In this paper we present a method to derive explicit representations of strong solutions of forwa... more In this paper we present a method to derive explicit representations of strong solutions of forward stochastic differential equations driven by a Brownian motion. These representations open new perspectives in the study of important topics like large time behaviour or the flow property of solutions of such equations.
Fractional Brownian motion with Hurst parameter $$H<\frac{1}{2}$$ H < 1 2 is used widely, f... more Fractional Brownian motion with Hurst parameter $$H<\frac{1}{2}$$ H < 1 2 is used widely, for instance, to describe ‘rough’ volatility data in finance. In this paper, we examine a generalised Ait-Sahalia-type model driven by a fractional Brownian motion with $$H<\frac{1}{2}$$ H < 1 2 and establish theoretical properties such as an existence-and-uniqueness theorem, regularity in the sense of Malliavin differentiability and higher moments of the strong solutions.
The purpose of this paper is to study the following topics and the relation between them: (i) Opt... more The purpose of this paper is to study the following topics and the relation between them: (i) Optimal singular control of mean-field stochastic differential equations with memory, (ii) reflected advanced mean-field backward stochastic differential equations, and (iii) optimal stopping of mean-field stochastic differential equations. More specifically, we do the following: - We prove the existence and uniqueness of the solutions of some reflected advanced memory backward stochastic differential equations (AMBSDEs), - we give sufficient and necessary conditions for an optimal singular control of a memory mean-field stochastic differential equation (MMSDE) with partial information, and - we deduce a relation between the optimal singular control of a MMSDE, and the optimal stopping of such processes.
Journal of Mathematical Analysis and Applications, 2014
This paper is concerned with the existence and uniqueness of weak solutions to the Cauchy-Dirichl... more This paper is concerned with the existence and uniqueness of weak solutions to the Cauchy-Dirichlet problem of backward stochastic partial differential equations (BSPDEs) with nonhomogeneous terms of quadratic growth in both the gradient of the first unknown and the second unknown. As an example, we consider a non-Markovian stochastic optimal control problem with cost functional formulated by a quadratic BSDE, where the corresponding value function satisfies the above quadratic BSPDE.
In this paper we demonstrate how concepts of white noise analysis can be used to give an explicit... more In this paper we demonstrate how concepts of white noise analysis can be used to give an explicit solution to a stochastic transport equation driven by Lévy white noise.
In this paper we develop a white noise framework for the study of stochastic partial differential... more In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a d-parameter (pure jump) Lévy white noise. As an example we use this theory to solve the stochastic Poisson equation with respect to Lévy white noise for any dimension d. The solution is a stochastic distribution process given explicitly. We also show that if d ≤ 3, then this solution can be represented as a classical random field in L 2 (µ), where µ is the probability law of the Lévy process. The starting point of our theory is a chaos expansion in terms of generalized Charlier polynomials. Based on this expansion we define Kondratiev spaces and the Lévy Hermite transform.
In this paper we introduce a new technique to construct unique strong solutions of SDEs with sing... more In this paper we introduce a new technique to construct unique strong solutions of SDEs with singular coefficients driven by certain Levy processes. Our method which is based on Malliavin calculus does not rely on a pathwise uniqueness argument. Furthermore, the approach, which provides a direct construction principle, grants the additional insight that the obtained solutions are Malliavin differentiable.
In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms R^d∋ x ... more In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms R^d∋ x ϕ_s,t(x)∈R^d, s,t∈R for a stochastic differential equation (SDE) of the form dX_t=b(t,X_t) dt+dB_t, s,t∈R,X_s=x∈R^d. The above SDE is driven by a bounded measurable drift coefficient b:R×R^d→R^d and a d-dimensional Brownian motion B. More specifically, we show that the stochastic flow ϕ_s,t(·) of the SDE lives in the space L^2(Ω;W^1,p(R^d,w)) for all s,t and all p∈ (1,∞), where W^1,p(R^d,w) denotes a weighted Sobolev space with weight w possessing a pth moment with respect to Lebesgue measure on R^d. From the viewpoint of stochastic (and deterministic) dynamical systems, this is a striking result, since the dominant "culture" in these dynamical systems is that the flow "inherits" its spatial regularity from that of the driving vector fields. The spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the (...
