Stochastic differential equations (SDEs) describe the evolution of systems subject to random influences. The equation $ \frac{d X_{t}}{d t}=b(t,X_{t})+\sigma(t,X_{t})W_{t} $, where $ W_{t} $ is white noise, is interpreted via Itô calculus as a stochastic integral equation. This leads to the differential form $ d X_{t}=b(t,X_{t})d t+\sigma(t,X_{t})d B_{t} $, where $ B_{t} $ is a Brownian motion. The key to solving SDEs is Itô's formula, which allows the transformation of functions of stochastic processes into differential equations. An example is the population growth model $ \frac{d N_{t}}{d t}=a_{t}N_{t} $, where $ a_{t}=r_{t}+\alpha W_{t} $. Assuming $ r_{t}=r $, the Itô interpretation gives $ d N_{t}=r N_{t}d t+\alpha N_{t}d B_{t} $, which can be transformed using Itô's formula to find the solution $ N_{t}=N_{0}\exp((r-\frac{1}{2}\alpha^{2})t+\alpha B_{t}) $. In contrast, the Stratonovich interpretation would yield $ \overline{N}_{t}=N_{0}\exp(r t+\alpha B_{t}) $. The solutions to such equations are processes of the form $ X_{t}=X_{0}\exp(\mu t+\alpha B_{t}) $, where $ \mu $ and $ \alpha $ are constants. These solutions highlight the importance of the Itô interpretation in stochastic calculus, as it accounts for the non-differentiable nature of Brownian motion. The study of existence and uniqueness of solutions to SDEs is an important topic, which will be addressed in Section 5.2.Stochastic differential equations (SDEs) describe the evolution of systems subject to random influences. The equation $ \frac{d X_{t}}{d t}=b(t,X_{t})+\sigma(t,X_{t})W_{t} $, where $ W_{t} $ is white noise, is interpreted via Itô calculus as a stochastic integral equation. This leads to the differential form $ d X_{t}=b(t,X_{t})d t+\sigma(t,X_{t})d B_{t} $, where $ B_{t} $ is a Brownian motion. The key to solving SDEs is Itô's formula, which allows the transformation of functions of stochastic processes into differential equations. An example is the population growth model $ \frac{d N_{t}}{d t}=a_{t}N_{t} $, where $ a_{t}=r_{t}+\alpha W_{t} $. Assuming $ r_{t}=r $, the Itô interpretation gives $ d N_{t}=r N_{t}d t+\alpha N_{t}d B_{t} $, which can be transformed using Itô's formula to find the solution $ N_{t}=N_{0}\exp((r-\frac{1}{2}\alpha^{2})t+\alpha B_{t}) $. In contrast, the Stratonovich interpretation would yield $ \overline{N}_{t}=N_{0}\exp(r t+\alpha B_{t}) $. The solutions to such equations are processes of the form $ X_{t}=X_{0}\exp(\mu t+\alpha B_{t}) $, where $ \mu $ and $ \alpha $ are constants. These solutions highlight the importance of the Itô interpretation in stochastic calculus, as it accounts for the non-differentiable nature of Brownian motion. The study of existence and uniqueness of solutions to SDEs is an important topic, which will be addressed in Section 5.2.