + a \frac{\partial g}{\partial y} + \frac{1}{2} b^2 \frac{\partial^2 g}{\partial y^2} \right)(t, g^{-1}(t, z))\]

• $\eqref{eq01:concept}​$ 와 같이 한 이유는

\[dg(t, y) = \frac{1}{b(t, x)} dx \biggr\vert_{x = y} \Rightarrow \frac{dg}{dy}(t, y) = \frac{1}{b(t, x)}\]

가 되도록 하여 diffusion coefficient가 1이 되도록 하기 위함이다.

Let $y = g^{-1}(t,z)$ is the inverse of $z = g(t, y)$. Then, we define a process \(Z_t = g(t, Y_t)\)

which is a diffusion procerss with drift $\bar{a}(t,z)$ and diffusion coefficient 1 such that,

\[dZ_t = \frac{\partial g}{\partial t}dt + \frac{\partial g}{\partial y}dY_t + \frac{1}{2}\frac{\partial^2 g}{\partial y^2}(dY_t)^2.\]

Since $Y_t$ includes a drift $a(t, y)$ and diffusion coefficient $b(t, y)$, i.e.

\[dY_t = a(t, y) dt + b(t, y)dW_t\] \[dZ_t = \left( \frac{\partial g}{\partial t} + a\frac{\partial g}{\partial y} + \frac{1}{2} b^2 \frac{\partial^2 g}{\partial y^2} \right) (t, y) dt+ b\frac{\partial g}{\partial y} (t, y) dW_t \label{eq03:concept} \tag{3}\]

and a process.

\[\tilde{W}_t = Z_t - Z_0 - \int_0^t \bar{a} (S, Z_s) ds \label{eq04:concept} \tag{4}\]

which will turn out to be a **Wiener process. ** (It reuires proof).

Consequently, $\eqref{eq04:concept}​$ will be equivalent to the stochastic differential equation

\[dZ_t = \bar{a}(t, Z_t)dt + 1 d\bar{W}_t\]