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

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Global optimization

Assume that I want to find the global minimum of a non-linear, non-convex, multidimensional function subject to several restrictions. Could you recommend me any deterministic strategy which can ...
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156 views

Stochastic inequality, true?

Consider two stochastic processes $X$ and $Y$ satisfying the following SDEs (with the same drift!): $$X_t = x + \int_0^t b(X_s)ds + B_t$$ $$Y_t = y + \int_0^t b(Y_s)ds + B_t.$$ If $0<x<y$, is ...
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51 views

Continuity problem in derivation of general ito integral

This is part of the derivation of the Ito integral. In particular extending the definition to more general functions. I cannot understand why $g(.,\omega)$ is continuous for each $\omega$. $\psi$ ...
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33 views

Extensions of the Ito integral

This is an extract from Oksendal's Stochastic Differential Equations (end of chapter 3). I cannot understand why we have taken the intersection, surely the union would have been more appropriate?
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31 views

Continuity theorem in Itô integral explanation

What is the continuity theorem used here in the explanation of the Itô integral? I cannot seem to find anything that would be exactly useful in my measure and integration text.
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29 views

Expectations of certain Brownian motion equations

$B_t$ is Brownian motion. It is assumed that motion starts at $0$. I do not understand how the highlighted equalities hold true. Is the first one equivalent to ...
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14 views

What is the steady state of the objective function in the following equation?

If we assume that $u$ in the time interval $\Delta t$ follows $N(\mu\Delta t, \sigma^2\Delta t)$ in the following equation : $$ R_{t} + max_{u} (\mu - u) \frac{\partial R}{\partial V} + (\sigma^2/2) ...
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240 views

Rigorous Book on Stochastic Calculus

I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been ...
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103 views

Write the Hamilton Jacobi Bellman equation

Consider the following stochastic optimal control problem. \begin{equation} V(t,x) = \max_{u}\,\, \log \left(\mathbb{E}\left[\int_{0}^{T} u^{2}(t)dt\right]\right) \end{equation} subject to the ...
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65 views

Strong solution of stochastic differential equation

Consider the stochastic differenctial equation: $dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t$. How to find a strong solution?
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49 views

Is there an example that shows that the optional stopping theorem fails for finite (unbounded) stopping times?

Is there a martingale $M=(M_t)_{t\geq 0}$ and finite stopping times $S,T$ with $S \leq T$ a.s. such that $\mathrm{E}(|M_T|)<\infty$, but $M_S \neq \mathrm{E}(M_T|\mathcal{F}_S)$ a.s.? I found a ...
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How to write the Hamilton Jacobi Bellman equation

We consider the following optimal control problem \begin{equation} V(t,x)=\max_{u}\mathbb{E} ( \log [\int_{0}^{T}u^{2}(t)dt + U(X(T))]) \end{equation} subject to the state process \begin{equation} ...
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Most probable path of diffusion process

Suppose we have an Ito diffusion $X_{t}$ on $\mathbb{R}$ given by \begin{align*} dX_{t} = A(X_{t})dt + B(X_{t}) dW_{t} \qquad (1) \end{align*} where $W_{t}$ is a standard Brownian motion. If $B = 1$, ...
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29 views

Let Y be a random variable with $0\le Y\le 1.$ [duplicate]

Let Y be a random variable with $$0\le Y\le 1.$$Show that $$var(Y)\le 1/4 $$ and that $$var(Y)= 1/4 $$ if and only if P(0)=1/2=P(1).
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BMO martingale and exponential martingale

Consider the BSDE, $$ Y_{T}-Y_{t}=\sum_{i=1}^{n} \int_{t}^{T} Z_{s}^{i}dB_{s}^{i} - \frac{1}{2}\int_{t}^{T} \left| Z_{s}\right|^{2}ds $$ where $B$ is a standard Brownian motion on a complete ...
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106 views

Power spectral density of convolution of stochastic processes

I was wondering what it is the result of convolving two WSS processes in terms of power spectral densities. I know that, the output $Y(t)$ of a generic linear time invariant system with impulse ...
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39 views

SDE with no weak solution

I'm facing the followingd d-dimensional SDE: $$dY_t=\sigma(h_t)\,dB_t$$ In addition it holds, that: $h_t\in H$ and $H$ is compact (for example the simplex of $R^n$) the proces $h_t$ is progressivley ...
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70 views

Solutions of SDE do not explode when drift term is zero.

