Stochastic calculus provides a consistent theory of integration for stochastic processes and is used to model random systems. Its applications range from statistical physics to quantitative finance.

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2
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1answer
167 views

explicit characterization of the stochastic integrand

Let $V$ be a cadlag positive supermartingale with the following decomposition: $$V_t=V_0+\int_0^tH_sdX_s-K_t$$ where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with ...
0
votes
0answers
22 views

is the minimum envelope of two inrersecting convex functions convex? [on hold]

let C1 and C2 two convex curves representing a cost function and intersecting each other once (say). how to prove that there exists at least one point for which C2 > C1?
-1
votes
0answers
40 views

Martingale definition [on hold]

To prove that one process is Martingale, generally we prove 3 things : 1) X is adapted. 2)$$ \mathbf{E} ( \vert X_n \vert )< \infty $$ 3) $$\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n $$ I ...
-2
votes
0answers
34 views

positiv Martingale using Itô [on hold]

I would to like to prove that the process: $$e^{\int_{0}^{T}\theta _{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2ds}$$ is a martingale which is positiv and has a mean=1 $$\theta is continuous ...
2
votes
0answers
24 views

Deriving HJB equation (why $\frac{dZ_t}{dt}=0$?)

I am trying to derive the HJB equation in a stochastic setting. Let me exemplify my problem with the simplest case where there is no control, just one state variable. Assume the payoff is given by $$ ...
26
votes
9answers
5k views

Intuition and/or visualisation of Ito integral/Ito's lemma

Riemann-sums can e.g. be very intuitively visualized by rectangles that approximate the area under the curve. See e.g. Wikipedia:Riemann sum The Ito integral has due to the unbounded total variation ...
2
votes
1answer
277 views

weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$. $$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$ ...
0
votes
0answers
26 views

comparison principle for viscosity solution to linear nonlocal equation with drift

I met a problem about comparison principle for nonlocal equation when I study SDEs driven by Levy noise. Since, I have no background about PDEs, it may be a stupid question: $$Iu=\lambda ...
0
votes
0answers
34 views

Strong Markov Property of the joint process $(B_t,L_t)_{t\ge 0}$

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion and $L=(L_t)_{t\ge 0}$ be its local time in zero. Given two strictly increasing functions $\phi_1$, $\phi_2: \mathbb R_+\to\mathbb R$ such that ...
2
votes
1answer
180 views

Time integral of a diffusion

Define $\bar\sigma^2_t=\frac{1}{t}\int_0^t\sigma^2(X_s)ds$ where $\sigma(x)\geq0$ is a measurable function and $X_t$ a diffusion process defined by \begin{equation} ...
1
vote
1answer
136 views

Question about the stochastic integral of martingales

Let $M=(M_t)_{t\ge 0}$ be a continuous martingale defined on some filtered probability space taking values in $R$. Let $H=(H_t)_{t\ge 0}$ be some bounded progressively measurable process, i.e. ...
1
vote
0answers
100 views

Joint law of a standard Brownian motion and its local time at a nonzero level

Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is $$ P\left(B_t\in d y, L_t^0\in d v\right) = ...
0
votes
0answers
37 views

Recursive parameter estimation for partially observed Ito SDEs

I'm trying to get my head around online (recursive) maximum-likelihood parameter estimation in the language of stochastic processes and in the context of stochastic filtering, i.e. where we have a ...
7
votes
1answer
460 views

Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail. Here is what I mean exactly. ...
5
votes
0answers
163 views

Quadratic variation and predictable quadratic variation for martingales

Let $(M_{t})_{0\le t\le 1}$ be a continuous martingale with respect to the filtration $(\mathcal{F}_{t})_{0\le t\le 1}$. Assume that $E M_1^2<\infty$. Fix $N$ and consider now a discrete version ...
2
votes
1answer
118 views

Stochastic differential equation associated with an optimal control problem

We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process $X_t$ is controlled up until it is stopped at a stopping time ...
2
votes
1answer
171 views

Is there a theory of SDEs whose coefficients are themselves adapted processes (i.e. “may depend on the past”)?

