A PDE approach to jump-diffusions

Carr, Peter; Cousot, Laurent

doi:10.1080/14697688.2010.531042

Cited by 13 publications

(9 citation statements)

References 40 publications

(46 reference statements)

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“…Also, SIEs corresponding to other BTP processes we introduced in [10] may also be studied by adapting and generalizing our approach here. We believe BTPs, their PDEs, their SIEs, and their discretized cousins (the BTCs and their equations) can play a useful role by adding new, currently unavailable, insights and models to the ever growing mathematical finance theory (see [19] for an example). We also hope to explore these aspects in future papers.…”

Section: Discussionmentioning

confidence: 99%

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Brownian-time Brownian motion SIEs on $\mathbb{R}_{+}$ × $\mathbb{R}^d$: Ultra regular direct and lattice-limits solutions and fourth order SPDEs links

Allouba¹

2013

Discrete &Amp; Continuous Dynamical Systems - A

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We delve deeper into the compelling regularizing effect of the Brownian-time Brownian motion density, K BTBM d t;x,y , on the space-time-whitenoise-driven stochastic integral equation we call BTBM SIE:which we recently introduced in [3]. In sharp contrast to traditional second order heat-operator-based SPDEs-whose real-valued mild solutions are confined to d = 1-we prove the existence of solutions to (0.1) in d = 1, 2, 3 with dimension-dependent and striking Hölder regularity, under both less than Lipschitz and Lipschitz conditions on a. In space, we show an unprecedented nearly local Lipschitz regularity for d = 1, 2-roughly, U is spatially twice as regular as the Brownian sheet in these dimensions-and we prove nearly local Hölder 1/2 regularity in d = 3. In time, our solutions are locally γ-Hölder continuous with exponent γ ∈ 0, 4−d 8 , 1 ≤ d ≤ 3. To investigate (0.1) under less than Lipschitz conditions on a, we (a) introduce the Brownian-time random walka special case of lattice processes we call Brownian-time chains-and we use it to formulate the spatial lattice version of (0.1); and (b) develop a delicate variant of Stroock-Varadhan martingale approach, the K-martingale approach, tailor-made for a wide variety of kernel SIEs including (0.1) and the mild forms of many SPDEs of different orders on the lattice. Solutions to (0.1) are defined as limits of their lattice version. Along the way, we prove interesting aspects of Brownian-time random walk, including a fourth order differential-difference equation connection. We also prove existence, pathwise uniqueness, and the same Hölder regularity for (0.1), without discretization, in the Lipschitz case. The SIE (0.1) is intimately connected to intriguing fourth order SPDEs in two ways. First, we show that (0.1) is connected to the diagonals of a new unconventional fourth order SPDE we call parametrized BTBM SPDE. Second, replacing K BTBM d t;x,y by the intimately connected kernel of our recently-introduced imaginary-Brownian-time-Brownian-angle process (IBTBAP), (0.1) becomes the mild form of a Kuramoto-Sivashinsky (KS) SPDE with linearized PDE part. Ideas and tools developed here are adapted in separate papers to give an entirely new approach, via our explicit IBTBAP representation, to many linear and nonlinear KS-type SPDEs in multi-spatial dimensions.1991 Mathematics Subject Classification. 60H20, 60H15, 60H30, 45H05, 45R05, 35R11, 35R60, 35G99, 60J45, 60J35, 60J60, 60J65.

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Section: Discussionmentioning

confidence: 99%

“…where C d is dimension-dependent 19 . Then there is a δ * > 0 such that, whenever δ ≤ δ * , we obtain…”

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confidence: 99%

Brownian-time Brownian motion SIEs on $\mathbb{R}_{+}$ × $\mathbb{R}^d$: Ultra regular direct and lattice-limits solutions and fourth order SPDEs links

Allouba¹

2013

Discrete &Amp; Continuous Dynamical Systems - A

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“…By changing the values of ε 1 and ε 2 , we are simply changing the intensities of the smoothing L-KS spatial operator − 1 8 (∆ + 2ϑ) 2 and the roughening noise term, respectively. Theorem 1.1 gives us existense, uniqueness, sharp Hölder regularity, 20 These lattice arguments have their roots in our second order SPDE works [14,8]. 21 Our work in a separate article indicates that, when a(u) = |u| α , there is a dimensiondependent critical blowup exponent α .16)).…”

Section: 2mentioning

confidence: 94%

“…In the recent multiparameter-time case the reader is referred to [4,3]. The BTBM scaling and its nonstandard PDEs connection have now attracted a lot of attention, even outside probability and PDEs, as evidenced by the recent physics and mathematical finance articles [19,20]. 19 We remind the reader again that the compact support condition on u 0 may be replaced with more relaxed integrability conditions like those given for the Brownian-time Brownian sheet in [3].…”

Section: 2mentioning

confidence: 99%

L-Kuramoto–Sivashinsky SPDEs in one-to-three dimensions: L-KS kernel, sharp Hölder regularity, and Swift–Hohenberg law equivalence

