This section discusses the solutions to stochastic differential equations (SDEs) of the form: \[ \frac{dX_t}{dt} = b(t, X_t) + \sigma(t, X_t)W_t, \quad b(t, x) \in \mathbf{R}, \sigma(t, x) \in \mathbf{R} \] where \( W_t \) is 1-dimensional "white noise". The Itô interpretation of this SDE is given by the stochastic integral equation: \[ X_t = X_0 + \int_0^t b(s, X_s)ds + \int_0^t \sigma(s, X_s)dB_s \] or in differential form: \[ dX_t = b(t, X_t)dt + \sigma(t, X_t)dB_t \] The key to solving many SDEs is the Itô formula. The section begins by addressing how to solve such equations through simple examples, specifically focusing on the population growth model: \[ \frac{d N_{t}}{d t}=a_{t} N_{t}, \quad N_{0} \text { given } \] where \( a_t = r + \alpha W_t \) and \( W_t \) is white noise. By assuming \( r \) is constant, the equation is transformed into: \[ d N_{t}=r N_{t} d t+\alpha N_{t} d B_{t} \] Using the Itô formula for the function \( g(t, x) = \ln x \), the solution is derived as: \[ \ln \frac{N_{t}}{N_{0}}=(r-\frac{1}{2} \alpha^{2}) t+\alpha B_{t} \] Thus, the solution is: \[ N_{t}=N_{0} \exp ((r-\frac{1}{2} \alpha^{2}) t+\alpha B_{t}) \] For comparison, the Stratonovich interpretation would yield a different solution: \[ \bar{N}_{t}=N_{0} \exp (r t+\alpha B_{t}) \] Both solutions belong to the class of processes of the form: \[ X_{t}=X_{0} \exp (\mu t+\alpha B_{t}) \quad(\mu, \alpha \text { constants }) \]This section discusses the solutions to stochastic differential equations (SDEs) of the form: \[ \frac{dX_t}{dt} = b(t, X_t) + \sigma(t, X_t)W_t, \quad b(t, x) \in \mathbf{R}, \sigma(t, x) \in \mathbf{R} \] where \( W_t \) is 1-dimensional "white noise". The Itô interpretation of this SDE is given by the stochastic integral equation: \[ X_t = X_0 + \int_0^t b(s, X_s)ds + \int_0^t \sigma(s, X_s)dB_s \] or in differential form: \[ dX_t = b(t, X_t)dt + \sigma(t, X_t)dB_t \] The key to solving many SDEs is the Itô formula. The section begins by addressing how to solve such equations through simple examples, specifically focusing on the population growth model: \[ \frac{d N_{t}}{d t}=a_{t} N_{t}, \quad N_{0} \text { given } \] where \( a_t = r + \alpha W_t \) and \( W_t \) is white noise. By assuming \( r \) is constant, the equation is transformed into: \[ d N_{t}=r N_{t} d t+\alpha N_{t} d B_{t} \] Using the Itô formula for the function \( g(t, x) = \ln x \), the solution is derived as: \[ \ln \frac{N_{t}}{N_{0}}=(r-\frac{1}{2} \alpha^{2}) t+\alpha B_{t} \] Thus, the solution is: \[ N_{t}=N_{0} \exp ((r-\frac{1}{2} \alpha^{2}) t+\alpha B_{t}) \] For comparison, the Stratonovich interpretation would yield a different solution: \[ \bar{N}_{t}=N_{0} \exp (r t+\alpha B_{t}) \] Both solutions belong to the class of processes of the form: \[ X_{t}=X_{0} \exp (\mu t+\alpha B_{t}) \quad(\mu, \alpha \text { constants }) \]