In this article we prove path-by-path uniqueness in the sense of Davie \cite{Davie07} and Shaposh... more In this article we prove path-by-path uniqueness in the sense of Davie \cite{Davie07} and Shaposhnikov \cite{Shaposhnikov16} for SDE's driven by a fractional Brownian motion with a Hurst parameter $H\in(0,\frac{1}{2})$, uniformly in the initial conditions, where the drift vector field is allowed to be merely bounded and measurable.\par Using this result, we construct weak unique regular solutions in $W_{loc}^{k,p}\left([0,1]\times\mathbb{R}^d\right)$, $p>d$ of the classical transport and continuity equations with singular velocity fields perturbed along fractional Brownian paths.\par The latter results provide a systematic way of producing examples of singular velocity fields, which cannot be treated by the regularity theory of DiPerna-Lyons \cite{DiPernaLions89}, Ambrosio \cite{Ambrosio04} or Crippa-De Lellis \cite{CrippaDeLellis08}.\par Our approach is based on a priori estimates at the level of flows generated by a sequence of mollified vector fields, converging to the ori...
In this paper we construct a new type of noise of fractional nature that has a strong regularizin... more In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation with this noise with a highly irregular coefficient. We employ a new method to prove existence and uniqueness of global strong solutions where classical methods fail because of the "roughness" and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. This opens a fundamental question on studying certain classes of interesting partial differential equations perturbed by this noise.
In this article we introduce a new method for the construction of unique strong solutions of a la... more In this article we introduce a new method for the construction of unique strong solutions of a larger class of stochastic delay equations driven by a discontinuous drift vector field and a Wiener process. The results obtained in this paper can be regarded as an infinite-dimensional generalization of those of A. Y. Veretennikov [42] in the case of certain stochastic delay equations with irregular drift coefficients. The approach proposed in this work rests on Malliavin calculus and arguments of a "local time variational calculus", which may also be used to study other types of stochastic equations as e.g. functional It\^{o}-stochastic differential equations in connection with path-dependent Kolmogorov equations [15].
We consider the problem of utility indifference pricing of a put option written on a non-tradeabl... more We consider the problem of utility indifference pricing of a put option written on a non-tradeable asset, where we can hedge in a correlated asset. The dynamics are assumed to be a two-dimensional geometric Brownian motion, and we suppose that the issuer of the option have exponential risk preferences. We prove that the indifference price dynamics is a martingale with respect to an equivalent martingale measure (EMM) Q after discounting, implying that it is arbitrage-free. Moreover, we provide a representation of the residual risk remaining after using the optimal utility-based trading strategy as the hedge. Our motivation for this study comes from pricing interest-rate guarantees, which are products usually offered by companies managing pension funds. In certain market situations the life company cannot hedge perfectly the guarantee, and needs to resort to sub-optimal replication strategies. We argue that utility indifference pricing is a suitable method for analysing such cases. W...
In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to sin... more In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to singular stochastic differential equations (SDE's) with additive noise given by a multi-dimensional fractional Brownian motion with Hurst parameter $H<1/2$. "Singular" here means that the drift vector field of such equations is allowed to be merely bounded and integrable. As an application we use this representation formula for the study of price sensitivities of financial claims based on a stock price model with stochastic volatility, whose dynamics is described by means of fractional Brownian motion driven SDE's. Our approach for obtaining these results is based on Malliavin calculus and arguments of a recently developed "local time variational calculus".
We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean-Vlasov ... more We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean-Vlasov equations with random coefficients, which are allowed to be nonLipschitz continuous. In the case of deterministic coefficients we also obtain the existence of unique strong solutions. By using our approach, which is based on an extension of the Yamada-Watanabe ansatz to the multidimensional setting and which does not rely on the construction of Lyapunov functions, we prove first moment and pathwise α-exponential stability of solutions for α > 0. Furthermore, we are able to compute Lyapunov exponents explicitly. MSC2010 classification: 60H20, 60H30, 60F25, 37H30, 45M10.
