Research Videos
Videos and presentations from my YouTube channel Prob-AM about the research in my group.
Non-asymptotic bounds for forward processes in denoising diffusions: Ornstein-Uhlenbeck is hard to beat (Part I: setting and key results)
Miha describes our recent results on lower bounds for the convergence of the forward process in denoising diffusion probabilistic models. For details see our paper "Non-asymptotic bounds for forward processes in denoising diffusions: Ornstein-Uhlenbeck is hard to beat", M. Brešar and A. Mijatović
DenoisingDiffusionsPresentation_Results___YT_.pdfSlides from Part I
This presentation is based on the paper arXiv:2408.13799 , co-authored by Miha Brešar at Warwick.
Non-asymptotic bounds for forward processes in denoising diffusions: Ornstein-Uhlenbeck is hard to beat (Part II: general results and proofs)
The second video discusses the proof of our general result in the paper "Non-asymptotic bounds for forward processes in denoising diffusions: Ornstein-Uhlenbeck is hard to beat", M. Brešar and A. Mijatović
DenoisingDiffusionsPresentation_Proofs___YT_.pdfSlides from Part II
This presentation is based on the paper arXiv:2408.13799 , co-authored by Miha Brešar at Warwick.
Subexponential lower bounds for f-ergodic Markov processes (Part I: Results)
This video describes a criterion for establishing lower bounds on the rate of convergence in $f$-variation of a continuous-time ergodic Markov process to its invariant measure.
LowerBoundsPresentationPart1.pdfSlides from Part I
This presentation is based on the paper arXiv:2403.14826, co-authored by Miha Brešar at Warwick.
Subexponential lower bounds for f-ergodic Markov processes (Part II: Proofs)
The second video discusses the proofs of the main results in the paper "Subexponential lower bounds for f-ergodic Markov processes", M. Brešar and A. Mijatović
LowerBoundsPresentationPart2.pdfSlides from Part II
The second presentation is also based on the paper arXiv:2403.14826, co-authored by Miha Brešar at Warwick.
Superdiffusive limits for Bessel-driven stochastic kinetics
In this presentation I explain why a Bessel-driven kinetic model exhibits anomalous diffusive behaviour. The model can be viewed as a Bessel process with a stochastic dimension. Unlike the Bessel process, at large times, this model exhibits superdiffusive growth at rates greater than square root.
The first half of the presentation sets out the model and the results. The second half discusses the proofs.
Bessel_Kinetics.pdfSlides from the video
This presentation, used in the YouTube video, is based on the paper "Superdiffusive limits for Bessel-driven stochastic kinetics", M. Brešar, C. da Costa, A. Mijatović, A. Wade
Fast exact simulation of the first-passage event of a subordinator
This presentation describes our exact simulation algorithm for the first-passage event of a general subordinator. It is based on the paper "Fast exact simulation of the first-passage event of a subordinator", J. González Cázares, F. Lin, A. Mijatović arXiv:2306.06927
First_passage_event_of_subordinator_slides.pdfSlides from the video
The slides used, in used in the YouTube presentation, explain our new algorithm for the exact simulation of the first-passage event for general subordinators and disucss our complexity results. For elements of the proof and applications of this algorithm, please see Part II of this presentation.
GitHub repository with implementation of exact simulation algorithm in Python and Julia
Fast exact simulation of the first-passage event of a subordinator (Part II)
This presentation describes our exact simulation algorithm for the first-passage event of a general subordinator. It is based on the paper "Fast exact simulation of the first-passage event of a subordinator", J. González Cázares, F. Lin, A. Mijatović arXiv:2306.06927
First_passage_event_of_subordinator_slides_Part2.pdfSlides from Part II
The slides used, in used in the YouTube presentation, explain the ideas behind the proof of the complexity theorem for the expected running time of our new algorithm for the exact simulation of the first-passage event for general subordinators. A numerical example is also discussed.
GitHub repository with implementation of exact simulation algorithm in Python and Julia
Fast exact simulation of the first passage of a tempered stable subordinator across a non-increasing function (Part 1)
This presentation describes our exact simulation algorithm for the first passage event. It is based on the recent paper "Fast exact simulation of the first passage of a tempered stable subordinator across a non-increasing function", J. González Cázares, F. Lin, A. Mijatović arXiv:2303.11964
First_passage_slides_Part_1.pdfSlides from Part 1
The slides used in the presentation explaining our new algorithm for the exact simulation of the first passage event for tempered stable subordinators. For applications of this algorithm, please see Part 2 of this presentation.
