Stochastic Differential Equations & Ito's Lemma
Stochastic Differential Equations & Ito's Lemma
Because Brownian paths have infinite variation, classical calculus fails: we cannot define $\int f\,dW_t$ as a Riemann-Stieltjes integral. Ito's stochastic integral extends integration to such integrands using an $L^2$ isometry and leads to a calculus with an extra second-order correction term. The result — Ito's lemma — is the stochastic chain rule and the cornerstone of quantitative finance and stochastic control.
A stochastic differential equation (SDE) $dX_t=\mu(t,X_t)\,dt+\sigma(t,X_t)\,dW_t$ describes the evolution of $X_t$ under drift $\mu$ and diffusion $\sigma$. Under Lipschitz and linear growth conditions on $\mu,\sigma$, a unique strong solution exists.
Ito Integral & Isometry
For adapted $H\in L^2([0,T]\times\Omega)$: $$\int_0^T H_s\,dW_s = \lim_{n\to\infty}\sum_k H_{t_k}(W_{t_{k+1}}-W_{t_k})$$ in $L^2$. Ito isometry: $E\!\left[\left(\int_0^T H_s\,dW_s\right)^2\right]=E\!\left[\int_0^T H_s^2\,ds\right].$ The Ito integral is a martingale.
Ito's Lemma
Let $X_t$ satisfy $dX_t=\mu_t\,dt+\sigma_t\,dW_t$ and $f\in C^{1,2}$. Then: $$df(t,X_t)=\left(\partial_t f + \mu_t\partial_x f + \tfrac{1}{2}\sigma_t^2\partial_{xx}f\right)dt + \sigma_t\partial_x f\,dW_t.$$ The $\frac{1}{2}\sigma^2 f''$ term arises because $dW_t^2=dt$ (quadratic variation rule).
Example 1 — GBM via Ito
Let $S_t$ satisfy $dS_t=\mu S_t\,dt+\sigma S_t\,dW_t$. Set $f(x)=\ln x$. Ito's lemma: $d(\ln S_t)=(\mu-\sigma^2/2)\,dt+\sigma\,dW_t$. Integrating: $S_t=S_0\exp\!((\mu-\sigma^2/2)t+\sigma W_t)$.
Example 2 — Ornstein-Uhlenbeck Process
$dX_t=-\theta X_t\,dt+\sigma\,dW_t$. By integrating factor $e^{\theta t}$: $X_t=X_0 e^{-\theta t}+\sigma\int_0^t e^{-\theta(t-s)}dW_s$. Thus $X_t\sim\mathcal{N}(X_0 e^{-\theta t},\,\frac{\sigma^2}{2\theta}(1-e^{-2\theta t}))$. Stationary distribution: $\mathcal{N}(0,\sigma^2/(2\theta))$.
Practice
- Apply Ito's lemma to $f(W_t)=W_t^3$ and identify the drift and diffusion.
- Derive the SDE for $Y_t=e^{W_t}$ and confirm it is a local martingale.
- State the existence-uniqueness theorem conditions for strong SDE solutions.
Show Answer Key
1. With $f(x)=x^3$: $f'=3x^2$, $f''=6x$. Itô: $d(W_t^3)=3W_t^2\,dW_t+3W_t\,dt$. Drift $=3W_t$, diffusion $=3W_t^2$.
2. $Y_t=e^{W_t}$. Itô: $dY_t=e^{W_t}\,dW_t+\tfrac{1}{2}e^{W_t}\,dt=Y_t\,dW_t+\tfrac{1}{2}Y_t\,dt$. It has a positive drift so is a strict local martingale (but $E[Y_t]=e^{t/2}\neq Y_0=1$ shows it is not a true martingale).
3. If $b(t,X_t)$ and $\sigma(t,X_t)$ satisfy global Lipschitz and linear growth conditions in $X$ uniformly in $t$, then the SDE $dX_t=b\,dt+\sigma\,dW_t$ has a unique strong solution.