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This is a linear SDE, whose explicit solution is straightforward to obtain. Exactly analogous to linear ODEs, one first finds a general solution to the homogeneous SDE $$ \Phi_t = \exp \left( (a - \frac{1}{2} \sigma_2^2) t + \sigma_2 W_{2t} \right) $$ which is a geometric Brownian motion. Given an initial condition $x_0$, then the solution to the SDE (3) can be written as: $$ X_t = \Phi_t \left( x_0 + \sigma_1 \int_0^t \Phi_s^{-1} dW_{1s} \right) \tag{$\star$} $$ To prove this, just apply integration-by-parts for Itô processes $$ d(A_t B_t) = A_t d B_t + A_t d B_t + [ A, B ]_t $$ with $A_t=X_t$ and $B_t = \Phi_t^{-1}$ to obtain: \begin{align*} & d ( X_t \Phi_t^{-1} ) \\ & = X_t \Phi_t^{-1} \left( (-a+ \sigma_2^2) dt - \sigma_2 d W_{2 t} \right) + \Phi_t^{-1} \left( \vphantom{\frac{1}{2}} a X_t dt + \sigma_1 d W_{1t} + \sigma_2 X_t dW_{2 t} \right) - X_t \Phi_t^{-1} \sigma_2^2 dt \\ &= \sigma_1 \Phi_t^{-1} dW_{1t} \end{align*} and then integrate what remains to get ($\star$). Here we used the fact that the covariation of the two Itô processes $X_t$ and $\Phi_t^{-1}$ is $[X, \Phi^{-1}]_t =-\sigma_2^2 \int_0^t X_s \Phi_s^{-1} ds$. This follows from the fact that $W_{1t}$ and $W_{2t}$ are independent standard Brownian motions.

Note that when $\sigma_1=0$, one recovers a geometric Brownian motion, and when $\sigma_2=0$, one obtains an Ornstein-Uhlenbeck process.

This solution is adapted from more general results for linear SDEs which may be found in, e.g., Chapter 5 of Lawrence C. Evans' AMS book entitled An Introduction to Stochastic Differential Equations.

This is a linear SDE, whose explicit solution is straightforward to obtain. Exactly analogous to linear ODEs, one first finds a fundamental solution, i.e., a solution of the homogeneous version of (3) with the initial condition $X_0=1$, $$ \Phi_t = \exp \left( (a - \frac{1}{2} \sigma_2^2) t + \sigma_2 W_{2t} \right) $$ which is a geometric Brownian motion. Given an initial condition $x_0$, then the solution to (3) can be written as: $$ X_t = \Phi_t \left( x_0 + \sigma_1 \int_0^t \Phi_s^{-1} dW_{1s} \right) \tag{$\star$} $$ To prove this, just apply integration-by-parts for Itô processes $$ d(A_t B_t) = A_t d B_t + A_t d B_t + [ A, B ]_t $$ with $A_t=X_t$ and $B_t = \Phi_t^{-1}$ to obtain: \begin{align*} & d ( X_t \Phi_t^{-1} ) \\ & = X_t \Phi_t^{-1} \left( (-a+ \sigma_2^2) dt - \sigma_2 d W_{2 t} \right) + \Phi_t^{-1} \left( \vphantom{\frac{1}{2}} a X_t dt + \sigma_1 d W_{1t} + \sigma_2 X_t dW_{2 t} \right) - X_t \Phi_t^{-1} \sigma_2^2 dt \\ &= \sigma_1 \Phi_t^{-1} dW_{1t} \end{align*} and then integrate what remains to get ($\star$). Here we used the fact that the covariation of the two Itô processes $X_t$ and $\Phi_t^{-1}$ is $[X, \Phi^{-1}]_t =-\sigma_2^2 \int_0^t X_s \Phi_s^{-1} ds$. This follows from the fact that $W_{1t}$ and $W_{2t}$ are independent standard Brownian motions.

Note that when $\sigma_1=0$, one recovers a geometric Brownian motion, and when $\sigma_2=0$, one obtains an Ornstein-Uhlenbeck process.

This solution is adapted from more general results for linear SDEs which may be found in, e.g., Chapter 5 of Lawrence C. Evans' AMS book entitled An Introduction to Stochastic Differential Equations.

This is a linear SDE, whose explicit solution is straightforward to obtain. Exactly analogous to linear ODEs, one first finds a general solution to the homogeneous SDE $$ \Phi_t = \exp \left( (a - \frac{1}{2} \sigma_2^2) t + \sigma_2 W_{2t} \right) $$ which is a geometric Brownian motion. Given an initial condition $x_0$, then the solution to the SDE (3) can be written as: $$ X_t = \Phi_t \left( x_0 + \sigma_1 \int_0^t \Phi_s^{-1} dW_{1s} \right) \tag{$\star$} $$ To prove this, just apply integration-by-parts for Itô processes $$ d(A_t B_t) = A_t d B_t + A_t d B_t + [ A, B ]_t $$ with $A_t=X_t$ and $B_t = \Phi_t^{-1}$ to obtain: \begin{align*} & d ( X_t \Phi_t^{-1} ) \\ & = X_t \Phi_t^{-1} \left( -(a-\frac{1}{2} \sigma_2^2 t) dt - \sigma_2 d W_{2 t} \right) + \Phi_t^{-1} \left( \vphantom{\frac{1}{2}} a X_t dt + \sigma_1 d W_{1t} + \sigma_2 X_t dW_{2 t} \right) - \frac{1}{2} \sigma_2^2 t \\ &= \sigma_1 \Phi_t^{-1} dW_{1t} \end{align*} and then integrate what remains to get ($\star$). Here we used the fact that the covariation of the two Itô processes $X_t$ and $\Phi_t^{-1}$ is $[X, \Phi^{-1}]_t =-1/2 \sigma_2^2 t$. This follows from the fact that $W_{1t}$ and $W_{2t}$ are independent standard Brownian motions.