Is there an existence and uniqueness theorem for SDEs of the following type: $dW_{t}=d\tilde{W}_{t}+\mu\left(\left(W_{s}\right)_{0\le s\le t},t\right)dt$, where $\tilde{W}_{t}$ is say ...
5
votes
0answers
225 views

Squaring random Schwartz distributions

Let $\mu$ denote the centered Gaussian measure on $S'(\mathbb{R}^d)$ with covariance $$ \mathbb{E} [\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)} ...
1
vote
0answers
32 views

Question about the characteristics of semimartingales

Let $D=D([0,1,R)$ be the space of cadlag (right-continuous with left limits) functions defined on [0,1] and $X:=(X_t)_{t\in [0,1]}$ be the canonical process on $D$, i.e. $X_t(x)=x(t)$ for all $x\in ...
4
votes
2answers
202 views

Average Value of Area Closed by Brownian Motion

Two dimensional brownian motion will intersect its own path infinitly many times. What is the average value of area, closed by curve during an intersection in brownian motion?
1
vote
2answers
56 views

SDEs: Bounding the variance of a solution

I've been thinking about something that would seem intuitive, but I haven't really been able to dig a direct answer to. This is a rough draft of it. Let $$X_t = \mu_{X,t} \mathrm{d}t + \sigma_{X,t} ...
0
votes
1answer
335 views

A question on Ito integral

Let $W$ be a standard one-dimensional Brownian motion and $0<T<+\infty$. Then $$\lim_{\beta\to+\infty}\sup_{0\le t\le T}|e^{-\beta t}\int_0^te^{\beta s}\mathrm{d}W_s|=0\quad \text{a.s.}$$ Could ...
1
vote
2answers
214 views

Existence of strong solution to SDEs with non-Lipschitzian drift

Consider the SDE: $$dX_t=b(X_t)dt+dW_t\quad X_0=x$$ If $b$ is bounded Borel function, using Zvonkin's Transform, one can prove there exists a unique strong solution. I want to know if we assume $b$ ...
3
votes
1answer
232 views

Analytic Solution to SDEs

Are there any example of SDEs with constant diffusion terms, other than the Ornstein Uhlenbeck process, which have exact solutions? I'm thinking of something of the form: \begin{equation} dX_t = ...
1
vote
1answer
175 views

Change of time variable in Wiener process

I'm following a solution of an SDE from here http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf Start with the SDE $$ dX_t = \delta dt + 2\sqrt{X_t} dW_t $$ consider a deterministic time change ...
4
votes
3answers
321 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
1
vote
2answers
208 views

$\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ for continuous semimartingales?

I am trying to prove the following Lemma, which seems intuitive, but I still have doubts: Lemma Given a Brownian motion $\{W_t,\mathcal F_t:0\le t \le1\}$, two bounded processes, $\mu$ and $\sigma$, ...
0
votes
0answers
99 views

When an integral with respect to a Poisson point process is finite?

Let $N(ds,dv)$ be a Poisson measure on $\mathbb{R} _+ \times \mathbb{R} _+$ with intensity $dsdv$. Let $N = \sum\limits \delta_{(s_i,v_i)}$. Assume that $N$ is compatible with a filtration $\{ ...
0
votes
1answer
237 views

Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution? $$ \int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau} $$ where ...
0
votes
0answers
108 views

Expected value of a stochastic integral expression

I am wondering if the following expression can be processed a bit analytically, $$ E \left[ e^{aX} \int_0^X e^{bu}dW(u)\right], $$ where $W_u$ is the normal Brownian motion (1D Wiener process), and ...
0
votes
0answers
40 views

Is a conditional copula invariant under strictly increasing transformations?

currently I am working on conditional copulas and I have a theoretical question. In "An Introduction to Copulas", Nelsen (2006) there is a theorem (2.4.3) which says: Let $X$ and $Y$ be continuous ...
0
votes
2answers
258 views

Version of Ito's lemma applied to a stochastic function

The Ito's formula stated in most books in stochastic calculus is in the form $F(t,X_t)$, where $F: \mathbb{R}^{d+1} \rightarrow \mathbb{R}$ is a $d+1-$dimensional deterministic $C^{1,2}$ function and ...
2
votes
2answers
231 views

Uniqueness in martingale representation theorem

Dudley's martingale representation theorem states that if $W=\{W_t,\mathcal{F}_t;0\le t<+\infty\}$ is a standard one-dimensional Brownian motion, $0<T<+\infty$ and $\xi$ is ...
5
votes
2answers
129 views

Origins and Industrial Applications of stochastic processes (eg. Brownian motion) on Riemannian manifolds

I am studying BM on Riemannian manifolds and I am curious how this theory started. In the references below (esp. in Hsu's exposition), you will find many applications of that theory such as a ...
2
votes
0answers
68 views

Sobolev Bundle on Wiener Space

Right now I am learning about analysis of stochastic processes and the Malliavin calculus. It seems though, that most of the theory works for Brownian motion in $\mathbb{R}^n$, and it seems ...
5
votes
1answer
293 views

Itô's article “A measure-theoretic approach to Malliavin calculus”

Apart from citations all over the internet, the following paper appears to be off-the-grid. K. Itô, A measure-theoretic approach to Malliavin calculus, in 'New Trends in Stochastic Analysis', Proc. ...
3
votes
1answer
100 views

Certain construction of the Itô integral on manifolds

Let $M$ be a compact Riemannian manifold and let $X \in \mathfrak{X}(\mathbb{R}\times M)$ be a time-dependent vector field on $M$. I want to construct the Itô integral $$ I(X) = \int_0^T \langle X(t, ...
4
votes
2answers
517 views

Converse to Girsanov's theorem?

