Articles of stochastic analysis

Demonstrate that every martingale is a local martingale.

The Original Question: Demonstrate that every martingale is a local martingale. Attempt at a Solution: Consider the standard setup of this problem: $\mathscr{F}_t$ is the filtration that satisfies the normal conditions, $(\sigma_n)_{n\in \mathbb{N}}$ is the monotone increasing sequence of stopping times with $\text{lim}_{n \to \infty}\sigma_n = \infty$ and $\{X_t\}_{t \in [0,\infty]}$ is a martingale. To […]

A book/text in Stochastic Differential Equations

Somebody know a book/text about Stochastic Differential Equations? I’m in the last period of the undergraduate course and I have interest in this field, but my university don’t have a specialist in this area. So, I want a book that can introduce me in this field without many difficulty and that permite me study still […]

Slight generalisation of the distribution of Brownian integral

I think I have seen once that if the processes $\sigma$ and $W$, a Brownian motion, are independent then one has that $$ E \left[\exp \left(iu\int_t^T \sigma_s \, dW_s\right) \mid \mathcal{F} \right] = E\left[\exp \left(- \frac{1}{2} u^2 \int_t^T \sigma_s^2 \, ds \right) \mid \mathcal{F} \right] $$ for $\mathcal{F}$ the $\sigma$-algebra generated by $(\sigma_s)_{ s \leq […]

Weak convergence of stochastic integral

Consider a sequence of processes $Z_t^n$ and a procoss $Z_t$, $t\in[0,1]$ such that all $\int_0^1 Z^n dW$ and $\int_0^1 Z dW$ are martingales. Assume $$\int_0^1 Z_t^n \mathrm dW_t \xrightarrow{d} \int_0^1 Z_t \mathrm dW_t$$ in distribution. Do we have $$\int_0^1 (Z_t^n)^2 \mathrm dt \xrightarrow{d} \int_0^1 Z_t^2 \mathrm dt$$ in distribution? Intuitively, this should be true as […]

Strong solutions SDE inequality with an application of Gronwall's inequality

Suppose that we have a general SDE on a probability space $(\Omega,\mathcal{F},P)$ defined by: $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) d W_t, $$ where $W$ is a Brownian motion and $b$ and $\sigma$ are Borel-measurable functions. Furthermore suppose that we have a strong solution $X$ to the SDE with initial condition $\xi$ if $X_0 […]

Quadratic variation – Semimartingale

We know that any Semimartingale has Quadratic variation. I am interested to know if the converse is also true i.e. if a process has quadratic variation then it is semimartingale. Can some one prove/disprove it. Thank you very much.

Some Kind of Generalized Brownian Bridge

Let $\displaystyle X(t) = \int_0^t f(s)dB(s)$ where $B(t)$ is a Brownian motion and $f(t)\in L^2[0,1]$. What is a simple representation for $Y(t):=(X(t)|X(1))$ in terms of $B(t)$? Note, I am not asking for the (conditional) probability distribution of $Y(t)$ I am thinking of this as a generalization of the Brownian bridge, which has several representation in […]

Equivalence of Itō and Stratonovich equations and how we ensure that the latter are well-defined

Let $T>0$ $I:=[0,T]$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\in I}$ be a complete and right-continuous filtration of $\mathcal A$ $B$ be an $\mathcal F$-Brownian motion on $(\Omega,\mathcal A,\operatorname P)$ $b,\sigma:I\times\mathbb R\to\mathbb R$ be Borel measurable Consider the Itō equation $${\rm d}X_t=\underbrace{b(t,X_t)}_{=:\:\varphi_t}{\rm d}t+\underbrace{\sigma(t,X_t)}_{=:\:\Phi_t}{\rm d}B_t\;\;\;\text{for all }t\in I\tag1$$ and the Stratonovich equation $${\rm d}X_t=b(t,X_t){\rm […]

Product rule with stochastic differentials

I am encountering difficulty in seeing how this relationship holds: with $S_T$ being stock price at time $T$, I want to find the sde for $S_t e^{-rt}$ $$dS_t = rS_tdt + \sigma S_t d\hat B_t$$ Where $ d\hat B_t$ is brownian motion such that $$X_t = S_t e^{-rt} = X_0 e^{\sigma \hat B_t – \frac{1}{2} […]

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(W_t)_{t\ge 0}$ be a $Q$-Wiener process on $(\Omega,\mathcal A,\operatorname P)$ $X_0$ be a $H$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$ $v:[0,\infty)\times H\to H$ be continuously Fréchet differentiable with respect to […]