Hey I found two definitions of semimartingale. One states that semimartingale is a sum of local martingale and FV process and the second one states that semimartingale is a sum of FV process and locally square integrable martingale. Are these definitions equivalent? I would like to define stochastic integral of caglad process with respect to semimartingale. Can I define this integral as the sum of Lebesgue-Stieltjes integral and integral w.r.t locally square integrable process? Or I have to define this integral using the first definition? What is the difference? Is this integral well defined? I would be thankful for any help.
2026-04-06 22:15:55.1775513755
Definition of semimartingale and stochastic integration
416 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in STOCHASTIC-PROCESSES
- Interpreting stationary distribution $P_{\infty}(X,V)$ of a random process
- Probability being in the same state
- Random variables coincide
- Reference request for a lemma on the expected value of Hermitian polynomials of Gaussian random variables.
- Why does there exists a random variable $x^n(t,\omega')$ such that $x_{k_r}^n$ converges to it
- Compute the covariance of $W_t$ and $B_t=\int_0^t\mathrm{sgn}(W)dW$, for a Brownian motion $W$
- Why has $\sup_{s \in (0,t)} B_s$ the same distribution as $\sup_{s \in (0,t)} B_s-B_t$ for a Brownian motion $(B_t)_{t \geq 0}$?
- What is the name of the operation where a sequence of RV's form the parameters for the subsequent one?
- Markov property vs. transition function
- Variance of the integral of a stochastic process multiplied by a weighting function
Related Questions in STOCHASTIC-CALCULUS
- Interpreting stationary distribution $P_{\infty}(X,V)$ of a random process
- Reference request for a lemma on the expected value of Hermitian polynomials of Gaussian random variables.
- Why does there exists a random variable $x^n(t,\omega')$ such that $x_{k_r}^n$ converges to it
- Compute the covariance of $W_t$ and $B_t=\int_0^t\mathrm{sgn}(W)dW$, for a Brownian motion $W$
- Mean and variance of $X:=(k-3)^2$ for $k\in\{1,\ldots,6\}$.
- 4th moment of a Wiener stochastic integral?
- Unsure how to calculate $dY_{t}$
- What techniques for proving that a stopping time is finite almost surely?
- Optional Stopping Theorem for martingales
- $dX_t=\alpha X_t \,dt + \sqrt{X_t} \,dW_t, $ with $X_0=x_0,\,\alpha,\sigma>0.$ Compute $E[X_t] $ and $E[Y]$ for $Y=\lim_{t\to\infty}e^{-\alpha t}X_t$
Related Questions in STOCHASTIC-INTEGRALS
- Meaning of a double integral
- 4th moment of a Wiener stochastic integral?
- Cross Variation of stochatic integrals
- Stochastic proof variance
- Solving of enhanced Hull-White $dX_t = \frac{e^t-X_t}{t-2}dt + tdW_t$
- Calculating $E[exp(\int_0^T W_s dW_s)]$?
- Applying Ito's formula on a $C^1$ only differentiable function yielding a martingale
- what does it mean by those equations of random process?
- Why aren't the sample paths of this stochastic process defined?
- Is the solution to this (simple) Stochastic Differential Equation unique?
Related Questions in STOCHASTIC-ANALYSIS
- Cross Variation of stochatic integrals
- Solution of an HJB equation in continuous time
- Initial Distribution of Stochastic Differential Equations
- Infinitesimal generator of $3$-dimensional Stochastic differential equation
- On the continuity of Gaussian processes on the interval [0,1] depending on the continuity of the covariance function
- Joint Markov property of a Markov chain and its integral against Brownian Motion
- How can a martingale be a density process?
- Show that for a continuous Gaussian martingale process $M$ that $\langle M, M \rangle_t = f(t)$ is continuous, monotone, and nondecreasing
- Laplace transform of hitting time of Brownian motion with drift
- Is the solution to this (simple) Stochastic Differential Equation unique?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Regarding your two definitions: I know the first one for classical semimartingales (or just semimartingales). The second one is the defintion of decomposable processes (e.g. in the book of Protter Ch.II,3).
The two definitions are equivalent: As every process which is locally a square integrable martingale is of course also a local martingale, all processes satisfying the second definition also satisfy the first definition. For the reverse you have to use the Fundamental Theorem of Local Martingales (e.g. in the book of Protter Ch.III,6,Th.25). This states that every local martingale can be written as the sum of two local martingales, whereas one of them is a FV process and the other has bounded jumps. The local martingale with bounded jumps is a locally bounded martingale und thus locally a square integrable martingale.
To your second question: As the FV part can be used for a Lebesgue-Stieltjes integral the answer is YES (for a FV process the stochastic and the Lebesgue-Stieltjes integral are indistinguishable, see Protter Ch.II,5,Th.17). Since you are only integrating càglàd processes the integral is well defined.