Roughly speaking, Girsanov's theorem says that if we have a Brownian motion $W$ on $[0,T]$, we can construct a new process with a modified drift that has an equivalent law to $W$ (subject to ...
0
votes
0answers
63 views

Extension of functions on Cameron-Martin space

Edit: The following is more or less nonsense: Let $\mu$ be the Standard Gaussian measure on $\mathbb{R}^\infty$ (i.e. the measure such that the projections $p_j$ are independent $\mathcal{N}(0, ...
3
votes
1answer
83 views

Could quadratic variation determine distribution?

Let $M=\{M_t,\mathcal{F}_t;0\le t<+\infty\}$, $N=\{N_t,\mathcal{F}_t;0\le t<+\infty\}$ be two continuous local martingales with $M_0=N_0=0\text{ a.s.}$. If $\langle M\rangle=\langle N\rangle$, ...
6
votes
2answers
274 views

A version of Wald identity

Let $W$ be a standard one-dimensional Brownian motion. Let $T$ be a stopping time with $\mathbb{E}\sqrt{T}<+\infty$. Then $$\mathbb{E}W_T=0\quad \mathbb{E}W^2_T=\mathbb{E}T$$ I can prove these ...
2
votes
0answers
64 views

Existence of 1-1 mapping/homeomorphism

Let $B$ be a standard 2-D Brownian motion, and $\sigma: \Omega\times \mathbb R^{+} \mapsto \mathbb R^{2 \times 2}$ is an $\mathcal F_{t}$ adapted process satisfying, for some constants ...
2
votes
0answers
52 views

What is the probability of B.M. hitting two disjoint spheres $(d\geq 3)$?

The hitting probability for spheres centered at origin is $P_{x}(T_{B_{r}(0)}<\infty)=\frac{r^{d-2}}{|x|^{d-2}}>0$, where $|x|>r$. 1)So I was wondering how can one compute ...
8
votes
1answer
455 views

Joint law of the time integral of Brownian motion and its maximum

Suppose $W_t$ is a standard one dimensional Brownian motion. Let $M_t$ and $I_t$ be its running maximum and time integral, respectively: $$M_t=\max_{0\leq s\leq t}\,W_s$$ ...
2
votes
1answer
161 views

Onsager-Machlup function and most probable path of a diffusion process

Let $X_{t}$ be a real, one-dimensional diffusion process satisfying the stochastic differential equation \begin{equation} dX_{t} = f(X_{t})dt + dW_{t}, \end{equation} where $f \in C_{b}^{2}(R)$ is a ...
0
votes
0answers
83 views

What is the sigma field of the derivative of a process?

When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in ...
1
vote
1answer
169 views

On the solution of a stochastic partial differential equation

Consider a simple SPDE as follows: $\partial_t u(t,x)=\partial_x^2 u(t,x)+V(u(t,x))+\dot{W}(t,x)$, $t>0$, $x\in(0,1)$, $u(t,0)=u(t,1)=0$, $u(0,x)=v(x)$, where $V$ is a bounded, smooth ...
3
votes
2answers
301 views

Convergence of iterated stochastic matrices

It is well-known that for a stochastic aperiodic matrix $M$, the sequence $(M^n)_n$ converges. Here I would like to a have a more precise analysis. Consider now a sequence of stochastic matrices ...
1
vote
1answer
48 views

question about the optimal decomposition of supermartingale

Given a filtered probability space $(\Omega, \mathbb{F}, \{\mathcal{F}_t\}_{0\le t\le 1}, \mathbb{P})$, let $X$ be a cadlag martingale and $V$ be cadlag supermartingale. Suppose $V$ has the following ...
1
vote
1answer
94 views

Perturbation of a Bessel process of dimension 2

Bessel process of dimension 2 is defined to be solution of $$ dX_t=dB_t+\frac{1}{2X_t}dt,\quad X_0=x_0>0 $$ where $B$ is a standard 1-dimensional Brownian motion. $X$ can be viewed as the norm of a ...