GitHub repository with implementation of exact simulation algorithm in Python and Julia
Fast exact simulation of the first passage of a tempered stable subordinator across a non-increasing function (Part 2)
Jorge describes Monte Carlo estimation, using our exact simulation algorithm for the first-passage event of a tempered stable subordinator, of a solution of a fractional PDE. See Part 1 of this presentation for the description of the exact simulation algorithm.
First_passage_slides_Part_2.pdfSlides from Part 2
The presentation used in this video is based on our recent paper "Fast exact simulation of the first passage of a tempered stable subordinator across a non-increasing function", J. González Cázares, F. Lin, A. Mijatović arXiv:2303.11964
GitHub repository with implementation of exact simulation algorithm in Python and Julia
Brownian motion with normal reflection in unbounded domains: transience to stability 1
In this video I discuss how a normally reflecting Brownian particle in an unbounded generalized parabolic domain can be positive recurrent. The main result described in the presentation states sharp polynomial rates of decay of the tails of the invariant distribution and polynomial convergence rates of the total variation distance of the reflected diffusion to stationarity.
For an intuitive description of the proofs in this paper see Part 2 of this video.
DiffusionsWithAsymptoticallyNormalReflectionPresentation_Part_I.pdfSlides for part 1
This presentation is based on the recent paper by M. Bresar, A. Mijatovic, and A.R. Wade, "Brownian motion with asymptotically normal reflection in unbounded domains: from stability to transience", (March 2023). arXiv link is arXiv:2303.06916
Brownian motion with normal reflection in unbounded domains: transience to stability 2
Part 2 of the presentation on the paper "Brownian motion with asymptotically normal reflection in unbounded domains: from stability to transience", M. Bresar, A. Mijatovic, and A.R. Wade (March 2023), discusses the ideas behind the proof. arXiv link is arXiv:2303.06916
DiffusionsWithAsymptoticallyNormalReflectionPresentation_Part_II.pdfSlides for part 2
The presentation used in part 2 video on the paper "Brownian motion with asymptotically normal reflection in unbounded domains: from stability to transience", M. Bresar, A. Mijatovic, and A.R. Wade (March 2023), arXiv link is arXiv:2303.06916
Explosion and superdiffusivity of reflected Brownian motion (Part 1)
In this video I describe what happens with an obliquely reflecting Brownian particle in a generalized parabolic domain.
This is a based on our recent paper by M. Menshikov, A. Mijatovic & A.R. Wade, "Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity", (2022), to appear in Ann. Inst. Henri Poincar\'e Probab. Stat.; arXiv link is here
explosions.pdfSlides for Part 1
Based on paper by M. Menshikov, A. Mijatovic & A.R. Wade, "Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity", (2022), to appear in Ann. Inst. Henri Poincar\'e Probab. Stat.; arXiv link is here
Explosion and superdiffusivity of reflected Brownian Motion (Part 2)
Based on paper by M. Menshikov, A. Mijatovic & A.R. Wade, "Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity", (2022), to appear in Ann. Inst. Henri Poincar\'e Probab. Stat.; arXiv link is here
explosions_part2.pdfSlides for Part 2
Based on paper by M. Menshikov, A. Mijatovic & A.R. Wade, "Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity", (2022), to appear in Ann. Inst. Henri Poincar\'e Probab. Stat.; arXiv link is here
Hölder continuity of the convex minorant of a Lévy process
In this video David gives a brief description of the characterisation of the Hölder continuity of the convex minorant of any Levy process. Our characterisation covers all Hölder exponents and nearly all Levy processes. This talk is based on the paper ``Hölder continuity of the convex minorant of a Levy process'', coauthored by David Bang (my PhD student at Warwick), Jorge Gonzalez Cazares (research fellow at Warwick) and AM.
Hoelder_continuity_slides.pdfSlides from the video
The paper "Hölder continuity of the convex minorant of a Lévy process?'' is on arXiv at: https://arxiv.org/abs/2207.12433
The related Stick-Breaking Representation Theorem video is here https://youtu.be/hEg4YmxOgXA
How smooth can the convex hull of a Lévy path be?