Suppose we have $dX_t = \sigma(X_t) dW_t$ where $\sigma : \mathbb{R} \rightarrow \mathbb{R}$ is Borel and $W_t$ is a standard one-dimensional Brownian motion. I am trying to show that $X_t$ cannot ...
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54 views

Random variables independent

We said that two random variables $X,Y$ are independent iff we have that for $Z = X+Y$: $$P_Z(B)=\int_{\mathbb{R}}P_X(B-s)dP_Y(s) = \int_{\mathbb{R}}P_Y(B-s)dP_X(s).$$ But I still don't get this ...
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38 views

convergence of Ito integral

Suppose there is a deterministic process $\phi$ in $L^2(R)$. Need to prove that $\int_0^n \phi_u dW_u$ converges in $L^2(P)$ to some $X\in L^2(P)$ as $n\rightarrow\infty$. Also need to show that ...
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31 views

Integrability and exponential integrability

I'm working on a paper, and I don't know if there is some kind of typo or if I just don't get what seems obvious to the author. Note : I'll be working with probabilities, but I guess this would be ...
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85 views

Brownian motion transition density question

Let $Y_t = M_t - W_t$ where $M_t$ is the running maximum of brownian motion and $W_t$ is brownian motion. I want to show that $P^0[Y_{t+s} \in dy| Y_t = x] = p(s,x,y)+p(s,x,-y)$ where $p$ is the ...
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127 views

Sample continuity of Brownian motion

I wanted to know if the Brownian motion and the fractional Brownian motion are almost surely sample continuous or not? Many thanks.
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53 views

Is this a Brownian motion

I am learning SDE, and here is some basic things I have trouble with, Let $B(t)$ be a Brownian motion, and $F \in \mathcal L^2$ is any stochastic process and I know $\int_0^tF(s)dB(t)$ is Ito process ...
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62 views

Integrate over different measures

In Probability theory the expected value of a random variables $X : \Omega \rightarrow \mathbb{R}$ is defined as $E(X) = \int_\Omega X dP$ Now, if $\Omega \subset \mathbb{R}$ and has a density ...
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Continuity in $x$ of $E^x \int_0^{\tau} f(X_t)dt$

Suppose I have a stochastic diffusion $X$. I am studying an expression of the form $u(x):=E^x\int_0^\tau f(X_t)dt$ where $\tau$ is the exit time of $X$ from my bounded open domain $D$. I am also ...
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34 views

Distribution of Levy driven O-U process

Is there a way to find an analytical expression for $E\left[\exp\left(-\int_0^T \gamma_s ds\right)\right]$, where $d\gamma_t=k(\theta-\gamma_t)dt+\sigma dL_t$, and $L_t$ is a symmetric alpha ...
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43 views

Elementary Malliavin Derivative question about definition.

I am reading a book that defines the Malliavin derivative $D_tF$ as follows: If $F = \sum_{n=0}^{\infty} I_n(f_n)$ is the Wiener Chaos expansion. $F$ is in the brownian filtration and $F \in ...
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72 views

Reflected process - Brownian motion

I am still new to stochastic processes and I tried to do this exercise. I don't know how to go on. Define the maximum process \begin{align*} M_t = \max_{0 \leqslant s \leqslant t} W_s, \end{align*} ...
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Construct an arbitrage opportunity in a multi-period model

I am currently revising for my exam in Financial Mathematics, and I could not solve this question: For $T > 1$, consider a $T$-period model with a single risky asset and a bank account which pays ...
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69 views

Application of Ito's Lemma to integral expression

I have a problem applying Ito's lemma. I know that if: $dX_t= \mu_t \, dt + \sigma_t \, dB_t$ then for $f(t,x)$: $df(t,X_t) =\left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial ...
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78 views

Ornstein-Uhlenbeck operator and divergence operator

So I'm still struggling with Malliavin calculus, and this time about the divergence operator. We are working in the classical Wiener space $(W,H,\mu)$ where $W$ is the Wiener space ...
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38 views

Stochastic integral with respect to a stochastic integral

[From Bass R.F. Stochastic processes. Exercise 10.4] Let $N_t = \int_0^tH_sdM_s$ where $M$ is a continuous square integrable martingale and H is predictable and integrable and $L_t = \int_0^tK_sdN_s$ ...
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86 views

Density of cylindrical random variables in classical Wiener space

I'm currently working on Malliavin calculus, and a theorem in my class notes is bothering me : Denote W the Wiener space of continuous functions from $[0,1]$ to $\mathbb{R}$, and $\mu$ the associated ...
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Integrating a function of a random variable; $\int g(X) dP$

