Bernt Øksendal

University of Oslo · Department of Mathematics

We investigate conditional McKean-Vlasov equations driven by time-space white noise, motivated by the propagation of chaos in an N-particle system with space-time Ornstein-Uhlenbeck dynamics. The framework builds on the stochastic calculus of time-space white noise, utilizing tools such as the two-parameter Ito formula, Malliavin calculus, and orth...

In this paper, we study the probability distribution of solutions of McKean-Vlasov stochastic differential equations (SDEs) driven by fractional Brownian motion. We prove the associated Fokker-Planck equation, which governs the evolution of the probability distribution of the solution. For the case where the distribution is absolutely continuous, w...

We study a linear filtering problem where the signal and observation processes are described as solutions of linear stochastic differential equations driven by time-space Brownian sheets. We derive a stochastic integral equation for the conditional value of the signal given the observation, which can be considered a time-space analogue of the class...

We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based upon a rigorous stochastic calculus derivation of the optimal hedging portfolio, optimal option price, and the corresponding equivalent martingale measure through the means of the Stackelberg game approac...

We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based on a rigorous stochastic calculus derivation of the optimal hedging portfolio, optimal option price, and the corresponding equivalent martingale measure through the means of the Stackelberg game approach....

In this paper, we consider impulse control problems involving conditional McKean–Vlasov jump diffusions, with the common noise coming from the σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-...

In this paper, we study a Pontryagin type stochastic maximum principle for the optimal control of a system, where the state dynamics satisfy a stochastic partial differential equation (SPDE) driven by a two-parameter (time-space) Brownian motion (also called Brownian sheet). We first discuss some properties of a Brownian sheet driven linear SPDE wh...

The aim of this paper is to analyse a WIS-stochastic differential equation driven by fractional Brownian motion with H>0.5. For this, we summarise the theory of fractional white noise and prove a fundamental L^2-estimate for WIS-integrals. We apply this to prove the existence and uniqueness of a solution in L^2(P) of a WIS-stochastic differential e...

We study the stochastic time-fractional stochastic heat equation $$\begin{aligned} \frac{\partial ^{\alpha }}{\partial t^{\alpha }}Y(t,x)=\lambda \varDelta Y(t,x)+\sigma W(t,x);\; (t,x)\in (0,\infty )\times \mathbb {R}^{d}, \end{aligned}$$ (0.1) where \(d\in \mathbb {N}=\{1,2,...\}\) and \(\frac{\partial ^{\alpha }}{\partial t^{\alpha }}\) is the C...

The purpose of this paper is to establish a stochastic differential equation for the Donsker delta measure of the solution of a McKean-Vlasov (mean-field) stochastic differential equation. If the Donsker delta measure is absolutely continuous with respect to Lebesgue measure, then its Radon-Nikodym derivative is called the Donsker delta function. I...

We consider optimal control of a new type of non-local stochastic partial differential equations (SPDEs). The SPDEs have \emph{space interactions}, in the sense that the dynamics of the system at time t and position in space x also depend on the space-mean of values at neighbouring points. This is a model with many applications, e.g. to population...

This paper establishes a verification theorem for impulse control problems involving conditional McKean-Vlasov jump diffusions. We obtain a Markovian system by combining the state equation of the problem with the stochastic Fokker-Planck equation for the conditional probability law of the state. We derive sufficient variational inequalities for a f...

We study the time-fractional stochastic heat equation driven by time-space white noise with space dimension $d\in\mathbb{N}=\{1,2,...\}$ and the fractional time-derivative is the Caputo derivative of order $\alpha \in (0,2)$. We consider the equation in the sense of distribution, and we find an explicit expression for the $\mathcal{S}'$-valued solu...

In the first part of this paper I give the historical background to my initial interest in stochastic analysis and to the writing of my book Stochastic Differential Equations. The first edition of this book was published by Springer in 1985, with the highly appreciated support of Catriona Byrne. In the second part I present a motivation for modelli...

In the first part of this paper I give the historical background to my initial interest in stochastic analysis and to the writing of my book Stochastic Differential Equations. The first edition of this book was published by Springer in 1985, with the highly appreciated support of Catriona Byrne.In the second part I present a motivation for modellin...

