This dissertation explores various topics related to stochastic dynamics with jumps, which find their initial motivation in numerical applications. It undertakes two distinct research trajectories, which ultimately converge in the concluding chapter of the thesis, where a potential interconnection between them is established from a theoretical perspective.In the first line of study, the concepts of cylindrical Wiener process subordinated to a strictly α−stable Lévy process, with α ∈ (0, 1), and of the corresponding stochastic convolution are introduced in an infinitedimensional, separable Hilbert space. The related OrnsteinUhlenbeck (OU) process is then analyzed, with a focus on the regularizing properties of the associated Markov transition semigroup. In particular, an original formula –which is not of BismutElworthyLi’s type– for the Gateaux derivatives of the functions generated by this semigroup is provided, together with an estimate for the norm of their gradients.Taking α ∈ (1/2,1), these results are applied to the study of semilinear, N−dimensional stochastic differential equations (SDEs) driven by the same additive, isotropic, stable Lévy noise. An important connection between the timedependent Markov transition semigroup associated with their solutions and Kolmogorov backward equations in mild integral form is established via regularizationbynoise techniques. Such a link is the starting point for an iterative method which allows to approximate probabilities related to the SDEs with a single batch of Monte Carlo simulations as several parameters change, bringing a compelling computational advantage over the standard Monte Carlo approach. This method also pertains to the numerical computation of solutions of highdimensional integrodifferential Kolmogorov backward equations. The scheme, and in particular the first order approximation it provides, is then tested for two nonlinear vector fields in dimension N = 100 and shown to offer satisfactory results, especially when compared with the OU approximation.Within this analysis, one of the concepts employed is the stochastic flow generated by SDEs with additive Lévy noise. In this dissertation, an extension to the case of multiplicative noise of some results known in the existing literature for the additive case is presented. More specifically, the existence of a sharp stochastic flow (Xs,x)t for an SDE of Itô’s type with multiplicative noise which, P−a.s., is simultaneously continuous in x (starting point) and càdlàg in s (starting time) and t (time) is proved. Remarkably, the study encompasses SDEs that include both compensated and noncompensated jump components, thereby addressing both small and large jumps, alongside a Brownian diffusion term. The theory is further expanded to cover controlled SDEs. Using the resulting sharp stochastic flow, a new dynamic programming principle is established with an argument that stands as an independently significant point of interest.The second area of research regards the theory of affine processes, which has been recently extended to stochastic Volterra equations (SVEs) with continuous trajectories. These socalled affine Volterra processes overcome modeling shortcomings of classical affine processes because they may possess pathdependent features which introduce memory structures into the models. Furthermore, they can have trajectories whose regularity is different from the paths of Brownian motion. In particular, singular kernels yield rough affine processes. In this thesis, a generalization of the abovementioned theory by considering affine SVEs with jumps is studied. This extension is not straightforward because the jump structure together with possible singularities of the kernel may induce explosions of the trajectories. Nonetheless, the extended framework enables to obtain semiexplicit formulas for the conditional FourierLaplace transforms of the solutions via deterministic RiccatiVolterra equations. This study also provides exponential affine expressions for the conditional transforms of marked Hawkes processes.Building upon this analysis, an extension of the rough Heston stochastic volatility model is introduced. In this extension, called the rough Hawkes Heston stochastic volatility model, the instantaneous spot variance is modeled as the solution of an affine SVE of convolution type with jumps. This setting takes into account both rough volatility and jump clustering phenomena, and employs the affine structure of the SVE to price options on the underlying (SPX) and on the related volatility index (VIX) via Fourier inversion techniques. A calibration of the model featuring a fractional kernel and an exponential distribution for the jumps is carried out, demonstrating its ability to accurately and simultaneously capture the volatility smiles of both SPX and VIX options. This is a remarkable result, especially considering the few parameters of the proposed model, namely five for the evolution of the dynamics and two for the term structure. Moreover, it proves the relevance, under an affine framework, of rough volatility and selfexciting jumps in order to describe the joint evolution of SPX and VIX.Lastly, an exploration of the theoretical interconnection between the subjects of the two preceding areas of research is investigated. More precisely, a Volterra convolution equation in Rd perturbed with an additive fractional Brownian motion of Riemann–Liouville type characterized by a Hurst parameter H ∈ (0, 1) is considered. The solution of this equation is shown to satisfy a stochastic partial differential equation (SPDE) in the Hilbert space of squareintegrable functions. This particular equation serves as the motivation for the study of an unconventional class of SPDEs, necessitating an original extension of the drift operator and its Fréchet differentials. It is demonstrated that these SPDEs generate a Markov stochastic flow which is twice Fréchet differentiable with respect to the initial data. This stochastic flow is subsequently employed to solve, in the classical sense of infinitedimensional calculus, the pathdependent Kolmogorov equation corresponding to the SPDEs. In particular, a timedependent infinitesimal generator is associated with the fractional Brownian motion. Certain challenges arise in the analysis of the mild formulation of the Kolmogorov equation for SPDEs driven by the same infinitedimensional noise. This issue, which is relevant to the theory of regularizationbynoise, remains an open area for future research.
On semilinear SDEs driven by 𝛂–stable noise, affine Volterra processes with jumps and their applications / Bondi, Alessandro; relatore: FLANDOLI, FRANCO; Scuola Normale Superiore, ciclo 35, 19Dec2024.
