arXiv subjects

Bernt Øksendal

Publications and source records attributed to Bernt Øksendal.

At least 37 records · Page 2Linked to original sources

arXiv2020

A financial market with singular drift and no arbitrage

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 & Protter for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas & Shreve (in the continuous setting), this allows for arbitrages in the market. However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system. In this paper we consider a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a delay θ>0 in the information flow available for the trader. We allow the stock price dynamics to depend on both a continuous process (Brownian motion) and a jump process (Poisson random measure). We believe that jumps and delays are essential in order to get more realistic financial market models. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as the delay θ> 0. This implies that there is no arbitrage in the market in that case. However, when θgoes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay. Our model is also relevant for high frequency trading issues. This is because high frequency trading often leads to intensive trading taking place on close to infinitesimal lengths of time, which in the limit corresponds to trading on time sets of measure 0. This may in turn lead to a singular drift in the pricing dynamics. See e.g. Lachapelle et al and the references therein.

q-fin.MF↗

arXiv2019

A new approach to optimal stopping for Hunt processes

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 illustrate our results with some examples of optimal stopping of reflected diffusions and absorbed diffusions.

math.OC↗

arXiv2019

Strategic Insider Trading Equilibrium with a Non-fiduciary Market Maker

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 the order flow, the market maker is able to obtain a profit of the order of magnitude, and even better than, a perfectly informed insider. Our model also indicates why speculative prices are more volatile than predicted by fundamentals.

q-fin.TR↗

arXiv2019

Mean-Field Stochastic Control with Elephant Memory in Finite and Infinite Time Horizon

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. We study both the finite horizon case and the infinite time horizon case. - In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem. - For infinite horizon, we derive sufficient and necessary maximum principles. As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.

math.OC↗

arXiv2019

Singular control of SPDEs with space-mean dynamics

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 backward stochastic partial differential equation (BSPDE) with space-mean dependence. We prove existence and uniqueness results for such equations. As an application we study optimal harvesting from a population modelled as an SPDE with space-mean dependence.

math.OC↗

arXiv2019

Introduction to White Noise, Hida-Malliavin Calculus and Applications

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 a natural extension of the classical Malliavin derivative defined on the subspace $\mathbb{D}_{1,2}$ of $L^2(P)$. The Hida-Malliavin calculus allows us to prove new results under weaker assumptions than could be obtained by the classical theory. In particular, we prove the following: (i) A general integration by parts formula and duality theorem for Skorohod integrals, (ii) a generalised fundamental theorem of stochastic calculus, and (iii) a general Clark-Ocone theorem, valid for all $F \in L^2(\mathcal{F}_T,P)$. As applications of the above theory we prove the following: A general representation theorem for backward stochastic differential equations with jumps, in terms of Hida-Malliavin derivatives; a general stochastic maximum principle for optimal control; backward stochastic Volterra integral equations; optimal control of stochastic Volterra integral equations and other stochastic systems.

math.OC↗

arXiv2019

Mean-field backward stochastic differential equations and applications

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[φ(Y(t),Z(t),K(t,.))])dt+Z(t)dB(t) +\int_{R_{0}}K(t,ζ)\tilde{N}(dt,dζ), where B is a Brownian motion, \tilde{N} is the compensated Poisson random measure. Under some mild conditions, we prove the existence and uniqueness of the solution triplet (Y,Z,K). It is commonly believed that there is no comparison theorem for general mean-field bsde. However, we prove a comparison theorem for a subclass of these equations. When the mean-field bsde is linear, we give an explicit formula for the first component Y(t) of the solution triplet. Our results are applied to solve a mean-field recursive utility optimization problem in finance.

math.OC↗

arXiv2018

New approach to optimal control of stochastic Volterra integral equations

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 application we solve a problem of optimal consumption from a cash flow modelled by an SVIE.

math.OC↗

arXiv2018

A Hida-Malliavin white noise calculus approach to optimal control

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. Subsequently it was discovered by Shige Peng (still assuming no jumps) that one could also allow the diffusion coefficient to depend on the control, provided that the corresponding adjoint backward stochastic differential equation (BSDE) for the first order derivative was extended to include an extra BSDE for the second order derivatives. In this paper we present an alternative approach based on Hida-Malliavin calculus and white noise theory. This enables us to handle the general case with jumps, allowing both the diffusion coefficient and the jump coefficient to depend on the control, and we do not need the extra BSDE with second order derivatives. The result is illustrated by an example of a constrained mean-variance portfolio problem.

math.OC↗

arXiv2018

Singular control and optimal stopping of memory mean-field processes

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.

math.OC↗

arXiv2018

Model Uncertainty Stochastic Mean-Field Control

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 probability measure $\mathbb{P}$. This is the classical case when there is no model uncertainty. But it could also be the law $\mathcal{L}_{\mathbb{Q}}(X(t))$ with respect to some other probability measure $\mathbb{Q}$ or, more generally, any random measure $μ(t)$ on $\mathbb{R}$ with total mass $1$. We represent this model uncertainty control problem as a stochastic differential game of a mean-field related type stochastic differential equation (SDE) with two players. The control of one of the players, representing the uncertainty of the law of the state, is a measure valued stochastic process $μ(t)$ and the control of the other player is a classical real-valued stochastic process $u(t)$. This control with respect to random probability processes $μ(t)$ on $\mathbb{R}$ is a new type of stochastic control problems that has not been studied before. By introducing operator-valued backward stochastic differential equations, we obtain a sufficient maximum principle for Nash equilibria for such games in the general nonzero-sum case, and saddle points for zero-sum games. As an application we find an explicit solution of the problem of optimal consumption under model uncertainty of a cash flow described by a mean-field related type SDE.