We study the problem of optimal stopping of conditional McKean-Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean-Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal. To achieve this, we com...

This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations, where the solution Xu,ξ(t) = X(t) is given by $$X(t) = \phi (t) + \int_0^t {b(t,s,X(s),u(s))\,{\rm{d}}s + \int_0^t {\sigma (t,s,X(s),u(s))\,{\rm{d}}B(s) + \int_0^t {h(t,s)\,{\rm{d}}\xi (s).} } } $$ Here dB(s) denotes the Brownian motion...

В этой статье мы преследуем двоякую цель. Во-первых, мы распространяем хорошо известное соотношение между оптимальной остановкой и рандомизированной остановкой заданного случайного процесса на ситуацию, когда доступный поток информации - это фильтрация, которая априори не предполагается как-либо связанной с фильтрацией случайного процесса. В этом с...

We study optimal control of McKean-Vlasov (mean-field) stochastic differential equations with jumps. - First we prove a Fokker-Planck equation for the law of the state. - Then we study the situation when the law is absolute continuous with respect to Lebesgue measure. In that case the Fokker-Planck equation reduces to a deterministic integro-differ...

We study a financial market where the risky asset is modelled by a geometric Itô-Lévy process, with a singular drift term. This can for example model a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier. See e.g. Jarrow and Protter (J Bank Finan 29:2803–2820, 2005...

We study option prices in financial markets where the risky asset prices are modelled by jump diffusions. For simplicity we put the risk free asset price equal to 1. Such markets are typically incomplete, and therefore there are in general infinitely many arbitrage-free option prices in these markets. We consider in particular European options with...

We study a financial market where the risky asset is modelled by a stochastic differential equation driven by a partially reflected Brownian motion. This models a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier in order to prevent it from going below that barri...

This paper deals with optimal combined singular and regular control for stochastic Volterra integral equations, where the solution $X^{u,\xi}(t)=X(t)$ is given by $X(t) =\phi(t)+\int_0^t b(t,s,X(s),u(s)) ds+\int_0^t\sigma(t,s,X(s),u(s)) dB(s)+\int_0^t h(t,s) d\xi(s)$. Here $\xi$ denotes the singular control and $u$ denotes the regular control. Such...

In this paper we present a new verification theorem for optimal stopping problems for Hunt processes. The approach is based on the Fukushima-Dynkin formula, and its advantage is that it allows us to verify that a given function is the value function without using the viscosity solution argument. Our verification theorem works in any dimension. We i...

The continuous-time version of Kyle's (1985) model is studied, in which market makers are not fiduciaries. They have some market power which they utilize to set the price to their advantage, resulting in positive expected profits. This has several implications for the equilibrium, the most important being that by setting a modest fee conditional of...

Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case. • In the finite horizon case, results about...

Fix an open set \(\mathcal S\subset \mathbb {R}^k\) (the solvency region ) and let Y(t) be a jump diffusion in \(\mathbb {R}^k\) given by \( \mathrm{d}Y(t)=b(Y(t))\mathrm{d}t+\sigma (Y(t))\mathrm{d}B(t) +\int _{\mathbb {R}^\ell } \!\gamma (Y(t^-), z) \bar{N}(\mathrm{d}t,\mathrm{d}z),\quad Y(0)=y\!\in \!\mathbb {R}^\ell , \) where \(b:\mathbb {R}^k\...

Choose \(f\in C^2(\mathbb {R})\) and put \(Y(t)=f(X(t))\). Then by the Itô formula.

Consider the general situation in Chap. 9, except that now we assume that we, in addition, are free at any state \(y\in \mathbb {R}^k\) to choose a Markov control \(u(y)\in U\), where U is a given closed convex set in \(\mathbb {R}^p\). Let \(\mathcal{U}\) be a given family of such Markov controls u.

In this section we present the dynamic programming approach to stochastic differential games. We only present the case for zero sum games. For the extension to non-zero sum games, we refer to [MØ].

In general it is not possible to reduce impulse control to optimal stopping, because the choice of the first intervention time \(\tau _1\) and the first impulse \(\zeta _1\) will influence the next and so on. However, if we allow only (up to) a fixed finite number n of interventions, then the corresponding impulse control problem can be solved by s...

