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11
votes
0answers
668 views

surprisingly difficult filtration problem

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky: Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
6
votes
0answers
278 views

Stochastic Integration via Skorohod Representation

I am interested to know if Ito integrals against Brownian motion can also be constructed via Skorohod representation. By this I mean the following: let $S_n$ be a simple random walk started at zero; ...
5
votes
0answers
281 views

Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem $$ \partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x) $$ ...
5
votes
0answers
75 views

Stochastic calculus for several inputs

In "On the Gap Between Deterministic and Stochastic Ordinary Differential Equations," The Annals of Probability, Vol. 6, No. 1 (Feb., 1978), pp. 19-41, Hector J. Sussmann showed that a stochastic ...
5
votes
0answers
376 views

When is an ODE a good approximation to an SDE?

Suppose $X_t$ is a weak solution to a stochastic differential equation in the form $$d X_t = \sigma(X_t) d W_t + \lambda(X_t) dt$$ for smooth functions $\sigma: \mathbb R^d \to L(\mathbb R^d,\mathbb ...
5
votes
0answers
987 views

Levy jump measure vs. Levy measure vs. sum of jumps

This question might be a bit basic, but I am struggling to understand the connection between various versions of the Ito's lemma for Levy processes (and semimartingales in general). Could someone ...
4
votes
0answers
78 views

Reference request: Stochastic integration and martingale theory on the whole real line

I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...
4
votes
0answers
182 views

Malliavin calculus w.r.t $G$-Brownian motion

I wonder if it is possible to define a Malliavin calculus w.r.t $G$-Brownian motion defined on a Sublinear Expectation Space available on this link. G–Brownian motion has a very rich and interesting ...
4
votes
0answers
171 views

Integrating a Bessel Bridge

Preliminaries An order-3 Bessel Process is the one-dimensional stochastic process $X$ described by $X(t) = \sqrt{W_1(t)^2 + W_2(t)^2 + W_3(t)^2}$, where each $W_k$ is an independent Brownian Motion. ...
4
votes
0answers
468 views

Dynamic programming principle (DPP)

In stochastic control problem, one shall use the measurable selection theorem to prove DPP. It was discussed in discrete time case in [Bertsekas and Shreve 1978]. Is there unified framework in ...
3
votes
0answers
56 views

How can one do change of variables for solutions to a staochastic partial differential equation?

isHow can one do change of variables for solutions to a staochastic partial differential equation? For example, let us consider the following stochastic transport equation: $$ dy(t,x) + y_x(t,x) + ...
3
votes
0answers
128 views

Expectation of running maximum of diffusion processes

Let $X$ be a one-dimensional Ito diffusion $$X_t=x+ \int_0^t b(X_s)ds + \int_0^t \sigma(X_s)dW_s,$$ where $b,\sigma$ satisfy the usual Lipschitz continuity and linear growth conditions. Define the ...
3
votes
0answers
81 views

The distribution of Jump gaps of Levy process

Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps ...
3
votes
0answers
172 views

Time reversibility of Stratonovich Diffusion: Reference Request

Please consider the Stratonovich stochastic differential equation (SDE) $$ dX = b(X)\circ dB $$ where $B$ is standard Brownian motion and $X(0)=X_0$. This corresponds to the Ito (SDE) $$ dX = ...
3
votes
0answers
175 views

Joint distribution of Ito integral and its quadratic varation

Any idea on solving the joint distribution of $X_T=\int_0^T \alpha_t dZ_t$ and $Y_T=\int_0^T \alpha_t^2 dt$ ? Here $X_T$ is an Ito integral and $Z_t$ is a standard Brownian process. When $\alpha_t$ ...
3
votes
0answers
148 views

stochastic control / geometric mean

Consider the following problem: Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...
3
votes
0answers
220 views

Observing drift of a Levy process

It is a well known fact, that it is very difficult to estimate the drift of a Brownian motion with drift from looking at a single path over a finite interval $[0, T]$. Is it the case with Levy ...
3
votes
0answers
145 views