Note that when $\sigma_1=0$, one recovers a geometric Brownian motion, and when $\sigma_2=0$, one obtains an Ornstein-Uhlenbeck process.

This solution is adapted from more general results given in Chapter 5 of Lawrence C. Evans' AMS book entitled An Introduction to Stochastic Differential Equations.

This is a linear SDE, whose explicit solution is straightforward to obtain. Exactly analogous to linear ODEs, one first finds a general solution to the homogeneous SDE $$ \Phi_t = \exp \left( (a - \frac{1}{2} \sigma_2^2) t + \sigma_2 W_{2t} \right) $$ which is a geometric Brownian motion. Given an initial condition $x_0$, then the solution to the SDE (3) can be written as: $$ X_t = \Phi_t \left( x_0 + \sigma_1 \int_0^t \Phi_s^{-1} dW_{1s} \right) \tag{$\star$} $$ To prove this, just apply integration-by-parts for Itô processes $$ d(A_t B_t) = A_t d B_t + A_t d B_t + [ A, B ]_t $$ with $A_t=X_t$ and $B_t = \Phi_t^{-1}$ to obtain: \begin{align*} & d ( X_t \Phi_t^{-1} ) \\ & = X_t \Phi_t^{-1} \left( (-a+ \sigma_2^2) dt - \sigma_2 d W_{2 t} \right) + \Phi_t^{-1} \left( \vphantom{\frac{1}{2}} a X_t dt + \sigma_1 d W_{1t} + \sigma_2 X_t dW_{2 t} \right) - X_t \Phi_t^{-1} \sigma_2^2 dt \\ &= \sigma_1 \Phi_t^{-1} dW_{1t} \end{align*} and then integrate what remains to get ($\star$). Here we used the fact that the covariation of the two Itô processes $X_t$ and $\Phi_t^{-1}$ is $[X, \Phi^{-1}]_t =-\sigma_2^2 \int_0^t X_s \Phi_s^{-1} ds$. This follows from the fact that $W_{1t}$ and $W_{2t}$ are independent standard Brownian motions.

Note that when $\sigma_1=0$, one recovers a geometric Brownian motion, and when $\sigma_2=0$, one obtains an Ornstein-Uhlenbeck process.

This solution is adapted from more general results for linear SDEs which may be found in, e.g., Chapter 5 of Lawrence C. Evans' AMS book entitled An Introduction to Stochastic Differential Equations.

This is a linear SDE, whose pathwise unique solution is known. Let $Y_t = (a - \frac{1}{2} \sigma_2^2) t + \sigma_2 W_t$. Given an initial condition $x_0$, the solution is explicitly: $$ X_t = \exp(Y_t) \left( x_0 - \sigma_1 \sigma_2 \int_0^t \exp(-Y_s) ds + \sigma_1 \int_0^t \exp(- Y_s ) dW_s \right) $$ Note that when $\sigma_1=0$, one recovers a geometric Brownian motion, and when $\sigma_2=0$, one obtains an OU process. The general case can be checked by using Itô's formula.

More general results can be found in, e.g., Chapter 5 of Lawrence C. Evans' AMS book entitled An Introduction to Stochastic Differential Equations.

This is a linear SDE, whose explicit solution is straightforward to obtain. Exactly analogous to linear ODEs, one first finds a general solution to the homogeneous SDE $$ \Phi_t = \exp \left( (a - \frac{1}{2} \sigma_2^2) t + \sigma_2 W_{2t} \right) $$ which is a geometric Brownian motion. Given an initial condition $x_0$, then the solution to the SDE (3) can be written as: $$ X_t = \Phi_t \left( x_0 + \sigma_1 \int_0^t \Phi_s^{-1} dW_{1s} \right) \tag{$\star$} $$ To prove this, just apply integration-by-parts for Itô processes $$ d(A_t B_t) = A_t d B_t + A_t d B_t + [ A, B ]_t $$ with $A_t=X_t$ and $B_t = \Phi_t^{-1}$ to obtain: \begin{align*} & d ( X_t \Phi_t^{-1} ) \\ & = X_t \Phi_t^{-1} \left( -(a-\frac{1}{2} \sigma_2^2 t) dt - \sigma_2 d W_{2 t} \right) + \Phi_t^{-1} \left( \vphantom{\frac{1}{2}} a X_t dt + \sigma_1 d W_{1t} + \sigma_2 X_t dW_{2 t} \right) - \frac{1}{2} \sigma_2^2 t \\ &= \sigma_1 \Phi_t^{-1} dW_{1t} \end{align*} and then integrate what remains to get ($\star$). Here we used the fact that the covariation of the two Itô processes $X_t$ and $\Phi_t^{-1}$ is $[X, \Phi^{-1}]_t =-1/2 \sigma_2^2 t$. This follows from the fact that $W_{1t}$ and $W_{2t}$ are independent standard Brownian motions.

Note that when $\sigma_1=0$, one recovers a geometric Brownian motion, and when $\sigma_2=0$, one obtains an Ornstein-Uhlenbeck process.

This solution is adapted from more general results given in Chapter 5 of Lawrence C. Evans' AMS book entitled An Introduction to Stochastic Differential Equations.