The uploaded image contains mathematical exercises related to stochastic processes, focusing on topics like Brownian motion, Itô calculus, and stochastic differential equations (SDEs). Let's go through the exercises step by step:

Exercise I

Calculate $d(e^{\alpha B_t})$ and $E(e^{\alpha B_t})$.

This involves applying Itô's Lemma to the exponential of a Brownian motion $B_t$:

1. Itô's Lemma for $f(t, B_t) = e^{\alpha B_t}$ gives: $df(B_t) = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dB_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} (dB_t)^2.$

   Substituting $f(B_t) = e^{\alpha B_t}$: $d(e^{\alpha B_t}) = \alpha e^{\alpha B_t} dB_t + \frac{1}{2} \alpha^2 e^{\alpha B_t} dt.$

2. For the expectation $E(e^{\alpha B_t})$:

   • The result follows from the Gaussian distribution of $B_t$: $B_t \sim N(0, t)$.
   • Using $E(e^{\alpha B_t}) = e^{\frac{1}{2} \alpha^2 t}$.

Exercise II

Calculate $dY_t$ for $Y_t = e^{2B_t} \int_0^t \cos(s) \, ds$ in two ways.

1. Direct Differentiation: Apply the product rule: $dY_t = d\left(e^{2B_t}\right) \int_0^t \cos(s) \, ds + e^{2B_t} d\left(\int_0^t \cos(s) \, ds\right).$ Use $d(e^{2B_t})$ from Itô's Lemma and $d\left(\int_0^t \cos(s) \, ds\right) = \cos(t) dt$.

2. Itô's Lemma on $Y_t$: Rewrite $Y_t = e^{2B_t}g(t)$, with $g(t) = \int_0^t \cos(s) \, ds$. Then: $dY_t = \frac{\partial Y_t}{\partial t} dt + \frac{\partial Y_t}{\partial B_t} dB_t + \frac{1}{2} \frac{\partial^2 Y_t}{\partial B_t^2} (dB_t)^2.$

Exercise III

Solve the SDE: $dX_t = a(t) X_t dt + b(t) dt + c(t) dB_t.$

1. Homogeneous Solution:

Set $b = c = 0$ and solve $dX_t = a(t)X_t dt$. The solution is: $X_t = X_0 e^{\int_0^t a(s) ds}.$

2. Transformation:

Let $Y(t) = e^{-\int_0^t a(s) ds} X_t$, simplifying the SDE using Itô's Lemma.

3. Integral Form:

Derive the integral form of $Y_t$.

4. Solve $dX_t = \frac{-X_t}{1-t} dt + dB_t$:

• This is a specific case of the general SDE.
• Use Itô's Lemma and calculate the variance of $Y_t$.

Exercise IV

For stopping times $S$ and $T$ of the stochastic process $(X_n)$: Prove that $S + T$, $S \wedge T$, and $S \vee T$ are stopping times.

1. Use the definition of stopping times, considering the information up to time $t$.
2. For each operation, verify whether it satisfies the property of stopping times.

Would you like me to solve any specific part in detail? Here are related questions to explore:

1. What are the key applications of Itô's Lemma in stochastic calculus?
2. How does the solution of an SDE depend on the choice of drift and diffusion terms?
3. Why is the expectation $E(e^{\alpha B_t}) = e^{\frac{1}{2}\alpha^2t}$?
4. How can stopping times be combined to create new stopping times?
5. What is the significance of solving SDEs in finance and physics?

Tip: When working with SDEs, always confirm if the coefficients are adapted processes to ensure the solution's validity.