Question

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Obtain the solution of the SDE \[ d X_{t}=\left(\frac{1}{4}-X_{t}\right) d t+\sqrt{X_{t}} d W_{t}, \quad X_{0}=x \geq 0 . \]

Answer 1

The student's question is about solving a stochastic differential equation with given initial conditions, which typically involves using Ito's Lemma and verifying the solution through time derivatives and substitution.

Step-by-step explanation:

The student is asking to solve a stochastic differential equation (SDE) defined by the expression , with the initial condition
`dX_t = (1/4 - X_t)dt + √X_t dW_t`. In solving this type of equation, one would typically apply Ito's Lemma for stochastic calculus which helps in finding the differential of functions of stochastic processes. To verify whether a function is the solution to the SDE, one must compute the first and second time derivatives of the proposed solution, and substitute them back into the SDE to check for consistency. This process ensures the proposed solution indeed satisfies the SDE. It's important to note that such solutions usually involve stochastic integration and may require advanced methods from stochastic analysis.