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

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
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 ...
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1answer
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 ...
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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 $H$. ...
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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 ...
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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 ...
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How to calculate the differential of the following stochastic integral?

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)\ ...
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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: ...
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Stopping of quadratic covariaton

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 ...
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1answer
40 views

Covariance of Ornstein-Uhlenbeck process

$U(t)=e^{-\mu t}W(\frac{\sigma^2e^{2\mu t}}{2\mu})$. The problem is to find $Cov[U(t),U(t+s)]$. I used the identity, $W(\frac{\sigma^2e^{2\mu t}}{2\mu})=W(\frac{\sigma^2e^{2\mu t}e^{2\mu s}}{2\mu ...
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Ito's representation for $L^1$ random variable

Given $(\Omega,\mathbb{F},P)$ where $\mathbb{F}$ is the $P$-complete filtration generated by Brownian motion $W$. Ito's representation says for $X\in L^2(\mathcal{F}_\infty,P)$ with zero mean,there ...
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Exact solution to nonlinear backward SDE

I have read a paper about numerical SDE. After deriving the method, it uses the method to calculate the following nonlinear cases: $$\begin{cases} dX_t=ud\tau+\sigma ...
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Expectation of quadratic variation

I got stuck in a step of a proof and need some help. The situation is the following: Let $M$ be a continuous local martingale (which satisfies $\mathbb{E}[\langle M\rangle(T)]<\infty$ - I don't ...
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Compute the Gibbs energy

I have a question about Gibbs distribution in Stochastic theory. In which, it defined a clique as a a subset $C$ in the whole image $\Omega$ if two different element of $C$ are neighbors. Figure 2 ...
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Deduce $\partial_tp=-\partial_x(b(x)p)+(1/2)\partial_{xx}(\sigma^2(x)p)$, for $p(x,t|y)$ of $X(t)$ and $dX=b(X)dt+\sigma(X)dW$, $X(0)=y$

I am stuck in this proof... I almost got it, but I must have made a mistake. It is part B that I am getting wrong. Thanks in advance for your help! QUESTION: Let $X$ satisfy the autonomous SDE ...
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1answer
41 views

Independence of a hitting time and the underlying stochastic process

While I was playing around with the Girsanov's Theorem I stumbled upon the following absurdity and I couldn't resolve it with the current knowledge of stochastic analysis that I have. $B$ being ...
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What can you tell me about backward Brownian motion?

I'm trying to understand "backward Brownian motion" and how it relates to standard Brownian motion. In this paper, they construct a solution to Burgers Equation (transformed via Cole-Hopf) with ...
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Help with proof of Girsanov Theorem

I'm studying this proof of Girsanov Theorem and trying the figure out the details however I need some help with this. I noticed there are, also here on stackexchange, a lot of different versions of ...
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Doob's inequalities for not necessarily right-continuous martingales

In Revuz and Yor, they denote $\mathbb{H}^2$ the space of $L^2$-bounded martingales, and $H^2$ the space of continuous $L^2$-bounded martingales. They state "... by Doob's inequality ... ...
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Is there a way to estimate moments of strong solution to SDE

Suppose the SDE $$\mathsf dX_t =b(t,X_t)\mathsf dt + \sigma(t,X_t)\mathsf dW_t,\; X_0 = x$$ where $t\in[0,T]$ has a strong solution. I know in general we can't find an explicit formula for the ...
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Stochastic Control

I would like to solve the following stochastic dynamic programming in the discrete-case and continuous case: Let the state variables have the following dynamics: \begin{align*} dS_t = \mu S_t dt + ...
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156 views

Quadratic variation of the Ornstein-Uhlenbeck process

Let $(X_t)_{t\geq 0}$ be the zero-mean Ornstein-Uhlenbeck process such that $X_0 = 0$ almost surely, i.e. $$X_t = \sigma e^{-\alpha t}\int_0^t e^{\alpha s}\,dB_s \quad \qquad (\triangle)$$ On the ...
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Burkholder's inequality for elementary stochastic integral

An elementary Burkholder's inequality for simple stochastic integral says that given nonnegative martingale $M$ and simple bounded predictable process $H$, it holds that for all $c>0$, the tail ...
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Find $\mathbb{E}_{X_0 = x} X_\tau$ for an Ornstein-Uhlenbeck process $(X_t)_{t \geq 0}$ where $\tau = \inf\{t>0 \mid X_t \notin [a,b]\}$

Let $X_t$ satisfy the following SDE: $dX_t = X_t dt + \sigma dB_t$, $\sigma$ is a constant and $B_t$ is Brownian Motion. Find $\mathbb{E}_{X_0 = x} X_\tau$ where $\tau = \inf\{t>0 \mid X_t \notin ...
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Fundamental theorem for Malliavin derivative and Lebesgue integral

I am interested in some kind of fundamental theorem of calculus for the Malliavin derivative: My notations are mainly taken from the Book Nualart: The Malliavin Calculus and Related Topics. Let ...
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Finding the flux of $\iint \vec F\hat n\;ds$

I need to find the flux $\displaystyle\iint \vec F\hat n\;ds$ of the vector feild $\vec F=4x \hat i-2y^2\hat j+z^2 \hat k$ throughe the surface $S=\{(x,y,z):x^2+y^2=4,z=0,z=3\}$ My attempt: (I'm ...
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Simple Stratonovich product for physical system

I was reading a physical textbook and they used the "Stratonovich product" referred to $v_1 \circ dW_1 = \frac{1}{2}[v_1 + (v_1+dv_1)]dW_1$. I think this product is from the Stochastic process, thus ...
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Analytic solution to stochastic differential equations

I need help to to find the analytic solution (if it exists) of the following system of SDE. Usually, I use Matlab as software but in this case I'm unable to use it in order to figure out the problem. ...
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Why are functions of semimartingales again semimartingales?

I am trying to prove the Itō's lemma, and need to show that if $X$ is a semimartingale and $f$ is a $\mathcal{C}^2$-function, then $f(X_t)$ is again a semimartingale. How do I do that? I cannot see ...
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1answer
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How to compute the solution of a differential equation involving Brownian local time

My problem is to compute numerically a function F. F is known to be convex and have kinks. It's also known to satisfy a "second order differential equation". Since the function is not everywhere ...
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62 views

Brownian bridge sde

The SDE for the Brownian bridge is the following: $dX_t = \dfrac{b-X_t}{1-t}dt+dB_t$ with the solution $X_t = a(1-t)+bt+(1-t)\int_{0}^t \dfrac{dB_s}{1-s}$. The expectation and covariance are: ...
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DRIFT MATRIX in Ornstein Uhlenbeck Process

The Weiner Process was unable to explain Brownian Motion and then there was the need of Ornstein-Uhlenbeck Process. The Ornstein-Uhlenbeck Process describes the Brownian Motion in the presence of ...
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Stochastic Integral of Simple Predictable Process is a Martingale

Take $H\in S$ to be a simple process defined as: $$H_t:=\sum_{i=1}^{n-1} H_i1_{(T_i,T_{i+1}]}(t),\ \ H_i\in \mathcal{F}_{T_i}, \ (T_1\leq...\leq T_n \ stopping\ times),$$ and $X$ a Martingale. I ...
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A Question About Probability of ratio of $\max(\cdot)$?

In My field , I reached to this problem. Assumptions: Consider $x_i,\hat{x}_i$ are iid (identical and independent) samples of a joint distribution (e.g., exponential). And also, assume we have $N$ ...