Allouba

2015

Journal of Differential Equations

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Abstract. Generalizing the L-Kuramoto-Sivashinsky (L-KS) kernel from our earlier work, we give a novel explicit-kernel formulation useful for a large class of fourth order deterministic, stochastic, linear, and nonlinear PDEs in multispatial dimensions. These include pattern formation equations like the SwiftHohenberg and many other prominent and new PDEs. We first establish existence, uniqueness, and sharp dimension-dependent spatio-temporal Hölder regularity for the canonical (zero drift) L-KS SPDE, driven by white noise on. The spatio-temporal Hölder exponents are exactly the same as the striking ones we proved for our recently introduced Browniantime Brownian motion (BTBM) stochastic integral equation, associated with time-fractional PDEs. The challenge here is that, unlike the positive BTBM density, the L-KS kernel is the Gaussian average of a modified, highly oscillatory, and complex Schrödinger propagator. We use a combination of harmonic and delicate analysis to get the necessary estimates. Second, attaching order parameters ε 1 to the L-KS spatial operator and ε 2 to the noise term, we show that the dimension-dependent critical ratio ε 2 /ε d/8 1 controls the limiting behavior of the L-KS SPDE, as ε 1 , ε 2 ց 0; and we compare this behavior to that of the less regular second order heat SPDEs. Finally, we give a change-ofmeasure equivalence between the canonical L-KS SPDE and nonlinear L-KS SPDEs. In particular, we prove uniqueness in law for the Swift-Hohenberg and the law equivalence-and hence the same Hölder regularity-of the SwiftHohenberg SPDE and the canonical L-KS SPDE on compacts in one-to-three dimensions.2010 Mathematics Subject Classification. 35R60, 60H15, 35R11, 35G99, 60H20, 60H30, 42A38, 45H05, 45R05, 60J45, 60J35, 60J60, 60J65.Key words and phrases. Fourth order (S)PDEs, L-Kuramoto-Sivashinsky (S)PDEs, SwiftHohenberg (S)PDE, linear and nonlinear fourth order (S)PDEs, L-KS kernel, imaginary-Browniantime-Brownian-angle process, Gaussian average of angle modified Shrödinger propagator, kernel stochastic integral equations.

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“…Lately, many phenomena in mathematical physics, fluids dynamics and turbulence models, mathematical finance, and the modern theory of stochastic processes have been related to and described through deterministic fractional and higher order evolution equations (e.g., see [3,4], [6]- [11], [19], [23]- [26], [31], [34]- [37], [40,41], [43]- [48], and [53]); and it is only natural to investigate these important equations under the influence of a driving random noise.…”

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confidence: 99%

Time-fractional and memoryful $\Delta^{2^{k}}$ SIEs on $\mathbb{R}_{+}\times\mathbb{R}^{d}$: How far can we push white noise?

Allouba¹

2013

Illinois J. Math.

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High order and fractional PDEs have become prominent in theory and in modeling many phenomena. Here, we focus on the regularizing effect of a large class of memoryful high-order or time-fractional PDEs-through their fundamental solutions-on stochastic integral equations (SIEs) driven by space-time white noise. Surprisingly, we show that maximum spatial regularity is achieved in the fourth-order-bi-Laplacian case; and any further increase in the spatial -Laplacian order is entirely translated into additional temporal regularization of the SIE. We started this program in [1,5], where we introduced two different stochastic versions of the fourth order memoryful PDE associated with the Brownian-time Brownian motion (BTBM): (1) the BTBM SIE and (2) the BTBM SPDE, both driven by space-time white noise. Under wide conditions, we showed the existence of random field locally-Hölder solutions to the BTBM SIE with striking and unprecedented time-space Hölder exponents, in spatial dimensions d = 1, 2, 3. In particular, we proved that the spatial regularity of such solutions is nearly locally Lipschitz in d = 1, 2. This gave, for the first time, an example of a space-time white noise driven equation whose solutions are smoother than the corresponding Brownian sheet in either time or space.In this paper, we introduce the 2β −1 -order β-inverse-stable-Lévy-time Brownian motion (β-ISLTBM) SIEs, β ∈ 1/2 k ; k ∈ N , driven by space-time white noise. Based on the dramatic regularizing effect of the BTBM density (β = 1/2), and since the kernels in these β-ISLTBM SIEs are fundamental solutions to higher order Laplacian PDEs; one may suspect that we get even more dramatic spatial regularity than the BTBM SIE case. We show, however, that the BTBM SIE spatial regularity and its random field third spatial dimension limit are maximal among all β-ISLTBM SIEs-no matter how high we take the order 1/β of the Laplacian. This gives a limit as to how far we can push the SIEs spatial regularity when driven by the rough white noise. Furthermore, we show that increasing the order of the Laplacian β −1 beyond the BTBM bi-Laplacian manifests entirely as increased temporal regularity of our random field solutions that asymptotically approaches that of the Brownian sheet as β ց 0. Our solutions are both direct and lattice limit solutions. We treat many stochastic fractional PDEs and their corresponding higher order SPDEs, including BTBM and β-inverse-stable-Lévy-time Brownian motion SPDEs, in separate articles.2000 Mathematics Subject Classification. 60H20, 60H15, 60H30, 45H05, 45R05, 35R60, 60J45, 60J35, 60J60, 60J65.

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A PDE approach to jump-diffusions

Cited by 13 publications

References 40 publications

Brownian-time Brownian motion SIEs on $\mathbb{R}_{+}$ × $\mathbb{R}^d$: Ultra regular direct and lattice-limits solutions and fourth order SPDEs links

Brownian-time Brownian motion SIEs on $\mathbb{R}_{+}$ × $\mathbb{R}^d$: Ultra regular direct and lattice-limits solutions and fourth order SPDEs links

L-Kuramoto–Sivashinsky SPDEs in one-to-three dimensions: L-KS kernel, sharp Hölder regularity, and Swift–Hohenberg law equivalence

Time-fractional and memoryful $\Delta^{2^{k}}$ SIEs on $\mathbb{R}_{+}\times\mathbb{R}^{d}$: How far can we push white noise?

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