In this chapter we discuss combined optimal stopping and stochastic control problems and their associated Hamilton–Jacobi–Bellman (HJB) variational inequalities. This is a subject which deserves to be better known because of its many applications. A thorough treatment of such problems (but without the associated HJB variational inequalities) can be...

We illustrate singular control problems by the following example, studied in [FØS2].

Suppose that—if there are no interventions—the state \(Y(t)\in \mathbb {R}^k\) of the system we consider is a jump diffusion.

The main results of Chaps. 3– 9 and 11 and are all verification theorems. Any function \(\phi \) which satisfies the given requirements is necessarily the value function \(\varPhi \) of the corresponding problem. These requirements are made as weak as possible in order to include as many cases as possible. For example, except for the singular contr...

It has been argued (see, e.g., [EK,B-N,Sc,Eb, CT]) that Lévy processes are relevant in mathematical finance, in particular in the modeling of stock prices. In this chapter we give a brief introduction to financial markets where the asset prices are represented by Itô–Lévy processes. For more information we refer to [CT].

In this chapter we give an introduction to backward stochastic differential equations (BSDEs) with jumps, and we relate them to the concepts of recursive utilities and convex risk measures. This section is based on the papers [ØS2] and [QS]. For a similar introduction in the Brownian motion case, we refer the reader to the survey paper on BSDEs by...

In this chapter we present the basic concepts and results needed for the applied calculus of jump diffusions. Since there are several excellent books which give a detailed account of this basic theory, we will just briefly review it here and refer the reader to these books for more information.

The purpose of these lectures is threefold: We first give a short survey of the Hida white noise calculus, and in this context we introduce the Hida-Malliavin derivative as a stochastic gradient with values in the Hida stochastic distribution space $(\mathcal{S}% )^*$. We show that this Hida-Malliavin derivative defined on $L^2(\mathcal{F}_T,P)$ is...

We use a white noise approach to study the problem of optimal insider control of a stochastic delay equation driven by a Brownian motion B and a Poisson random measure N. In particular, we use Hida-Malliavin calculus and the Donsker delta functional to study the problem. We establish a sufficient and a necessary maximum principle for the optimal co...

We consider the problem of optimal singular control of a stochastic partial differential equation (SPDE) with space-mean dependence. Such systems are proposed as models for population growth in a random environment. We obtain sufficient and necessary maximum principles for such control problems. The corresponding adjoint equation is a reflected bac...

By a memory mean-field process we mean the solution $X(\cdot)$ of a stochastic mean-field equation involving not just the current state $X(t)$ and its law $\mathcal{L}(X(t))$ at time $t$, but also the state values $X(s)$ and its law $\mathcal{L}(X(s))$ at some previous times $s<t$. Our purpose is to study stochastic control problems of memory mean-...

The original version of this article unfortunately contained a few mistakes in Theorems and notation. The corrected information is given below. © 2018 Springer Science+Business Media, LLC, part of Springer Nature

The main purpose of the book is to give a rigorous introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusions and their applications. Both the dynamic programming method and the stochastic maximum principle method are discussed, as well as the relation between them. Correspondi...

We consider optimal control of a new type of non-local stochastic partial differential equations (SPDEs). The SPDEs have space interactions, in the sense that the dynamics of the system at time $t$ and position in space x also depend on the space-mean of values at neighbouring points. This is a model with many applications, e.g. to population growt...

Our purpose of this paper is to study stochastic control problem for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history in both cases, finite and infinite time horizon. - In the finite horizon case, results about existence and uniqueness of...

We study optimal control for mean-field stochastic partial differential equations (stochastic evolution equations) driven by a Brownian motion and an independent Poisson random measure, in case of partial information control. One important novelty of our problem is represented by the introduction of general mean-field operators, acting on both the...

We present an explicit solution triplet (Y,Z,K) to the backward stochastic Volterra integral equation (BSVIE) of linear type, driven by a Brownian motion and a compensated Poisson random measure. The process Y is expressed by an integral whose kernel is explicitly given. The processes Z and K are expressed by Hida–Malliavin derivatives involving Y.