In this article we will present a new perspective on the variable order fractional calculus, whic... more In this article we will present a new perspective on the variable order fractional calculus, which allows for differentiation and integration to a variable order, i.e. one differentiates (or integrates) a function along the path of a regularity function. The concept of multifractional calculus has been a scarcely studied topic within the field of functional analysis in the last 20 years. We develop a multifractional derivative operator which acts as the inverse of the multifractional integral operator. This is done by solving the Abel integral equation generalized to a multifractional order. With this new multifractional derivative operator, we are able to analyze a variety of new problems, both in the field of stochastic analysis and in fractional and functional analysis, ranging from regularization properties of noise to solutions to multifractional differential equations. In this paper, we will focus on application of the derivative operator to the construction of strong solution...
The purpose of this paper is two-fold: We extend the well-known relation between optimal stopping... more The purpose of this paper is two-fold: We extend the well-known relation between optimal stopping and randomized stopping of a given stochastic process to a situation where the available informatio ...
In this paper, we show the existence of unique Malliavin differentiable solutions to SDE‘s driven... more In this paper, we show the existence of unique Malliavin differentiable solutions to SDE‘s driven by a fractional Brownian motion with Hurst parameter H < 12 and singular, unbounded drift vector fields, for which we also prove a stability result. Further, using the latter results, we propose a stock price model with rough and correlated volatility, which also allows for capturing regime switching effects. Finally, we also derive a Bismut-Elworthy-Li formula with respect to our stock price model for certain classes of vector fields.
A general market model with memory is considered in terms of stochastic functional differential e... more A general market model with memory is considered in terms of stochastic functional differential equations. We aim at representation formulae for the sensitivity analysis of the dependence of option prices on the memory. This implies a generalization of the concept of delta.
In this paper we present a method to derive explicit representations of strong solutions of forwa... more In this paper we present a method to derive explicit representations of strong solutions of forward stochastic differential equations driven by a Brownian motion. These representations open new perspectives in the study of important topics like large time behaviour or the flow property of solutions of such equations.
Fractional Brownian motion with Hurst parameter $$H<\frac{1}{2}$$ H < 1 2 is used widely, f... more Fractional Brownian motion with Hurst parameter $$H<\frac{1}{2}$$ H < 1 2 is used widely, for instance, to describe ‘rough’ volatility data in finance. In this paper, we examine a generalised Ait-Sahalia-type model driven by a fractional Brownian motion with $$H<\frac{1}{2}$$ H < 1 2 and establish theoretical properties such as an existence-and-uniqueness theorem, regularity in the sense of Malliavin differentiability and higher moments of the strong solutions.
The purpose of this paper is to study the following topics and the relation between them: (i) Opt... more The purpose of this paper is to study the following topics and the relation between them: (i) Optimal singular control of mean-field stochastic differential equations with memory, (ii) reflected advanced mean-field backward stochastic differential equations, and (iii) optimal stopping of mean-field stochastic differential equations. More specifically, we do the following: - We prove the existence and uniqueness of the solutions of some reflected advanced memory backward stochastic differential equations (AMBSDEs), - we give sufficient and necessary conditions for an optimal singular control of a memory mean-field stochastic differential equation (MMSDE) with partial information, and - we deduce a relation between the optimal singular control of a MMSDE, and the optimal stopping of such processes.
Journal of Mathematical Analysis and Applications, 2014
This paper is concerned with the existence and uniqueness of weak solutions to the Cauchy-Dirichl... more This paper is concerned with the existence and uniqueness of weak solutions to the Cauchy-Dirichlet problem of backward stochastic partial differential equations (BSPDEs) with nonhomogeneous terms of quadratic growth in both the gradient of the first unknown and the second unknown. As an example, we consider a non-Markovian stochastic optimal control problem with cost functional formulated by a quadratic BSDE, where the corresponding value function satisfies the above quadratic BSPDE.