math.OC↗

arXiv2018

A white noise approach to optimal insider control of systems with delay

We use a white noise approach to study the problem of optimal inside 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 control when the trader from the beginning has inside information about the future value of some random variable related to the system.These results are applied to the problem of optimal inside harvesting control in a population modelled by a stochastic delay equation. Next, we apply a direct white noise method to find the optimal insider portfolio in a financial market where the risky asset price is given by a stochastic delay equation. A classical result of Pikovski and Karatzas shows that when the inside information is B(T), where T is the terminal time of the trading period, then the market is not viable. Our results show that with this inside information the market is not viable even if there is delay in the equations.

math.OC↗

arXiv2018

A continuous auction model with insiders and random time of information release

In a unified framework we study equilibrium in the presence of an insider having information on the signal of the firm value, which is naturally connected to the fundamental price of the firm related asset. The fundamental value itself is announced at a future random (stopping) time. We consider two cases. First when the release time of information is known to the insider and then when it is unknown also to her. Allowing for very general dynamics, we study the structure of the insider's optimal strategies in equilibrium and we discuss market efficiency. In particular, we show that in the case the insider knows the information release time, the market is fully efficient. In the case the insider does not know this random time, we see that there is an equilibrium with no full efficiency, but where the sensitivity of prices is decreasing in time according with the probability that the announcement time is greater than the current time. In other words, the prices become more and more stable as the announcement approaches.

q-fin.PR↗

arXiv2018

Viable Insider Markets

We consider the problem of optimal inside portfolio $π(t)$ in a financial market with a corresponding wealth process $X(t)=X^π(t)$ modelled by \begin{align}\label{eq0.1} \begin{cases} dX(t)&=π(t)X(t)[α(t)dt+β(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 insider at time $t$ has access to market information $\varepsilon_t>0$ units ahead of time, in addition to the history of the market up to time $t$. The problem is to find an insider portfolio $π^{*}$ which maximizes the expected logarithmic utility $J(π)$ of the terminal wealth, i.e. such that $$\sup_πJ(π)= J(π^{*}), \text {where } J(π)= \mathbb{E}[\log(X^π(T))].$$ The insider market is called \emph{viable} if this value is finite. We study under what inside information flow $\mathbb{H}$ the insider market is viable or not. For example, assume that for all $t<T$ the insider knows the value of $B(t+ε_t)$, where $t + ε_t \geq T$ converges monotonically to $T$ from above as $t$ goes to $T$ from below. Then (assuming that the insider has a perfect memory) at time $t$ she has the inside information $\mathcal{H}_t$, consisting of the history $\mathcal{F}_t$ of $B(s); 0 \leq s \leq t$ plus all the values of Brownian motion in the interval $[t+ε_t, ε_0]$, i.e. we have the enlarged filtration \begin{equation}\label{eq0.2} \mathbb{H}=\{\mathcal{H}_t\}_{t\in[0.T]},\quad \mathcal{H}_t=\mathcal{F}_t\veeσ(B(t+ε_t+r),0\leq r \leq ε_0-t-ε_t), \forall t\in [0,T]. \end{equation} Using forward integrals, Hida-Malliavin calculus and Donsker delta functionals we show that if $$\int_0^T\frac{1}{\varepsilon_t}dt=\infty,$$ then the insider market is not viable.

q-fin.MF↗

arXiv2017

Stochastic Control of Memory Mean-Field Processes

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-field processes. - We consider the space $\mathcal{M}$ of measures on $\mathbb{R}$ with the norm $|| \cdot||_{\mathcal{M}}$ introduced by Agram and Øksendal in \cite{AO1}, and prove the existence and uniqueness of solutions of memory mean-field stochastic functional differential equations. - We prove two stochastic maximum principles, one sufficient (a verification theorem) and one necessary, both under partial information. The corresponding equations for the adjoint variables are a pair of \emph{(time-) advanced backward stochastic differential equations}, one of them with values in the space of bounded linear functionals on path segment spaces. - As an application of our methods, we solve a memory mean-variance problem as well as a linear-quadratic problem of a memory process.

math.OC↗

arXiv2017

Optimal control of forward-backward stochastic Volterra equations

We study the problem of optimal control of a coupled system of forward-backward stochastic Volterra equations. We use Hida-Malliavin calculus to prove a sufficient and a necessary maximum principle for the optimal control of such systems. Existence and uniqueness of backward stochastic Volterra integral equations are proved. As an application of our methods, we solve a recursive utility optimisation problem in a financial model with memory.

math.OC↗

arXiv2017

Linear Volterra backward stochastic differential equations

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 $Y$.

math.PR↗

arXiv2017

Stochastic control of mean-field SPDEs with jumps

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-field} operators, acting on both the controlled state process and the control process. We first formulate a sufficient and a necessary maximum principle for this type of control. We then prove existence and uniqueness of the solution of such general forward and backward mean-field stochastic partial differential equations. We finally apply our results to find the explicit optimal control for an optimal harvesting problem.

math.OC↗