On semilinear SDEs driven by 𝛂–stable noise, affine Volterra processes with jumps and their applications
BONDI, Alessandro
2024
Abstract
This dissertation explores various topics related to stochastic dynamics with jumps, which find their initial motivation in numerical applications. It undertakes two distinct research trajectories, which ultimately converge in the concluding chapter of the thesis, where a potential interconnection between them is established from a theoretical perspective.In the first line of study, the concepts of cylindrical Wiener process subordinated to a strictly α−stable Lévy process, with α ∈ (0, 1), and of the corresponding stochastic convolution are introduced in an infinitedimensional, separable Hilbert space. The related OrnsteinUhlenbeck (OU) process is then analyzed, with a focus on the regularizing properties of the associated Markov transition semigroup. In particular, an original formula –which is not of BismutElworthyLi’s type– for the Gateaux derivatives of the functions generated by this semigroup is provided, together with an estimate for the norm of their gradients.Taking α ∈ (1/2,1), these results are applied to the study of semilinear, N−dimensional stochastic differential equations (SDEs) driven by the same additive, isotropic, stable Lévy noise. An important connection between the timedependent Markov transition semigroup associated with their solutions and Kolmogorov backward equations in mild integral form is established via regularizationbynoise techniques. Such a link is the starting point for an iterative method which allows to approximate probabilities related to the SDEs with a single batch of Monte Carlo simulations as several parameters change, bringing a compelling computational advantage over the standard Monte Carlo approach. This method also pertains to the numerical computation of solutions of highdimensional integrodifferential Kolmogorov backward equations. The scheme, and in particular the first order approximation it provides, is then tested for two nonlinear vector fields in dimension N = 100 and shown to offer satisfactory results, especially when compared with the OU approximation.Within this analysis, one of the concepts employed is the stochastic flow generated by SDEs with additive Lévy noise. In this dissertation, an extension to the case of multiplicative noise of some results known in the existing literature for the additive case is presented. More specifically, the existence of a sharp stochastic flow (Xs,x)t for an SDE of Itô’s type with multiplicative noise which, P−a.s., is simultaneously continuous in x (starting point) and càdlàg in s (starting time) and t (time) is proved. Remarkably, the study encompasses SDEs that include both compensated and noncompensated jump components, thereby addressing both small and large jumps, alongside a Brownian diffusion term. The theory is further expanded to cover controlled SDEs. Using the resulting sharp stochastic flow, a new dynamic programming principle is established with an argument that stands as an independently significant point of interest.The second area of research regards the theory of affine processes, which has been recently extended to stochastic Volterra equations (SVEs) with continuous trajectories. These socalled affine Volterra processes overcome modeling shortcomings of classical affine processes because they may possess pathdependent features which introduce memory structures into the models. Furthermore, they can have trajectories whose regularity is different from the paths of Brownian motion. In particular, singular kernels yield rough affine processes. In this thesis, a generalization of the abovementioned theory by considering affine SVEs with jumps is studied. This extension is not straightforward because the jump structure together with possible singularities of the kernel may induce explosions of the trajectories. Nonetheless, the extended framework enables to obtain semiexplicit formulas for the conditional FourierLaplace transforms of the solutions via deterministic RiccatiVolterra equations. This study also provides exponential affine expressions for the conditional transforms of marked Hawkes processes.Building upon this analysis, an extension of the rough Heston stochastic volatility model is introduced. In this extension, called the rough Hawkes Heston stochastic volatility model, the instantaneous spot variance is modeled as the solution of an affine SVE of convolution type with jumps. This setting takes into account both rough volatility and jump clustering phenomena, and employs the affine structure of the SVE to price options on the underlying (SPX) and on the related volatility index (VIX) via Fourier inversion techniques. A calibration of the model featuring a fractional kernel and an exponential distribution for the jumps is carried out, demonstrating its ability to accurately and simultaneously capture the volatility smiles of both SPX and VIX options. This is a remarkable result, especially considering the few parameters of the proposed model, namely five for the evolution of the dynamics and two for the term structure. Moreover, it proves the relevance, under an affine framework, of rough volatility and selfexciting jumps in order to describe the joint evolution of SPX and VIX.Lastly, an exploration of the theoretical interconnection between the subjects of the two preceding areas of research is investigated. More precisely, a Volterra convolution equation in Rd perturbed with an additive fractional Brownian motion of Riemann–Liouville type characterized by a Hurst parameter H ∈ (0, 1) is considered. The solution of this equation is shown to satisfy a stochastic partial differential equation (SPDE) in the Hilbert space of squareintegrable functions. This particular equation serves as the motivation for the study of an unconventional class of SPDEs, necessitating an original extension of the drift operator and its Fréchet differentials. It is demonstrated that these SPDEs generate a Markov stochastic flow which is twice Fréchet differentiable with respect to the initial data. This stochastic flow is subsequently employed to solve, in the classical sense of infinitedimensional calculus, the pathdependent Kolmogorov equation corresponding to the SPDEs. In particular, a timedependent infinitesimal generator is associated with the fractional Brownian motion. Certain challenges arise in the analysis of the mild formulation of the Kolmogorov equation for SPDEs driven by the same infinitedimensional noise. This issue, which is relevant to the theory of regularizationbynoise, remains an open area for future research.File  Dimensione  Formato  

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