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 information flow is a sub-filtration of the filtration of the process. We call these problems optimal stopping and randomized stopping with partial information. Fo...

By a memory mean-field stochastic differential equation (MMSDE) we mean a stochastic differential equation (SDE) where the coefficients depend on not just the current state $X(t)$ at time $t$, but also on previous values (history/memory) $X_{t}:=\{X(t-s)\}_{s\in\lbrack0,\delta]}$, as well as the law $M(t):=\mathcal{L}(X(t))$ and its history/memory...

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 specifical...

In this paper we investigate a logarithmic utility maximization problem of the terminal wealth for an insider portfolio, where the inside information consists of knowledge of some future values of the Brownian motion B(t) driving the financial market. More specifically, we assume that at time t the insider has access to market information at least...

We consider the problem of optimal inside portfolio $\pi(t)$ in a financial market with a corresponding wealth process $X(t)=X^{\pi}(t)$ modelled by \begin{align}\label{eq0.1} \begin{cases} dX(t)&=\pi(t)X(t)[\alpha(t)dt+\beta(t)dB(t)]; \quad t\in[0, T] X(0)&=x_0>0, \end{cases} \end{align} where $B(\cdot)$ is a Brownian motion. We assume that the in...

In this paper we study the mean-field backward stochastic differential equations (mean-field bsde) of the form \begin{align*} dY(t) =-f(t,Y(t),Z(t),K(t),\mathbb{E}[\varphi(Y(t),Z(t),K(t,\cdot ))])dt+Z(t)dB(t) + {\textstyle\int_{\mathbb{R}_{0}}} K(t,\zeta)\tilde{N}(dt,d\zeta), \end{align*} where $B$ is a Brownian motion, $\tilde{N}$ is the compensat...

In this paper we study the mean-field backward stochastic differential equations (mean-field bsde) of the form dY(t) =-f(t,Y(t),Z(t),K(t, . ),E[\varphi(Y(t),Z(t),K(t,.))])dt+Z(t)dB(t) +\int_{R_{0}}K(t,\zeta)\tilde{N}(dt,d\zeta), where B is a Brownian motion, \tilde{N} is the compensated Poisson random measure. Under some mild conditions, we prove t...

We study optimal control of stochastic Volterra integral equations (SVIE) with jumps by using Hida-Malliavin calculus. • We give conditions under which there exist unique solutions of such equations. • Then we prove both a sufficient maximum principle (a verification theorem) and a necessary maximum principle via Hida-Malliavin calculus. • As an a...

We study optimal control of stochastic Volterra integral equations (SVIE) with jumps by using Hida-Malliavin calculus. - We give conditions under which there exists unique solutions of such equations. - Then we prove both a sufficient maximum principle (a verification theorem) and a necessary maximum principle via Hida-Malliavin calculus. - As an a...

We present an explicit solution triplet $(Y, Z, K)$ to the backward stochastic Volterra integral equation (BSVIE) of linear type, driven by a Brownian motion and a compensated Poisson random measure. The process $Y$ is expressed by an integral whose kernel is explicitly given. The processes $Z$ and $K$ are expressed by Hida-Malliavin derivatives in...

This article studies singular mean field control problems and singular mean field two-players stochastic differential games. Both sufficient and necessary conditions for the optimal controls and for the Nash equilibrium are obtained. Under some assumptions the optimality conditions for singular mean-field control are reduced to a reflected Skorohod...

The classical maximum principle for optimal stochastic control states that if a control $\hat{u}$ is optimal, then the corresponding Hamiltonian has a maximum at $u=\hat{u}$. The first proofs for this result assumed that the control did not enter the diffusion coefficient. Moreover, it was assumed that there were no jumps in the system. Subsequentl...

The classical maximum principle for optimal stochastic control states that if a control $\hat{u}$ is optimal, then the corresponding Hamiltonian has a maximum at $u=\hat{u}$. The first proofs for this result assumed that the control did not enter the diffusion coefficient. Moreover, it was assumed that there were no jumps in the system. Subsequentl...