Characterizing polyhedron from Brownian particle collisions with a boundary

Please imagine that we have an ordinary 2-sphere, of radius $r_{sphere}$, and some three-dimensional polygon that has all of its points fixed at positions strictly internal to the sphere's surface. ...
2
votes
0answers
47 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
35 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 ...
2
votes
0answers
96 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
2
votes
0answers
62 views

Numerical Methods for stochastic PDE, from rough paths to backward equations

this question is about some literary references regarding the state of the art in terms of numerical methods for SPDE's. In particular, Have the numerical implications, if any, of the results in ...
2
votes
0answers
85 views

Generalization of Ito's formula

If $f:R\to R$ is a convex function then we have Ito-Tanaka formula. Now my question is that if we are given a function $u: R\times R_+\to R$ such that $u(s,\cdot)$ is smooth for every $s\in R$ and ...
2
votes
0answers
39 views

The distribution of maximum of fraction Brownian motion over finite time interval

Suppose that $\{B_t^H,\ t\geq 0\}$ is a fractional Brownian motion with Hurst exponent $H$, I wonder if there are explicit expressions for the joint distribution of $(\sup_{0\leq t\leq ...
2
votes
0answers
116 views

On the infinitesimal generator of a 1-dimensional stochastic heat equation: core and explicit form

Denote $E = C([0, 1])$. I am consider a 1-dimentional stochastic heat equation on $h$: $\partial_tu(t, x) = \partial_x^2u(t, x) - V'(u(t, x)) + \dot{W}(t, x)$, for all $(t, x) \in (0, ...
2
votes
0answers
88 views

a generalization of Monge-Kantorovich Problem

I am thinking about the martingale version of Monge-Kantorovich Problem. Let $\mu(x)$ and $\nu(y)$ denote two density laws on $\mathbb{R}$, and define $M(\mu,\nu)$ the set of densities $f(x,y)$ on ...
2
votes
0answers
237 views

distribution of integral of exponential of wiener process

I am absolute newbie to stochastic calculus and have to solve a weighted hazard rates integral, where the hazard rates are stochastic, their logarithm governed by arithmetic Ornstein-Uhlenbeck (OU) ...
2
votes
0answers
117 views

Lyapunov function of exponential growth for existence of a solution of an SDE

Let $$dX_t = a(X_t) dt + b(X_t) dW_t$$ be a one-dimensional stochastic differential equation, where the coefficients $a,b: \mathbb{R} \rightarrow \mathbb{R}$ satisfy for every ball $B_R$ the following ...
2
votes
0answers
115 views

Existence of predictable quadratic covariation for a special pair of local martingales

In Limit theorems for stochastic processes, by Jacod and Shiryaev we have the existence of a predictable quadratic covariation process stated as the following theorem $\mathbf{Theorem}$ To each ...
2
votes
0answers
63 views

Tail for the integral of a diffusion process

I would like to compute the following tail, $$ \mathbb{P}\left(\int_{0}^{T} f(X_t)\mathrm{dt}>x\right), $$ assuming $$ \mathbb{P}[f(X_t)>x] = x^{-\alpha} \log(x), $$ and $X$ is a diffusion ...
2
votes
0answers
143 views

Cameron-Martin like RKHS

Hello, I know that $k(x,y)=min(x,y)$ is the reproducing kernel of the Cameron Martin space of all i.i.d. RVs of Brownian motion at different times, with the $cov$ inner product. What is the RKHS ...
2
votes
0answers
152 views

Computing a density function for the integral of a stochastic process, given its transition function

$P$ is a one-dimensional Markov stochastic process that runs on time interval $[0, t_f]$. I know its transition function: $P(0) = x_0$ and for any $0 \le t_a < t_b \le t_f$, the function $f(x_b | ...
2
votes
0answers
201 views

Is this process strictly positive?

Let $W_t$ is standard Brownian motion under probability measure $P$. Consider 1-D stochastic differential equation $$ dY_t = dt + \sigma(Y_t) dW_t, \ Y_0 = y\ge 0.$$ We assume $\sigma(0) = 0$, and ...
2
votes
0answers
96 views

Does this series stopping times marching forward?