In this paper we demonstrate how concepts of white noise analysis can be used to give an explicit... more In this paper we demonstrate how concepts of white noise analysis can be used to give an explicit solution to a stochastic transport equation driven by Lévy white noise.
In this paper we develop a white noise framework for the study of stochastic partial differential... more In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a d-parameter (pure jump) Lévy white noise. As an example we use this theory to solve the stochastic Poisson equation with respect to Lévy white noise for any dimension d. The solution is a stochastic distribution process given explicitly. We also show that if d ≤ 3, then this solution can be represented as a classical random field in L 2 (µ), where µ is the probability law of the Lévy process. The starting point of our theory is a chaos expansion in terms of generalized Charlier polynomials. Based on this expansion we define Kondratiev spaces and the Lévy Hermite transform.
In this paper we introduce a new technique to construct unique strong solutions of SDEs with sing... more In this paper we introduce a new technique to construct unique strong solutions of SDEs with singular coefficients driven by certain Levy processes. Our method which is based on Malliavin calculus does not rely on a pathwise uniqueness argument. Furthermore, the approach, which provides a direct construction principle, grants the additional insight that the obtained solutions are Malliavin differentiable.
In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms R^d∋ x ... more In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms R^d∋ x ϕ_s,t(x)∈R^d, s,t∈R for a stochastic differential equation (SDE) of the form dX_t=b(t,X_t) dt+dB_t, s,t∈R,X_s=x∈R^d. The above SDE is driven by a bounded measurable drift coefficient b:R×R^d→R^d and a d-dimensional Brownian motion B. More specifically, we show that the stochastic flow ϕ_s,t(·) of the SDE lives in the space L^2(Ω;W^1,p(R^d,w)) for all s,t and all p∈ (1,∞), where W^1,p(R^d,w) denotes a weighted Sobolev space with weight w possessing a pth moment with respect to Lebesgue measure on R^d. From the viewpoint of stochastic (and deterministic) dynamical systems, this is a striking result, since the dominant "culture" in these dynamical systems is that the flow "inherits" its spatial regularity from that of the driving vector fields. The spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the (...
In this article we prove path-by-path uniqueness in the sense of Davie \cite{Davie07} and Shaposh... more In this article we prove path-by-path uniqueness in the sense of Davie \cite{Davie07} and Shaposhnikov \cite{Shaposhnikov16} for SDE's driven by a fractional Brownian motion with a Hurst parameter $H\in(0,\frac{1}{2})$, uniformly in the initial conditions, where the drift vector field is allowed to be merely bounded and measurable.\par Using this result, we construct weak unique regular solutions in $W_{loc}^{k,p}\left([0,1]\times\mathbb{R}^d\right)$, $p>d$ of the classical transport and continuity equations with singular velocity fields perturbed along fractional Brownian paths.\par The latter results provide a systematic way of producing examples of singular velocity fields, which cannot be treated by the regularity theory of DiPerna-Lyons \cite{DiPernaLions89}, Ambrosio \cite{Ambrosio04} or Crippa-De Lellis \cite{CrippaDeLellis08}.\par Our approach is based on a priori estimates at the level of flows generated by a sequence of mollified vector fields, converging to the ori...
In this paper we construct a new type of noise of fractional nature that has a strong regularizin... more In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation with this noise with a highly irregular coefficient. We employ a new method to prove existence and uniqueness of global strong solutions where classical methods fail because of the "roughness" and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. This opens a fundamental question on studying certain classes of interesting partial differential equations perturbed by this noise.
In this article we introduce a new method for the construction of unique strong solutions of a la... more In this article we introduce a new method for the construction of unique strong solutions of a larger class of stochastic delay equations driven by a discontinuous drift vector field and a Wiener process. The results obtained in this paper can be regarded as an infinite-dimensional generalization of those of A. Y. Veretennikov [42] in the case of certain stochastic delay equations with irregular drift coefficients. The approach proposed in this work rests on Malliavin calculus and arguments of a "local time variational calculus", which may also be used to study other types of stochastic equations as e.g. functional It\^{o}-stochastic differential equations in connection with path-dependent Kolmogorov equations [15].