Assume a random variable $X$ on probability space $\Omega$, taking values in $\mathbb{R}$ with some known distribution $F(dX)$. Assume also a function of the random variable, $g(X)$. Does then the ...
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65 views

Representation of Markov process adapted to given Filtration

Let $X$ be continuous Markov process adapted to a filtration generated by Brownian motion $B$. Does there exist a function $f$ such that $X_t = f(t,B_t)$? My guess is that it should have such ...
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If $dX_t=b_tdt+\sigma_tdW_t=\tilde{b}_tdt+\tilde{\sigma}_tdW_t$ then $b_t=\tilde{b}_t$ and $\sigma_t=\tilde{\sigma}_t$ a.s

Let $X_t$ be an Ito's process where $dX_t=b_tdt+\sigma_tdW_t=\tilde{b}_tdt+\tilde{\sigma}_tdW_t$. Prove $b_t=\tilde{b}_t$ and $\sigma_t=\tilde{\sigma}_t$ a.s Here my solution for ...
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99 views

About the increasing process in the Doob-Meyer decomposition

As we know, a RCLL submartingale on [0,T], $Y$, in class D can be decomposed as: $$Y_t=Y_0+M_t+A_t,\ a.s.,$$ where $M$ is a martingale and $A$ is an increasing previsible process. In my question, I ...
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131 views

Skorohod convergence (space of right continuous functions with left limit)

If $f_n$ is a sequence of functions of the Skorohod Space $D([0,\infty),E)$, where $E$ is a separable Banach space, such that $f_n \to f$ in the Skorohod topology. Is it possible that there exists a ...
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Tail of hitting times for Brownian motion on the circle

For $y\in \mathbb R/\mathbb Z$ and $\varphi\in C([0,\infty);\mathbb R/\mathbb Z)$ let $T_{y}(\varphi) \ := \ \inf\{t>0: \varphi_t = y \} \ \ \ $ (first time the path $\varphi$ hits $y$) ...
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164 views

Brownian Motion with Optional Stopping Theorem (OST)

Let $(B_t)_{t \geq 0}$ be a standard Brownian Motion and let $T:=\inf\{t \geq 0: B_t=at-b\}$ for some positive constant $a,b>0$. Calculate $\mathbb{E}[T]$. How do i begin it?
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64 views

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 ...
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91 views

Transforming semimartingale to local martingale by change of measure

Consider a continuous $\mathbb{P}$ - semimartingale X which can be decomposed as M+A (M is local martingale and A is bounded variation process). Is it possible to change measure to $\mathbb{Q}$ s.t. ...
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77 views

Is any FV-Process a special Semimartingale?

Any FV-Process can be represented as the difference of two increasing (or decreasing) processes and so any FV-Process is a quasimartingale. Due to Raos Theorem any FV-Process is a special ...
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29 views

Autocorrelation of Radial Stochastic Process with Planar Derivatives

I have a random field $h(\vec{r})$ that depends on $\vec{r}=(x,y)$, such that \begin{equation} \langle h(\vec{r})h(\vec{r}+\vec{r}') \rangle \sim \exp(-||\vec{r}-\vec{r}'||/a^2) \end{equation} where ...
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71 views

Optional Sampling a.s. finite stopping time

Given a uniformly integrable discrete martingale $M_n$ on prob. space $(\Omega, \mathcal{F}, \mathbb{P})$, and a.s. finite stopping times $T$ and $S$ with $T\geq S$. Show that $E[M_T|\mathcal{F}_S] = ...
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Are there “necessary” conditions for a solution to the multivariate, truncated Hausdorff moment problem?

I am looking for NECESSARY conditions for a solution to the multivariate, truncated Hausdorff moment problem (i.e., conditions under which a given finite sequence of numbers is the sequence of first ...
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238 views

Poisson Process Change of Measure

I have seen the following result stated in the literature: Let $N(t)$ be a (finite time horizon) Poisson process defined on a probability space $(\Omega, \mathbb{P})$ with constant intensity ...
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106 views

Limit of stochastic integrals?

Let $(W_t)t$ be a Wiener process. I want to find the limit for $\epsilon\to 0$ of $$\frac{W_t^2}{2\epsilon}\chi_{(-\epsilon,\epsilon)}(W_t)-\int_0^t ...
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69 views

What is wrong with my example where the Itô Integral and Riemann-Stieltjes Integral don't coincide?

I have an interesting question concerning those two integrals. Considering a Brownian motion $(B_t)_{t \geq 0}$ with start in $x$. We can choose an $\omega \in \Omega$ such that, $t \to B_t(\omega)$ ...