How to prove the martingale?

Question

How to prove that the integral $\int_{0}^{+\infty}\upsilon e^{-ru}S_{u}dW_{u}^{Q}$ is a martingale under Q where $S_{t}$ is a martingale under Q and $\mathbb{E}^{Q}[\int_{0}^{+\infty}|\upsilon e^{-ru}S_{u}|^{2}dt]<\infty$? After that, prove the expected value of the integral equals 0?

Answer 1

You should take a look in a book about stochastic processes/stochastic analysis. If you would know some basics about stochastic analysis this exercise would be (more or less) trivial. Since it's obviously not that easy for you, you should read more about this topic, because it's too much to explain it in detail.

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Here are some basic definitions/theorems you should know before thinking about this exercise: Let $(W_t)_{t \geq 0}$ a Brownian motion and $(\mathcal{F}_t)_{t \geq 0}$ a admissible filtration. Let $T>0$.

Definition 1 The $\sigma$-algebra $\mathcal{P}$ defined as $$\mathcal{P} := \{\Gamma \subseteq [0,T] \times \Omega; \forall t \leq t: \Gamma \cap ([0,t] \times \Omega) \in \mathcal{B}[0,t] \times \mathcal{F}_t\}$$ is called progressive $\sigma$-algebra.

Definition 2 Let $f:[0,T] \times \Omega \to \mathbb{R}$ such that

1. $[0,T] \times \Omega \ni (t,\omega) \mapsto f(t,\omega)$ is $\mathcal{P}$-measurable (so-called progressively measurable).
2. $\mathbb{E} \left( \int_0^T |f(t,\cdot)|^2 \, dt \right) < \infty$

We denote the set of these functions by $L_T^2$.

Theorem Let $f \in L^2_T$ and define $$X_t := \int_0^t f(u) \, dW_u$$ Then $(X_t)_{t \leq T}$ is a (continuous) martingale. In particular $$\mathbb{E}X_t = \mathbb{E}\underbrace{X_0}_{0} = 0$$

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Define $f(t,\omega) := v \cdot e^{-r \cdot t} \cdot S_t(\omega)$. You have to prove $f \in L_T^2$ (then you are done by applying the theorem above). Some hints how to prove the conditions in definition 2:

1. One can show that an adapted continuous process is progressively measurable. You can apply this statement since $S$ is a continuous martingale.
2. Use the assumption $\mathbb{E} \left(\int_0^{\infty} |v \cdot e^{-r \cdot u} \cdot S_u|^2 \, du \right)< \infty$.

These definitions and the theorem are contained in "Brownian motion - An introduction to stochastic processes" - René L. Schilling/Lothar Partzsch, chapter 14.