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4

First we show that that if $X\colon \Omega \rightarrow \mathbb{R}^n$ is normal $\mathcal{N}(m, C)$, $Y\colon \Omega \rightarrow \mathbb{R}^n$ is normal $\mathcal{N}(m^{\prime}, C^{\prime})$ and they are independent then $X+Y$ is normal $\mathcal{N}(m_1+m_2, C_1+C_2)$. Here $m$, $m^{\prime} \in \mathbb{R}^n$ and $C=[c_{jk}]$, $C^{\prime}=[c_{jk}^{\prime}]$ ...


2

We have $dX_t = dW_t - \frac{a}{2}dt$ and $d[X]_t = d[W]_t = dt$ since the deterministic piece $-at/2$ doesn't contribute to the quadratic variation. So, $$ f(X_t) = f(X_0)+\int_0^t f'(X_s)\, dW_s - \frac{a}{2} \int_0^t f'(X_s) \,ds + \frac{1}{2} \int_0^t f''(X_s) \, ds \\ = f(X_0)+\int_0^t f'(X_s)\, dW_s + \frac{1}{2} \int_0^t \big[f''(X_s) - af'(X_s) ...


2

This is in Feller volume 2, and on page 2 of this paper.


2

For any $\omega \in \Omega$, we can construct a sequence $(\Pi_n)_n=(\Pi_n(\omega))_n$ of partitions of $[0,T]$ such that the mesh size $|\Pi_n|$ tends to $0$ as $n \to \infty$ and $$\lim_{n \to \infty} \sum_{t_j \in \Pi_n} (B_{t_j}(\omega)-B_{t_{j-1}}(\omega))^2 = \infty.$$ This means in particular that $$[B,B](T)(\omega) := \sup_{\Pi} \sum_{t_j \in \Pi} ...


2

There are several definitions for the quadratic variation: For a "nice" process $(X_t)_{t \geq 0}$ ("nice" means semimartingale), the quadratic variation is defined by $$[X]_t := X_t^2-X_0^2 -2 \int_0^t X_{s-} \, dX_s.$$ For $X_t := B_t^2$, we know from Itô's formula that $$d(B_s^2)= 2 B_s \, dB_2+ \, ds$$ and $$B_t^4 = 4 \int_0^t B_s^3 \, dB_s + 6 ...


2

For every $t\geqslant0$, $(B_{t+s}^2)_{s\geqslant0}$ is distributed as $(X_s)_{s\geqslant0}$, where, for every $s\geqslant0$, $$X_s=B_t^2+2\sqrt{B_t^2}\cdot W_s+W_s^2,$$ where $(W_s)_{s\geqslant0}$ is a Brownian motion independent of $(B_u)_{0\leqslant u\leqslant t}$. Thus, indeed, $(B^2_t)_{t\geqslant0}$ is a Markov process. The discrete analogue of this ...


2

This notion is called tightness of a sequence of measures. You can apply it in probability theory with the sequence $P\#{X_n}$ of the image probabilities under the action of $X_n$: $$ P\#{X_n}(A) = P(X_n\in A) $$ You transfer the topology issues to the (metric, often polish) space $\mathcal X$ where $$ X:\Omega \to \mathcal X $$ Note also that in the ...


2

Surely not an exam proof, what is going to come. We will use Theorem 5.4.1 from Ethier and Kurtz, Markov Processes, Wiley, 1986 (EK86); the important part is quoted below. Let us first write down the generator $A_n$ of the process $X^{(n)}$ \begin{equation} A_n f(x) = \sin (nx) f'(x) + \frac{1}{2} f''(x) , x \in \mathbb{R}. \end{equation} The expected ...


2

Let $X_t=t\,B_{1/t}$. Every linear combination of the random variables $X_t$ is a linear combination of the random variables $B_t$ hence it is normal. Thus, the process $(X_t)$ is gaussian, in particular every increment $X_t-X_s$ is gaussian. More generally, let $Y_t=a(t)B_{c(t)}+b(t)$. For every functions $(a,b,c)$, the process $(Y_t)$ is gaussian.


1

As $D$ is open and $y \in D$, we have $$r := d(y,\partial D) := \inf\{z \in \partial D; |z-y| \}>0.$$ On the other hand, the boundedness of $D$ implies that we can choose $R>0$ such that $|z| \leq R$ for any $z \in \partial D$. Consequently, $$\log r \leq \log|B(T)(\omega)-y| \leq \log (R+|y|)$$ holds for any $\omega \in \Omega$. Taking expectation ...


1

Key-facts: The function $x\mapsto\|x\|^{-1}$ is harmonic on $\mathbb R^3\setminus\{0\}$ and the distribution of $\sqrt{t}\underline{X}$ is radially symmetric. Assume that $\underline{X}_0$ and $\underline{X}$ are independent (otherwise anything can happen) and note that the result is a consequence of the fact that, for every random variable $B$ whose ...


1

As mentionned by saz from Itô's lemma applied to $Y_t=B^2_t$ you get : $dY_t=finite_variation_ter.dt + 2.B_tdB_t$ so the quadratic variation is (using Itô's isometry) is equal to : $$<Y>_t=4\int_0^t B_s^2ds$$ Please note that this is a stochastic process. Best regards


1

Hint: under the risk neutral probability, the prices of the securities having an $L^2$ payoff are martingales. So you probably need to compute the Ito-differential of $$ (t,\omega)\to S^3_t e^{(2r+3σ^2)(T−t)} $$


1

As the process $(W(t))_{t\ge 0}$ has the same distribution than $(cW(\frac t{c^2}))_{t\ge 0}$, you get with $c = \sqrt t$: $$ P( W(t) > 0 ;\ \ W(2t) > 0) = P\left( cW(\frac t{c^2}) > 0 ;\ \ cW\left(2 \frac t{c^2}\right) > 0\right)\\ = P\left( W\left(\frac t{c^2}\right) > 0 ;\ \ W\left(2 \frac t{c^2}\right) > 0\right)= P( W(1) > 0 ; ...


1

Let $(P_t)_{t\ge0}$ denote a regular conditional distribution of $W$ given $T$. This means that each $P_t$ is a probability measure on $C[0,\infty)$, the set of continuous functions from $[0,\infty)$ to $\mathbb{R}$ endowed with the $\sigma$-algebra induced by the coordinate projections, such that $$ P(W\in A,T \in B) = \int_B P_t(A) d Q(t) $$ where $Q$ is ...


1

It suffices to prove that $\int_0^t \phi_s^2 \, ds$ is finite for all $t \in [0,\infty)$. This allows us to define the stochastic integral $$\int_0^t \phi(s) \, dB_s$$ for any $t \in [0,\infty)$ (as a local integral). This means that your proof is correct. In fact, the very argumentation shows that $\Lambda_{\text{loc}}^2$ contains all functions $f$ ...



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