# Solving SDE using Itô's lemma

I'm trying to solve stochastic differential equation: $$d X(t)=X(t) d t+d W(t)$$

Here, W(t) denotes a Brownian Motion. Comparing with Itô's lemma (with $$f\left(W_{t}, t\right) = X(t)$$), $$d X(t)=\partial_{w} X(t) d W_{t}+\frac{1}{2} \partial_{w}^{2} X(t) d t+\partial_{t} X(t) d t$$ I get two equations

a)

$$\partial_{w} X(t) = 1$$ $$\partial_{w} X(t) = 1 \Rightarrow X(t)= W + C(t)$$

b) $$\frac{1}{2} \partial_{w}^{2} X(t) +\partial_{t} X(t) = X(t)$$ substituting result from part (a) we have $$\frac{1}{2} \partial_{w}^{2} (W + C(t)) +\partial_{t} (W + C(t)) = W + C(t)$$ $$C'(t) = W + C(t)$$ which can be solved as $$C(t) = C_0e^t-W$$ The issue starts when I try to substitute this back $$X(t)= W + C_0e^t-W$$ $$X(0)= C_0e^0$$ $$X_0= C_0$$ $$X(t)= X_0e^t$$ Now equation (a) does not hold anymore, because $$\partial_{w} X(t)=\partial_{w} X_{0} e^{t}= 0$$ Anyone can point me what am I missing or what I'm solving wrong? Thanks.

## 1 Answer

After your clarifying comment, I see the following problem with your approach: You assume that the solution can be written in the form $$f(t,W_t)$$, but for this particular SDE this is not the case. So you need to use a slightly different method.

You can still use the approach of matching coefficients, but you should start with $$X_t = \alpha(t)Y(t)=\alpha(t)\left[X_0 + \int_0^t \beta(s)dW_s\right],$$ where $$\alpha$$ and $$\beta$$ are deterministic functions of time.

Now, applying Ito's Lemma we obtain $$dX_t=\alpha'(t)Y(t)dt+\alpha(t)\beta(t)dW(t)$$

Matching the coefficients we have $$\alpha'(t)Y(y)=X(t) \\ \alpha(t)\beta(t)=1$$ From the first equation, we conclude that $$\alpha(t)=e^t$$, which implies that $$\beta(t)=e^{-t}$$. Thus, $$X_t$$ can be written as $$X_t = e^t X_0 + e^t \int_0^t e^{-s}dW_s$$ As you can see X_t is not in the form of $$f(t,W_t)$$.

Alternatively, this equation can be solved by applying Ito's Lemma to $$e^{-t}X_t$$, where $$f(t,x)=e^{-t}x$$. Then write the resulting equation in the integral form and multiply both sides of it by $$e^t$$.

• I would disagree with the first part of your answer, you have just used altered notations. Itô's lemma says: $$d f\left(W_{t}, t\right)=\partial_{w} f\left(W_{t}, t\right) d W_{t}+\frac{1}{2} \partial_{w}^{2} f\left(W_{t}, t\right) d t+\partial_{t} f\left(W_{t}, t\right) d t$$ and $f\left(W_{t}, t\right) = X(t)$. I assume you mean I should write $X(W_t, t)$, but that shouldn't be an issue and all the textbooks that I see write it in the same manner. – Blade Apr 24 at 2:29
• I updated the answer based on your clarification. But to be honest, I have never seen notation $\partial_w X_t$ used for $\partial_w f(W_t,t)$ in this context. – Mdoc Apr 24 at 5:17
• Because it is a wrong notation if $\partial_\omega$ stands for $\dfrac{\partial}{\partial W_t}$. That's why Ito Lemma exists. – QFi Apr 24 at 6:08
• @QFi You can find same notation used in this document from some lecture at NYU: math.nyu.edu/faculty/goodman/teaching/StochCalc2012/notes/… look at eq.3 – Blade Apr 24 at 14:36
• @Mdoc Thanks for the update, is $$X_{t}=\alpha(t) Y(t)=\alpha(t)\left[X_{0}+\int_{0}^{t} \beta(s) d W_{s}\right]$$ a general rule or formula based off of the form of SDE? – Blade Apr 25 at 1:22