We consider the problem of utility indifference pricing of a put option written on a non-tradeabl... more We consider the problem of utility indifference pricing of a put option written on a non-tradeable asset, where we can hedge in a correlated asset. The dynamics are assumed to be a two-dimensional geometric Brownian motion, and we suppose that the issuer of the option have exponential risk preferences. We prove that the indifference price dynamics is a martingale with respect to an equivalent martingale measure (EMM) Q after discounting, implying that it is arbitrage-free. Moreover, we provide a representation of the residual risk remaining after using the optimal utility-based trading strategy as the hedge. Our motivation for this study comes from pricing interest-rate guarantees, which are products usually offered by companies managing pension funds. In certain market situations the life company cannot hedge perfectly the guarantee, and needs to resort to sub-optimal replication strategies. We argue that utility indifference pricing is a suitable method for analysing such cases. W...
In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to sin... more In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to singular stochastic differential equations (SDE's) with additive noise given by a multi-dimensional fractional Brownian motion with Hurst parameter $H<1/2$. "Singular" here means that the drift vector field of such equations is allowed to be merely bounded and integrable. As an application we use this representation formula for the study of price sensitivities of financial claims based on a stock price model with stochastic volatility, whose dynamics is described by means of fractional Brownian motion driven SDE's. Our approach for obtaining these results is based on Malliavin calculus and arguments of a recently developed "local time variational calculus".
We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean-Vlasov ... more We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean-Vlasov equations with random coefficients, which are allowed to be nonLipschitz continuous. In the case of deterministic coefficients we also obtain the existence of unique strong solutions. By using our approach, which is based on an extension of the Yamada-Watanabe ansatz to the multidimensional setting and which does not rely on the construction of Lyapunov functions, we prove first moment and pathwise α-exponential stability of solutions for α > 0. Furthermore, we are able to compute Lyapunov exponents explicitly. MSC2010 classification: 60H20, 60H30, 60F25, 37H30, 45M10.
In this article we will present a new perspective on the variable order fractional calculus, whic... more In this article we will present a new perspective on the variable order fractional calculus, which allows for differentiation and integration to a variable order, i.e. one differentiates (or integrates) a function along the path of a regularity function. The concept of multifractional calculus has been a scarcely studied topic within the field of functional analysis in the last 20 years. We develop a multifractional derivative operator which acts as the inverse of the multifractional integral operator. This is done by solving the Abel integral equation generalized to a multifractional order. With this new multifractional derivative operator, we are able to analyze a variety of new problems, both in the field of stochastic analysis and in fractional and functional analysis, ranging from regularization properties of noise to solutions to multifractional differential equations. In this paper, we will focus on application of the derivative operator to the construction of strong solution...
The purpose of this paper is two-fold: We extend the well-known relation between optimal stopping... more The purpose of this paper is two-fold: We extend the well-known relation between optimal stopping and randomized stopping of a given stochastic process to a situation where the available informatio ...
In this paper, we show the existence of unique Malliavin differentiable solutions to SDE‘s driven... more In this paper, we show the existence of unique Malliavin differentiable solutions to SDE‘s driven by a fractional Brownian motion with Hurst parameter H < 12 and singular, unbounded drift vector fields, for which we also prove a stability result. Further, using the latter results, we propose a stock price model with rough and correlated volatility, which also allows for capturing regime switching effects. Finally, we also derive a Bismut-Elworthy-Li formula with respect to our stock price model for certain classes of vector fields.
A general market model with memory is considered in terms of stochastic functional differential e... more A general market model with memory is considered in terms of stochastic functional differential equations. We aim at representation formulae for the sensitivity analysis of the dependence of option prices on the memory. This implies a generalization of the concept of delta.
Uploads
Papers by Frank Proske