Let $W_t$ is standard Brownian motion under probability measure $P$. Consider stochastic differential equation $$ dY_t = dt + Y_t dW_t, \ Y_0 = 0.$$ Note that, the above SDE has a strong non-negative ...
2
votes
0answers
261 views

How to deal with the vector norm item as a denominator in this expectation?

Hello, everyone. I want to calculate the expectation shown in the following formula, where $X$ follows a standard $d$-dimensional multi-variable normal distribution as ...
2
votes
0answers
291 views

Finding jump probabilities from mean-occupancy values for positions on a one-dimensional random walk

Please imagine a discrete random walk on a one-dimensional lattice. The lattice consists of a set of $L$ positions, $(x_0, x_1, ..., x_L) \in L$, where $x_0$ is the initial position of the walk (as ...
1
vote
0answers
173 views

Girsanov theorem with Geometric Brownian Motion

I am not a student in mathematics, but I am trying to use the following Theorem 8.6.6 (Girsanov theorem II) of Oksendal's SDE with geometric Brownian motion $S_{t}$ instead of the standard Brownian ...
1
vote
0answers
231 views

Inflated independent samples for Monte Carlo estimation

In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...
1
vote
0answers
56 views

question related to Tanaka Formulae

Supposse $X=(X_t)$ is a cadlag martingale taking values in $\mathbb{R}$. If $f:\mathbb{R}\to\mathbb{R}$ is a convex function, then we have Tanaka Formulae. Now let $g: ...
1
vote
0answers
145 views

Fundamental theorem of calculus for iterated stochastic integrals

I'm trying to find the rate (or a bound for it) with which an iterated integral of the type $$\int_{-h}^0 \int_{-h}^{t} A_s d B_s A_t d B_t$$ converges to zero (in probability/distribution) for $h ...
1
vote
0answers
36 views

a question about the modification of a supermartingale

Let $\mathbf{D}\subset\mathbf{D}([0,1],\mathbb{R}_+)$ denote the space of positive cadlag functions $\mathbf{x}$ defined on $[0,1]$ with $\mathbf{x}(0)=1$. Define the canonical process ...
1
vote
0answers
83 views

question about Doob-Meyer decomposition

Given a filtered probability space and let $X$ be a cadlag local martingale defined on this space. Let $V$ be a cadlag supermartingale and assume we know the following decomposition: ...
1
vote
0answers
71 views

Time change for non-homogeneous Markov processes

Background: Let $C$ be the space of continuous function on $[0,T]$, $f, \sigma \in C$ bounded with $\sigma^2 \geq \varepsilon > 0$ and let $X=(X_t)_{t\in [0,T]}$ be a diffusion process of ...
1
vote
0answers
73 views

stochastic calculus

I am trying to understand Ito's lemma for Poisson type processes and here is my question: Assume I have a jump process given by the stochastic equation: $dp(t,T_1)= r_1p(t,T_1)1_{\tau<=t}dt + ...
1
vote
0answers
110 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 ...
1
vote
0answers
80 views

Supermartingale inequality on a particular event

Say, I have a supermartingale $Y_t$ with respect to the filtration $F_t$. Let $T$ and $S$ two stopping times greater than $t>0$ such that on the event $A$, $T>S$, then since $Y_t$ is a ...
1
vote
0answers
282 views

How is Kolmogorov forward equation derived from the theory of semigroup of operators?

In Lamperti's Stochastic Processes, given a time-homogeneous Markov process $X(t), t\geq 0$ with Markov transition kernel $p_t(x,E)$ and state space being a measurable space $(S, \mathcal{F})$, a ...
1
vote
0answers
106 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
0answers
86 views

Attractors and solutions to these generalized Ornstein–Uhlenbeck processes?

This is a question about generalized Ornstein–Uhlenbeck processes I asked on MSE, but I haven't received replies about their attractors and solutions yet. So I would appreciate if someone could give ...
1
vote
0answers
187 views

What conditions on a filtration guarantee that a (sub)martingale has a continuous modification?

There is a theorem as follows: Theorem. Let $\mathcal{F}_t$ be a filtration which is right-continuous and complete. Assume $M_t$ is a submartingale adapted to $\mathcal{F}_t$ such that $t \mapsto ...