We study the problem of optimal control for mean-field stochastic partial differential equations (stochastic evolution equations) driven by a Brownian motion and an independent Poisson random measure, in the case of \textit{partial information} control. One important novelty of our problem is represented by the introduction of \textit{general mean-...

We study the problem of optimal insider control of an SPDE (a stochastic evolution equation) driven by a Brownian motion and a Poisson random measure. Our optimal control problem is new in two ways: • (i) The controller has access to inside information, i.e. access to information about a future state of the system. • (ii) The integro-differential...

A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities:(i) The optimal terminal wealth \(X^*(T) : = X_{\varphi ^*}(T)\) of the problem to maximize the expected U-utility of the terminal wealth \(X_{\varphi }(T)\) generated by admissible portfolios \(\varphi (t); 0 \le t \le T\) in...

By a memory mean-field process we mean the solution $X(\cdot)$ of a stochastic mean-field equation involving not just the current state $X(t)$ and its law $\mathcal{L}(X(t))$ at time $t$, but also the state values $X(s)$ and its law $\mathcal{L}(X(s))$ at some previous times $s<t$. Our purpose is to study stochastic control problems of memory mean-...

We consider a simple stochastic model for fluid flow in a porous medium. The permeability is modelled through a lognormal distribution with a certain correlation structure. Our equations can be interpreted in two ways. We first use ordinary products and then Wick products. The latter can be looked upon as a kind of renormalization procedure. We com...

We consider the problem of optimal control of a mean-field stochastic differential equation under model uncertainty. The model uncertainty is represented by ambiguity about the law $\mathcal{L}(X(t))$ of the state $X(t)$ at time $t$. For example, it could be the law $\mathcal{L}_{\mathbb{P}}(X(t))$ of $X(t)$ with respect to the given, underlying pr...

We consider the problem of optimal control of a mean-field stochastic differential equation under model uncertainty. The model uncertainty is represented by ambiguity about the law $\mathcal{L}(X(t))$ of the state $X(t)$ at time $t$. For example, it could be the law $\mathcal{L}_{\mathbb{P}}(X(t))$ of $X(t)$ with respect to the given, underlying pr...

We study the problem of optimal inside control of a stochastic delay equation driven by a Brownian motion and a Poisson random measure. We prove a sufficient and a necessary maximum principle for the optimal control when the trader from the beginning has inside information about the future value of some random variable related to the system. The re...

We study the problem of optimal inside control of an SPDE (a stochastic evolution equation) driven by a Brownian motion and a Poisson random measure. Our optimal control problem is new in two ways: (i) The controller has access to inside information, i.e. access to information about a future state of the system, (ii) The integro-differential operat...

We study the problem of optimal control of a coupled system of forward-backward stochastic Volterra equations. We prove existence and uniqueness of solutions of such systems, and using Hida-Malliavin calculus we prove two maximum principles for the optimal control. As an application of our methods, we solve a recursive utility optimisation problem...

We combine stochastic control methods, white noise analysis and Hida-Malliavin calculus applied to the Donsker delta functional to obtain new representations of semimartingale decompositions under enlargement of filtrations. The results are illustrated by explicit examples.

We study stochastic differential games of jump diffusions, where the players have access to inside information. Our approach is based on anticipative stochastic calculus, white noise, Hida-Malliavin calculus, forward integrals and the Donsker delta functional. We obtain a characterization of Nash equilibria of such games in terms of the correspondi...

We present a new approach to the optimal portfolio problem for an insider with logarithmic utility. Our method is based on white noise theory, stochastic forward integrals, Hida-Malliavin calculus and the Donsker delta function.

We study a coupled system of controlled stochastic differential equations (SDEs) driven by a Brownian motion and a compensated Poisson random measure, consisting of a forward SDE in the unknown process $X(t)$ and a \emph{predictive mean-field} backward SDE (BSDE) in the unknowns $Y(t), Z(t), K(t,\cdot)$. The driver of the BSDE at time $t$ may depen...

We introduce the concept of singular recursive utility. This leads to a kind of singular BSDE which, to the best of our knowledge, has not been studied before. We show conditions for existence and uniqueness of a solution for this kind of singular BSDE. Furthermore, we analyze the problem of maximizing the singular recursive utility. We derive suff...