In this video I describes our recent results on the growth rate of the derivative of boundary of the the convex hull of a path of a Levy process. The paper "How smooth can the convex hull of a Levy path be?" is coauthored by David Bang (my PhD student at Warwick), Jorge Gonzalez Cazares (research fellow at Warwick) and AM.
LiltimedependentSubordinator_Presentation.pdfSlides from the video
The paper "How smooth can the convex hull of a Lévy path be?'' is on arXiv at: https://arxiv.org/abs/2206.09928
Related Stick-Breaking Representation Theorem video https://youtu.be/hEg4YmxOgXA
When is the convex hull of a Lévy path smooth?
PART 1
Jorge describes our recent results on the characterisation of the smoothness of the convex hull of a path of a Lévy process. The paper "When is the convex hull of a Lévy path smooth?" is coauthored by González Cázares (Research fellow at Warwick) and David Bang (my PhD student at Warwick).
When is the convex hull of a Lévy path smooth?
PART 2
This is Part 2 of Jorge's presentation about our recent results on the characterisation of the smoothness of the convex hull of a path of a Lévy process.
SmoothnessOfCMPresentation - split.pdfSlides from the videos
The paper "When is the convex hull of a Lévy path smooth?'' is on arXiv at: https://arxiv.org/abs/2205.14416
Related Stick-Breaking Representation Theorem video https://youtu.be/hEg4YmxOgXA
Convex minorant and fluctuation theory of Levy processes
The stick-breaking representation theorem (SBRT) for convex minorants is stated at 8'38''. First half of the video discusses the derivation of the fluctuation theory using convex minorants. The second half is about the proof of SBRT.
Fluctuation_theory_for_Levy_processes___presentation (8).pdfSlides from the video
The paper "Convex minorants and the fluctuation theory of Lévy processes'' is on arXiv at: http://arxiv.org/abs/2105.15060
Asymptotic shape of the concave majorant of a Levy process
My PhD student David Bang gives a short presentation of the results in the paper "Asymptotic shape of the concave majorant of a Lévy process", coauthored by Jorge González Cázares and myself.
AsympLengthPresentation.pdfSlides from the video
The paper is available on arXiv: https://arxiv.org/pdf/2106.09066.pdf
Tempered stick-breaking algorithm (TSB-Alg)
Jorge Gonzalez Cazares describes our work on the geometrically convergent algorithm for the simulation of tempered stable processes (including CGMY) in the context of barrier options.
SBT___presentation.pdfSlides from the video
The paper is available on arXiv: https://arxiv.org/abs/2103.15310
It includes a central limit theorem for the MC and MLMC estimators for any payoff depending on the supremum, position at final time and the time the supremum is attained.
SB-Algorithm: geometrically convergent simulation of Levy extrema
This presentation describes the SB-Alg for the simulation of Levy extrema. Its bias is geometrically small and all it needs is the ability to sample increments of the Levy process. See SBG-Alg presentation when increments cannot be sampled and TSB-Alg talk for tempered stable processes.
Paper available on arXiv: https://arxiv.org/abs/1810.11039
LevySupSim_presentation.pdfSlides from the video
This presentation is based on the paper "Geometrically Convergent Simulation of the Extrema of Lévy Processes" Jorge Gonzalez Cazares, AM and Geronimo Uribe Bravo
SBG-Alg (SB-Alg with Gaussian approx) https://youtu.be/EL0v2QUb5tQ
TSB-Alg (SB-Alg with tempering) https://youtu.be/FJG6A3zk2lI
Stick-Breaking Representation Thm https://youtu.be/hEg4YmxOgXA
How to sample extrema of a Levy process when even the increments are out of reach? (SBG-Alg)
SBG-Alg is based on SB-Alg and the Gaussian approximation of a Levy process. SBG-alg can be analysed for locally Lipschitz and discontinuous payoffs. A comparison with other methods is possible for Lipschitz payoffs where SBG-Alg typically outperforms existing methodologies by orders of magnitude.
SBG_Talk.pdfSlides from the video
Based on the paper ``Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation'' by Jorge Gonzalez Cazares and AM, available on arXiv https://arxiv.org/abs/2011.06618
Julia code on GitHub https://github.com/jorgeignaciogc/SBG.jl