scholarly journals Joint distribution of a Lévy process and its running supremum

2018 ◽  
Vol 55 (2) ◽  
pp. 488-512 ◽  
Author(s):  
Laure Coutin ◽  
Monique Pontier ◽  
Waly Ngom
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Weak Solution ◽  
Diffusion Process ◽  
Lebesgue Measure ◽  
Joint Distribution ◽  
Random Variable ◽  
Jump Diffusion ◽  
Marginal Density ◽  
The Law

Abstract Let X be a jump-diffusion process and X* its running supremum. In this paper we first show that for any t > 0, the law of the pair (X*t, Xt) has a density with respect to the Lebesgue measure. This allows us to show that for any t > 0, the law of the pair formed by the random variable Xt and the running supremum X*t of X at time t can be characterized as a weak solution of a partial differential equation concerning the distribution of the pair (X*t, Xt). Then we obtain an expression of the marginal density of X*t for all t > 0.

scholarly journals Existence of a Unique Solution to a Fractional Partial Differential Equation and Its Continuous Dependence on Parameters

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 851
Author(s):  
Robert Stegliński
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Weak Solution ◽  
Variational Methods ◽  
Unique Solution ◽  
Continuous Dependence ◽  
Fractional Laplacian ◽  
Unique Weak Solution ◽  
Fractional Partial Differential Equation ◽  
Integrodifferential Operator

In the present paper we give conditions under which there exists a unique weak solution for a nonlocal equation driven by the integrodifferential operator of fractional Laplacian type. We argue for the optimality of some assumptions. Some Lyapunov-type inequalities are given. We also study the continuous dependence of the solution on parameters. In proofs we use monotonicity and variational methods.

scholarly journals The ruin problem for a Wiener process with state-dependent jumps

2020 ◽  
Vol 16 (1) ◽  
pp. 13-23
Author(s):  
M. Lefebvre
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Diffusion Process ◽  
Wiener Process ◽  
Jump Diffusion ◽  
Moment Generating Function ◽  
Continuous Part ◽  
State Dependent ◽  
The Moment ◽  
First Time

AbstractLet X(t) be a jump-diffusion process whose continuous part is a Wiener process, and let T (x) be the first time it leaves the interval (0,b), where x = X(0). The jumps are negative and their sizes depend on the value of X(t). Moreover there can be a jump from X(t) to 0. We transform the integro-differential equation satisfied by the probability p(x) := P[X(T (x)) = 0] into an ordinary differential equation and we solve this equation explicitly in particular cases. We are also interested in the moment-generating function of T (x).

Sojourns and extremes of a diffusion process on a fixed interval

1982 ◽  
Vol 14 (4) ◽  
pp. 811-832 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Diffusion Process ◽  
Lebesgue Measure ◽  
Stationary Distribution ◽  
Real Line ◽  
Diffusion Coefficients ◽  
Limit Theorems ◽  
Random Variable ◽  
Fixed Interval ◽  
The Real ◽  
And Diffusion

Let X(t), , be an Ito diffusion process on the real line. For u > 0 and t > 0, let Lt(u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt(u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t]X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.

scholarly journals On critical exponents for weak solutions to the Cauchy problem for one nonlinear equation with gradient non-linearity

2020 ◽  
Author(s):  
Maxim Korpusov ◽  
Alexandra Matveeva
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Weak Solution ◽  
Nonlinear Equation ◽  
Critical Exponents ◽  
Weak Sense ◽  
Order Partial Differential Equation ◽  
Third Order ◽  
The Cauchy Problem

In this paper, we consider the Cauchy problem for one nonclassical, third-order, partial differential equation with gradient non-linearity $|\nabla u(x,t)|^q$. The solution to this problem is understood in a weak sense. We show that for $1“3/2$ the existence of the only local-in-time weak solution of Cauchy’s problem.If $3/2”

Sojourns and extremes of a diffusion process on a fixed interval

1982 ◽  
Vol 14 (04) ◽  
pp. 811-832 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Diffusion Process ◽  
Lebesgue Measure ◽  
Stationary Distribution ◽  
Real Line ◽  
Diffusion Coefficients ◽  
Limit Theorems ◽  
Random Variable ◽  
The Real ◽  
Simple Limit ◽  
And Diffusion

Let X(t), , be an Ito diffusion process on the real line. For u > 0 and t > 0, let Lt (u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt (u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t] X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.

Weak star limits of probability measures of special type

1999 ◽  
Vol 127 (1) ◽  
pp. 173-191
Author(s):  
IOANNIS PAPADOPERAKIS
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Lebesgue Measure ◽  
Probability Measures ◽  
Compact Set ◽  
Winding Number ◽  
Compact Sets ◽  
Partial Differential ◽  
Accumulation Points ◽  
Weak Star

We study the weak star accumulation points of sequences of probability measures of the form [XKn(x) ρn(x)dσ(x)]/ [∫Kn ρn(t)dσ(t)], where ρ(x)>0 is continuous on Rκ, σ denotes Lebesgue measure in Rκ and the sequence of compact sets Kn⊂Rκ converges in the sense of Hausdorff towards a compact set K. The motivation of our study was given by a result relating interval averages with the winding number. Similar probability measures are considered in partial differential equation problems and we extend our study to this case.

Uniqueness of the Solution to a Difference-Partial Differential Equation for Finance

2003 ◽  
Vol 13 (07) ◽  
pp. 919-943
Author(s):  
C. Mancini
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Interest Rate ◽  
Jump Diffusion ◽  
Market Model ◽  
Contingent Claim ◽  
Partial Differential ◽  
Financial Model ◽  
Diffusion Modelling ◽  
Uniqueness Of The Solution

In this paper we study a difference partial differential equation, arising from a financial model, whose solution represents the price of a security linked to a dividend-paying stock. The market model consists of a jump-diffusion process modelling the stock evolution and an independent diffusion modelling the stochastic spot interest rate. We establish the desirable property of the uniqueness of solution to the equation and, since the specialized model is complete, we can consistently price any contingent claim.

scholarly journals Including Jumps in the Stochastic Valuation of Freight Derivatives

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 154
Author(s):  
Lourdes Gómez-Valle ◽  
Julia Martínez-Rodríguez
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Diffusion Process ◽  
Option Price ◽  
Jump Diffusion ◽  
Freight Rate ◽  
Parametric Models ◽  
Geometric Process ◽  
Rate Processes ◽  
Jump Diffusion Model

The spot freight rate processes considered in the literature for pricing forward freight agreements (FFA) and freight options usually have a particular dynamics in order to obtain the prices. In those cases, the FFA prices are explicitly obtained. However, for jump-diffusion models, an exact solution is not known for the freight options (Asian-type), in part due to the absence of a suitable valuation framework. In this paper, we consider a general jump-diffusion process to describe the spot freight dynamics and we obtain exact solutions of FFA prices for two parametric models. Moreover, we develop a partial integro-differential equation (PIDE), for pricing freight options for a general unifactorial jump-diffusion model. When we consider that the spot freight follows a geometric process with jumps, we obtain a solution of the freight option price in a part of its domain. Finally, we show the effect of the jumps in the FFA prices by means of numerical simulations.

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