| 1. |
- Agram, Nacira, 1987-, et al.
(författare)
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A financial market with singular drift and no arbitrage
- 2021
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Ingår i: Mathematics and Financial Economics. - : Springer. - 1862-9679 .- 1862-9660. ; 15, s. 477-500
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Tidskriftsartikel (refereegranskat)abstract
- We study a financial market where the risky asset is modelled by a geometric Ito-Levy 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) for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas and Shreve (Methods of Mathematical Finance, Springer, Berlin, 1998) (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 theta>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 theta>0. This implies that there is no arbitrage in the market in that case. However, when theta goes 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. (Math Finan Econom 10(3):223-262, 2016) and the references therein.
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| 2. |
- Agram, Nacira, 1987-, et al.
(författare)
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A Hida-Malliavin white noise calculus approach to optimal control
- 2018
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Ingår i: Infinite Dimensional Analysis Quantum Probability and Related Topics. - : World Scientific. - 0219-0257. ; 21:3
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Tidskriftsartikel (refereegranskat)abstract
- The classical maximum principle for optimal stochastic control states that if a control û is optimal, then the corresponding Hamiltonian has a maximum at u = û. 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 linear-quadratic optimal control.
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| 3. |
- Agram, Nacira, 1987-, et al.
(författare)
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A maximum principle for infinite horizon delay equations
- 2013
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Ingår i: SIAM Journal on Mathematical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1410 .- 1095-7154. ; 45:4, s. 2499-2522
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Tidskriftsartikel (refereegranskat)abstract
- We prove a maximum principle of optimal control of stochastic delay equations on infinite horizon. We establish first and second sufficient stochastic maximum principles as well as necessary conditions for that problem. We illustrate our results with an application to the optimal consumption rate from an economic quantity.
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| 4. |
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| 5. |
- Agram, Nacira, 1987-
(författare)
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Dynamic risk measure for BSVIE with jumps and semimartingale issues
- 2019
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Ingår i: Stochastic Analysis and Applications. - : Taylor & Francis. - 0736-2994 .- 1532-9356. ; 37:3, s. 361-376
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Tidskriftsartikel (refereegranskat)abstract
- Risk measure is a fundamental concept in finance and in the insuranceindustry. It is used to adjust life insurance rates. In this article,we will study dynamic risk measures by means of backward stochasticVolterra integral equations (BSVIEs) with jumps. We prove a comparisontheorem for such a type of equations. Since the solution of aBSVIEs is not a semimartingale in general, we will discuss some particularsemimartingale issues.
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| 6. |
- Agram, Nacira, Associate professor, 1987-, et al.
(författare)
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Impulse Control of Conditional McKean–Vlasov Jump Diffusions
- 2024
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Ingår i: Journal of Optimization Theory and Applications. - : Springer Nature. - 0022-3239 .- 1573-2878. ; 200:3, s. 1100-1130
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we consider impulse control problems involving conditional McKean–Vlasov jump diffusions, with the common noise coming from the σ-algebra generated by the first components of a Brownian motion and an independent compensated Poisson random measure. We first study the well-posedness of the conditional McKean–Vlasov stochastic differential equations (SDEs) with jumps. Then, we prove the associated Fokker–Planck stochastic partial differential equation (SPDE) with jumps. Next, we establish a verification theorem for impulse control problems involving conditional McKean–Vlasov jump diffusions. We obtain a Markovian system by combining the state equation with the associated Fokker–Planck SPDE for the conditional law of the state. Then we derive sufficient variational inequalities for a function to be the value function of the impulse control problem, and for an impulse control to be the optimal control. We illustrate our results by applying them to the study of an optimal stream of dividends under transaction costs. We obtain the solution explicitly by finding a function and an associated impulse control, which satisfy the verification theorem.
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| 7. |
- Agram, Nacira, 1987-, et al.
(författare)
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Infinite horizon optimal control of forward–backward stochastic differential equations with delay
- 2014
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 259:Part B, s. 336-349
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Tidskriftsartikel (refereegranskat)abstract
- We consider a problem of optimal control of an infinite horizon system governed by forward–backward stochastic differential equations with delay. Sufficient and necessary maximum principles for optimal control under partial information in infinite horizon are derived. We illustrate our results by an application to a problem of optimal consumption with respect to recursive utility from a cash flow with delay.
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| 8. |
- Agram, Nacira, 1987-, et al.
(författare)
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Introduction to White Noise, Hida-Malliavin Calculus and Applications
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- 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 (S. We show that this Hida-Malliavin derivative defined on L2(FT,P) is a natural extension of the classical Malliavin derivative defined on the subspace D1,2 of L2(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∈L2(FT,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.
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| 9. |
- Agram, Nacira, 1987-, et al.
(författare)
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Malliavin calculus and optimal control of stochastic Volterra equations
- 2015
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Ingår i: Journal of Optimization Theory and Applications. - : Springer. - 0022-3239 .- 1573-2878. ; 167:3, s. 1070-1094
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Tidskriftsartikel (refereegranskat)abstract
- Solutions of stochastic Volterra (integral) equations are not Markov processes, and therefore, classical methods, such as dynamic programming, cannot be used to study optimal control problems for such equations. However, we show that using Malliavin calculus, it is possible to formulate modified functional types of maximum principle suitable for such systems. This principle also applies to situations where the controller has only partial information available to base her decisions upon. We present both a Mangasarian sufficient condition and a Pontryagin-type maximum principle of this type, and then, we use the results to study some specific examples. In particular, we solve an optimal portfolio problem in a financial market model with memory.
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| 10. |
- Agram, Nacira, 1987-, et al.
(författare)
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Mean-field backward stochastic differential equations and applications
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we study the mean-field backward stochastic differential equations (mean-field bsde) of the formdY (t) = −f(t, Y (t), Z(t), K(t, ·), E[ϕ(Y (t), Z(t), K(t, ·))])dt + Z(t)dB(t) + R R0 K(t, ζ)N˜(dt, dζ),where B is a Brownian motion, 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.
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