Stochastic integration with respect to Volterra processes

By Laurent Decreusefond. 2003

72 citations recorded by the source corpus.

We construct the basis of a stochastic calculus for so-called Volterra processes, i.e., processes which are defined as the stochastic integral of a time-dependent kernel with respect to a standard Brownian motion. For these processes which are natural generalization of fractional Brownian motion, we construct a stochastic integral and show some of its main properties: regularity with respect to time and kernel, transformation under an absolutely continuous change of probability, possible approximation schemes and Ito formula.

Subfields: Applied Mathematics, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/4987ad00474f4a6ae1e15b8e5ae2ea007a5018d7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. T. Cass
2. Laure Coutin
3. Dung Nguyen Tien
4. Bruno Saussereau
5. Peter K. Friz
6. Nicolas Victoir
7. M. Folz

In Finance

1. Peter K. Friz
2. T. Cass
3. Bruno Saussereau
4. Dung Nguyen Tien
5. Laure Coutin
6. Nicolas Victoir
7. Martin Hairer
8. David Nualart

In Modeling and Simulation

1. T. Cass
2. Laure Coutin
3. Dung Nguyen Tien
4. Bruno Saussereau
5. Peter K. Friz

Citations in the corpus

• Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (2010, unknown): “[36, 59]) (1) 0 ¡ [0  ] R ¢ ⊂ H  (15.”
• Fractional stochastic differential equations with applications to finance (2013, methodology): “• The Malliavin calculus approach was introduced by Decreusefond and Üstünel in [7] for fBm and extended to more general Gaussian processes by Alos, Mazet, and Nualart in [1], Decreusefond in [8], etc.”
• Wiener integrals, Malliavin calculus and covariance measure structure (2006, unknown): “Recently, several papers were written in relation to fractional Brownian motion and Volterra processes of the type X t = t 0 G ( t, s ) dW s , where G is a deterministic kernel, see for instance [2, 9].”
• Transportation inequalities for stochastic differential equations driven by a fractional Brownian motion (2012, methodology): “We will use of the following property [7], Lemma 3.”
• Short-time near-the-money skew in rough fractional volatility models (2017, unknown): “Then it can be shown (see Decreusefond 2005, Section 3) that H is the image of L2 under the map K : ḟ → f̂ := ( t → ∫ t 0 K (t, s) ḟs ds ) and ‖Kḟ ‖H = ‖ḟ ‖L2 .”
• An Introduction to (Stochastic) Calculus with Respect to Fractional Brownian Motion (2007, methodology): “A stochastic integral for these processes is introduced in [ Dec05 ].”
• Densities for rough differential equations under Hormander's condition (2007, background): “3 one could consider Volterra processes [7], i.”