# Tagged Questions

Questions about stochastic analysis or stochastic calculus, for example the Ito integral. See https://en.wikipedia.org/wiki/Stochastic_calculus

53 views

### Proof of white noise expansion in Banach spaces

We're reading through Da Prato/Zabczyk's proof of the existence of a white noise expansion of Gaussian measures and there's a step where we are stuck: Theorem 2.12 [Da Prato/Zabczyk, Stochastic ...
329 views

### Itô's formula: Differential form

I've started a course on financial mathematics and I'm currently being introduced to stochastical analysis, spesifically Itô's formula. From the book: It is sometimes useful to use the following ...
29 views

### Tensor products of $L^2$ spaces

Consider a probability space, $(\Omega,\Sigma,P)$, and some arbitrary Hilbert space $H$ (in my case, a space of functions on $\mathbb{R}^d$). On an intuitive level is ...
43 views

### the convergence in probability of the stochastic integral

In Jacod's Limit Theorems for Stochastic Processes:page 47,thm 4.31 (iii) $X$ is a semimartingale , $H_n$ are predictable process converge pointwise to $H$ , and $|H^n|\le K$ , where $K$ is a locally ...
118 views

### Representation theorem for local martingales

I want to prove the following local martingale representation theorem. For the statement of the theorems to come we fix a filtered probability space $(\Omega,\mathcal{A},\mathcal{F},\mathbb{P})$ where ...
62 views

### Multiple Wiener Integral by Ito

In the context of Wiener-Ito chaos expansion, I had a look at Ito's paper "Multiple Wiener Integral", 1951. I am puzzled by his last result, theorem 5.1, that a multiple Wiener integral ...
4 views

### Are the elements of the set of adapted, almost surely increasing processes with $A_0=0$ finite variation processes?

$\mathbb{A}_c=\{A_t \mid A \text{ adapted ,a.s continuous and increasing process,} A_0=0\}$ $\mathbb{V}_C=\{V_t \mid V \text{ adapted ,a.s continuous and finite variation process,} V_0=0\}$ Is ...
16 views

### Verify a stochastic integral has normal distribution.

A well-known result is that:if $\sigma$ is a non-random process,then $$\int_0^T\sigma_t\,dW_t\sim N(0,\int_0^T\sigma_t^2\,dt)$$ ( from Shreve's "Stochastic Calculus for Finance" thm 4.4.9) by means of ...
60 views

### Why do we need optional stopping theorem?

For martingale,optional stopping theorem states: Let $(M_n)_{n\in \mathbb{N}}$ be adapted with $M_n\in L^1$ for all $n$ and if $(M_n)_{n\in \mathbb{N}}$ is a martingale, then $E[M_T]=E[M_0]$, for all ...
38 views

### A martingale characterization

I saw the following characterization of martingales (without proof) in some lecture notes I found on the web and I haven't been able to produce a proof it. Let $X$ be an adapted process. If ...
24 views

### proving $C_0$ group (strong continuity on the whole $\mathbb{R}$

Let $T$ be a $C_0$-semigroup on the Banach space $E$. Then $\exists$ $M\ge 0$ and $\omega\in\mathbb{R}$ such that a) \begin{equation*} \|T(t)\|\le M e^{\omega t}\quad\forall\quad t\ge 0 ...
44 views

### proving equivalence of strongly continuity

A semigroup $S(t)$ on a Banach space $E$ is a family of bounded linear operators $\{S(t)\}_{t\ge 0}$ with the property that $S(t)S(s)=S(t+s)$ for any $s,t\ge 0$ and that $S(0)=I$. A semigroup is ...
23 views

### How to derive the distribution of measurement noise in discrete Kalman filter which is transformed from continuous one?

With sampling time $T$, and a continuous measuring model: \begin{align} y(t) &= Cx(t)+v(t) \\ v(t) & \sim \text{N}(0,R_c) \end{align} we can change it into a practical discrete one, ...
55 views

### How to prove this martingale's jumps are not summable?

I'm working on a problem from Stochastic Integration and Differential Equations by Protter and am not sure how to proceed: Let $(N_t^i)_{t \ge 0}$ be an iid sequence of Poisson processes, each with ...
48 views

### How to find the distribution of the following stochastic integral of a geometric Brownian motion?

$K_{\phi,\lambda}(r)=\int_{0}^{r}\exp\{(r-s)\phi+\lambda(W_r-W_s)\}dB_s$ where $W$ and $B$ are independent standard Brownian motions, and $(\phi,\lambda) \in \mathbb{R} \times \mathbb{R}_+$ The ...
39 views

### pointwise convergence of semimartingales in probability

In a paper on stochastic finance I'm recently studying, the author defined a closure of some subspace of semimartingales by convergence in probability: $S^N_t\rightarrow S_t$ for each $t$, as ...
25 views

### application of Holder's inequality from Oksendal's book on SDEs

I am following the proof of the existence of solutions of SDE: let $b(t,x)$ and $\sigma(t,x)$ be Lipschitz continuous and consider the following SDE $dX_t=b(t,X_t)dt + \sigma(t,X_t)dB_t$. Define ...
128 views

### Probability: Pair of socks problem

I have a problem calculating the part b and c of the following problem, maybe because my professor post the problem in a very wrong way. I already calculate the first part. The problem is, A student ...
68 views

### Doob's inequalities: Going from discrete to continuous

After having read about Doob's inequalities in discrete time, I am trying to understand the move to continuous time. I know that the Martingale regularization theorem tells me that there is (under the ...
14 views

### Covariance of nonlinear sde

My problem is to compute the covariance of the following Ito process $$dX_t=AX_t+\sum_{k=1}^{n}B_kX_tdW_k,$$ where $A,B_k$ are nonlinear operators defined on a complex separable Hilbert space $H$. ...
69 views

### Convergence in $L^2$ of the stochastic integral $\int\limits^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s$

Let $e\in \mathbb{R}^+$ and $B_t$ 1-dimensional Brownian motion. Consider $$X_t=\int^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s.$$ How to show that $X_t \to 0$ in $L^2$ as $e\to0$? Obviously the ...
13 views

### How to compute this integral using Ito isometry? [duplicate]

I am trying to evaluate the following integral: $E\Bigg[\Bigg(\int^{t}_{0} \frac{B_s}{e}1\big(B_s\in(-e,e)\big)\Bigg)^2\Bigg]$ I cannot figure out how to apply Ito isometry when the indicator ...
23 views

Let $$Y_t=\int_t^T f(t,s)\ \mathsf dW_s$$ I want to compute $\mathsf dY_t$. This suggests me to consider how to find $\mathsf dY_t$ for $$Y_t=\int_t^T f(t,s)\ \mathsf dW_s$$ or $$Y_t=\int_t^T g(t,s)\ ... 1answer 46 views ### Probability that a stochastic process is below a special random level Given a stochastic process x(t) over time t \in [0,T], and a given (deterministic) \tau, where 0<\tau<T, define a random variable x^{*} as$$ x^{*} \triangleq \inf\bigg\{y: ...
I am given two local martingales $M$ and $N$ and a stopping time $\tau$. We work on a finite time interval $[0,T]$. I want to prove $$\langle M,N\rangle^{\tau}=\langle M^\tau,N\rangle$$ using the ...