Backward Stochastic PDE and Imperfect Hedging

We consider a problem of minimization of a hedging error, measured by a positive convex random function, in an incomplete financial market model, where the dynamics of asset prices is given by an Rd-valued continuous semimartingale. Under some regularity assumptions we derive a backward stochastic PDE for the value function of the problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As an example the case of mean-variance hedging is considered.

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RECURSIVE BACKWARD SCHEME FOR THE SOLUTION OF A BSDE WITH A NON LIPSCHITZ GENERATOR

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964816000498 ◽

2017 ◽

Vol 31 (2) ◽

pp. 207-225

Author(s):

Paola Tardelli

Keyword(s):

Unique Solution ◽

Financial Market ◽

Utility Maximization ◽

Value Function ◽

Jump Processes ◽

Maximization Problem ◽

Admissible Strategies ◽

Incomplete Financial Market ◽

Utility Maximization Problem ◽

The Value Function

On an incomplete financial market, the stocks are modeled as pure jump processes subject to defaults. The exponential utility maximization problem is investigated characterizing the value function in term of Backward Stochastic Differential Equations (BSDEs), driven by pure jump processes. In general, in this setting, there is no unique solution. This is the reason why, the value function is proven to be the limit of a sequence of processes. Each of them is the solution of a Lipschitz BSDE and it corresponds to the value function associated with a subset of bounded admissible strategies. Given a representation of the jump processes driving the model, the aim of this note is to give a recursive backward scheme for the value function of the initial problem.

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Viscosity Solution of Mean-Variance Portfolio Selection of a Jump Markov Process with No-Shorting Constraints

Journal of Applied Mathematics ◽

10.1155/2016/4543298 ◽

2016 ◽

Vol 2016 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Moussa Kounta

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Portfolio Selection ◽

Viscosity Solution ◽

Value Function ◽

Auxiliary Problem ◽

Partial Differential ◽

Mean Variance Portfolio ◽

Mean Variance ◽

The Value Function

We consider the so-called mean-variance portfolio selection problem in continuous time under the constraint that the short-selling of stocks is prohibited where all the market coefficients are random processes. In this situation the Hamilton-Jacobi-Bellman (HJB) equation of the value function of the auxiliary problem becomes a coupled system of backward stochastic partial differential equation. In fact, the value functionVoften does not have the smoothness properties needed to interpret it as a solution to the dynamic programming partial differential equation in the usual (classical) sense; however, in such casesVcan be interpreted as a viscosity solution. Here we show the unicity of the viscosity solution and we see that the optimal and the value functions are piecewise linear functions based on some Riccati differential equations. In particular we solve the open problem posed by Li and Zhou and Zhou and Yin.

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FINANCIAL MARKET MODEL WITH INFLUENTIAL INFORMED INVESTORS

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024905003219 ◽

2005 ◽

Vol 08 (06) ◽

pp. 693-716 ◽

Cited By ~ 8

Author(s):

AXEL GRORUD ◽

MONIQUE PONTIER

Keyword(s):

Brownian Motion ◽

Interest Rate ◽

Financial Market ◽

Asset Prices ◽

Optimization Problem ◽

Statistical Tests ◽

Market Model ◽

Financial Model ◽

Additional Information ◽

The Interest Rate

We develop a financial model with an "influential informed" investor who has an additional information and influences asset prices by means of his strategy. The prices dynamics are supposed to be driven by a Brownian motion, the informed investor's strategies affect the risky asset trends and the interest rate. Our paper could be seen as an extension of Cuoco and Cvitanic's work [4] since, as these authors, we solve the informed influential investor's optimization problem. But our main result is the construction of statistical tests to detect if, observing asset prices and agent's strategies, this influential agent is or not an informed trader.

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A Unified Characterization of 𝑞-Optimal and Minimal Entropy Martingale Measures by Semimartingale Backward Equations

Georgian Mathematical Journal ◽

10.1515/gmj.2003.289 ◽

2003 ◽

Vol 10 (2) ◽

pp. 289-310

Author(s):

M. Mania ◽

R. Tevzadze

Keyword(s):

Asset Prices ◽

Incomplete Market ◽

Market Model ◽

Hölder Condition ◽

Martingale Measures ◽

Continuous Semimartingale ◽

Special Cases ◽

Minimal Martingale Measures ◽

Reverse Hölder

Abstract We give a unified characterization of 𝑞-optimal martingale measures for 𝑞 ∈ [0, ∞) in an incomplete market model, where the dynamics of asset prices are described by a continuous semimartingale. According to this characterization the variance-optimal, the minimal entropy and the minimal martingale measures appear as the special cases 𝑞 = 2, 𝑞 = 1 and 𝑞 = 0 respectively. Under assumption that the Reverse Hölder condition is satisfied, the continuity (in 𝐿1 and in entropy) of densities of 𝑞-optimal martingale measures with respect to 𝑞 is proved.

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Robust Optimal Investment-Reinsurance Strategies for a Jump-Diffusion Risk Model with Mean-reverting Stock Return

Tobacco Regulatory Science ◽

10.18001/trs.7.6.88 ◽

2021 ◽

Vol 7 (6) ◽

pp. 6100-6114

Author(s):

Wu Yungao

Keyword(s):

Financial Market ◽

Stock Return ◽

Value Function ◽

Risk Model ◽

Optimal Investment ◽

Jump Diffusion ◽

Insurance Companies ◽

Insurance Company ◽

Terminal Time ◽

The Value Function

Objectives: This paper proposes a strategy of robust optimal investment reinsurance for insurance companies. It was assumed that the surplus procedure of the insurance company satisfies the jump-diffusion procedure. Insurance companies could invest their surplus funds in the financial market consisted of both risk assets and one risk-free asset. The price procedure of risk assets satisfies the stochastic procedure with a mean reversion rate. Considering the uncertainty of the model, the ambiguity-averse insurance firm aims to enhance the exponential utility of insurance surplus at terminal time. This paper has investigated the problem of robust optimal investment reinsurance and obtained the differential equation supported by the value function.

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An HJB approach to a general continuous-time mean-variance stochastic control problem

Random Operators and Stochastic Equations ◽

10.1515/rose-2018-0020 ◽

2018 ◽

Vol 26 (4) ◽

pp. 225-234

Author(s):

Georgios Aivaliotis ◽

A. Yu. Veretennikov

Keyword(s):

Dynamic Optimization ◽

Continuous Time ◽

Optimization Problem ◽

Value Function ◽

Dynamic Optimization Problem ◽

Terminal Time ◽

Diffusion Controlled ◽

Time Component ◽

Mean Variance ◽

The Value Function

Abstract A general continuous mean-variance problem is considered for a diffusion controlled process where the reward functional has an integral and a terminal-time component. The problem is transformed into a superposition of a static and a dynamic optimization problem. The value function of the latter can be considered as the solution to a degenerate HJB equation either in the viscosity or in the Sobolev sense (after a regularization) under suitable assumptions and with implications with regards to the optimality of strategies. There is a useful interplay between the two approaches – viscosity and Sobolev.

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