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Partial differential equations (PDE) on manifolds arise in many areas, including mathematics and many applied fields. Among all kinds of PDEs, the Poisson-type equations including the standard Poisson equation and the related eigenproblem…
We present an analytic approach to solve a degenerate parabolic problem associated to the Heston model, which is widely used in mathematical finance to derive the price of an European option on an risky asset with stochastic volatility. We…
In the first part of this thesis, we focus on American options in the Heston model. We first give an analytical characterization of the value function of an American option as the unique solution of the associated (degenerate) parabolic…
We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes,…
An efficient linear solver plays an important role while solving partial differential equations (PDEs) and partial integro-differential equations (PIDEs) type mathematical models. In most cases, the efficiency depends on the stability and…
The crossover among two or more types of diffusive processes represents a vibrant theme in nonequilibrium statistical physics. In this work we propose two models to generate crossovers among different L\'evy processes: in the first model we…
SABR models have been used to incorporate stochastic volatility to LIBOR market models (LMM) in order to describe interest rate dynamics and price interest rate derivatives. From the numerical point of view, the pricing of derivatives with…
Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs),…
Standard jump-diffusion models assume independence between jumps and diffusion components. We develop a multi-type jump-diffusion model where jump occurrence and magnitude depend on contemporaneous diffusion movements. Unlike previous…
We develop a mixed least squares Monte Carlo-partial differential equation (LSMC-PDE) method for pricing Bermudan style options on assets whose volatility is stochastic. The algorithm is formulated for an arbitrary number of assets and…
This work is focused on the solvability of initial-boundary value problems for degenerate parabolic partial differential equations that arise in the pricing of Asian options, and on the investigation of differential and certain qualitative…
As a simplified model for subsurface flows elliptic equations may be utilized. Insufficient measurements or uncertainty in those are commonly modeled by a random coefficient, which then accounts for the uncertain permeability of a given…
Several two-boundary problems are solved for a special L\'{e}vy process: the Poisson process with an exponential component. The jumps of this process are controlled by a homogeneous Poisson process, the positive jump size distribution is…
The change of numeraire gives very important computational simplification in option pricing. This technique reduces the number of sources of risks that need to be accounted for and so it is useful in pricing complicated derivatives that…
Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged as an alternative to classical numerical schemes for solving Partial Differential Equations (PDEs). They are very appealing at…
We consider a jump-diffusion process on a bounded domain with reflection at the boundary, and establish long-term results for a general additive process of its path. This includes the long-term behaviour of its occupation time in the…
This paper performs the numerical analysis and the computation of a Spread option in a market with imperfect liquidity. The number of shares traded in the stock market has a direct impact on the stock's price. Thus, we consider a…
We study solution techniques for parabolic equations with fractional diffusion and Caputo fractional time derivative, the latter being discretized and analyzed in a general Hilbert space setting. The spatial fractional diffusion is realized…
Partial differential equation (PDE) solvers underpin modern quantitative finance, governing option pricing and risk evaluation. Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving the forward and…
We consider an 1D partial integro-differential equation (PIDE) comprising of an 1D parabolic partial differential equation (PDE) and a nonlocal integral term. The control input is applied on one of the boundaries of the PIDE. Partitioning…