# American Institute of Mathematical Sciences

## A class of stochastic Fredholm-algebraic equations and applications in finance

 1 School of Economics, Sichuan University, Chengdu 610065, China 2 School of Mathematics, Sichuan University, Chengdu 610065, China

*Corresponding author: The research was supported by the NSF of China under grant 11971332 and 11931011

Received  October 2019 Revised  July 2020 Published  September 2020

A class of stochastic Fredholm-algebraic equations (SFAEs) is introduced and investigated. Like backward stochastic differential equations (BSDEs), its solution includes two parts. The interesting thing is that the first part is deterministic and constrained, even though the whole system is stochastic. Our study is mainly motivated by risk indifference pricing problem. Actually, the existing risk indifference price always keeps unchangeable with respect to initial wealth, which is economically unsatisfying. Nevertheless, here a new wealth dependent risk indifference price is proposed by particular SFAEs.

Citation: Zheng Liu, Tianxiao Wang. A class of stochastic Fredholm-algebraic equations and applications in finance. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020267
##### References:
 [1] P. Barrieu and N. El Karoui, Pricing, hedging and optimally designing derivatives via minimization of risk measures, Indifference Pricing: Theory and Applications. Princeton University Press, (2008), Princeton, USA. Google Scholar [2] S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Rev. Finan. Stud., 23 (2010), 2970-3016.  doi: 10.1093/rfs/hhq028.  Google Scholar [3] T. Björk, A. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x.  Google Scholar [4] C. Borell, Monotonicity properties of optimal investment strategies for log-Brownian asset prices, Math. Finance, 17 (2007), 143-153.  doi: 10.1111/j.1467-9965.2007.00297.x.  Google Scholar [5] Ł. Delong, Optimal investment for insurance company with exponential utility and wealth-dependent risk aversion coefficient, Math. Method. Oper. 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Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.  Google Scholar [10] R. J. Elliott and T. K. Siu, Risk-based indifference pricing under a stochastic volatility model, Commun. Stoch. Anal., 4 (2010), 51-73.  doi: 10.31390/cosa.4.1.05.  Google Scholar [11] R. J. Elliott and T. K. Siu, A BSDE approach to a risk-based optimal investment of an insurer, Automatica, 47 (2011), 253-261.  doi: 10.1016/j.automatica.2010.10.032.  Google Scholar [12] V. Henderson and D. Hobson, Utility indifference pricing-an overview, In Volume on Indifference Pricing, (ed. R. Carmona), (2004), Princeton University Press. Google Scholar [13] E. Karni, Risk aversion for state-dependent utility functions: Measurement and applications, Int. Econ. Review, 24 (1983), 637-647.  doi: 10.2307/2648791.  Google Scholar [14] S. Klöppel and M. Schweizer, Dynamic Utility Indifference Valuation via Convex Risk Measures, Nccr Finrisk working paper No. 209, (2005), ETH Zürich. Google Scholar [15] S. Klöppel and M. Schweizer, Dynamic indifference valuation via convex risk measures, Math. Finance, 17 (2007), 599-627.  doi: 10.1111/j.1467-9965.2007.00317.x.  Google Scholar [16] D. Kramkov and M. Sirbu, Sensitivity analysis of utility based prices and risk- tolerance wealth processes, Ann. Appl. Probab., 16 (2006), 2140-2194.  doi: 10.1214/105051606000000529.  Google Scholar [17] D. Kramkov and M. Sirbu, Asymptotic analysis of utility-based hedging strategies for small number of contingent claims, Stochastic Process Appl., 117 (2007), 1606-1620.  doi: 10.1016/j.spa.2007.04.014.  Google Scholar [18] J. P. Lepeltier and J. San Martin., Backward stochastic differential equations with continuous coefficient, Statist. Probab. Lett. 32 (1997), 425–430. doi: 10.1016/S0167-7152(96)00103-4.  Google Scholar [19] J. Li and Q. Wei, Optimal control problem of fully coupled FBSDEs and viscosity solutions of Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim. 52 (2014), 1622–1662. doi: 10.1137/100816778.  Google Scholar [20] B. Øksendal and A. Sulem, Risk indifference pricing in jump diffusion markets, Math. Finance 19 (2009), 619–637. doi: 10.1111/j.1467-9965.2009.00382.x.  Google Scholar [21] B. Øksendal and A. Sulem, Portfolio optimization under model uncertainty and BSDE games, Quant. Finance 11 (2011), 1665–1674. doi: 10.1080/14697688.2011.615219.  Google Scholar [22] E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), 55–61. doi: 10.1016/0167-6911(90)90082-6.  Google Scholar [23] E. Rosazza Gianin, Risk measures via g-expectations, Insurance Math. Econom., 39 (2006), 19–34. doi: 10.1016/j.insmatheco.2006.01.002.  Google Scholar [24] H. Wang, J. Sun and J. Yong, Recursive utility processes, dynamic risk measures and quadratic backward stochastic Volterra integral equations, to appear in Appl. Math. Optim. (2020). doi: 10.1007/s00245-019-09641-7.  Google Scholar [25] T. Wang and J. Yong, Comparison theorems for some backward stochastic Volterra integral equations, Stochastic Process. Appl. 125 (2015), 1756–1798. doi: 10.1016/j.spa.2014.11.013.  Google Scholar [26] T. Wang and H. Zhang, Optimal control problems of forward-backward stochastic Volterra integral equations with closed control regions, SIAM J. Control Optim., 55 (2017), 2574-2602.  doi: 10.1137/16M1059801.  Google Scholar [27] T. Wang and J. Yong, Backward stochastic Volterra integral equations–representaton of adpated solutions, Stochastic Process. Appl., 129 (2019), 4926–4964. doi: 10.1016/j.spa.2018.12.016.  Google Scholar [28] Z. Wu and Z. Yu, Probabilistic interpretation for a system of quasilinear parabolic partial differential equations combined with algebra equations, Stochastic Process. Appl. 124 (2014), 3921–3947. doi: 10.1016/j.spa.2014.07.013.  Google Scholar [29] J. Xia, Risk aversion and portfolio selection in a continuous-time model, SIAM J. Control Optim, 49 (2011), 1916–1937. doi: 10.1137/10080871X.  Google Scholar [30] M. Xu, Risk measure pricing and hedging in incomplete markets, Ann. Financ. 2 (2006), 51–71. doi: 10.1007/s10436-005-0023-x.  Google Scholar [31] J. Yong, Continuous-time dynamic risk measures by backward stochastic Volterra integral equations, Appl. Anal., 86 (2007), 1429–1442. doi: 10.1080/00036810701697328.  Google Scholar [32] J. Yong, Well-posedness and regularity of backward stochastic Volterra integral equations, Probab. Theory. Related Fields, 142 (2008), 21–77. doi: 10.1007/s00440-007-0098-6.  Google Scholar

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##### References:
 [1] P. Barrieu and N. El Karoui, Pricing, hedging and optimally designing derivatives via minimization of risk measures, Indifference Pricing: Theory and Applications. Princeton University Press, (2008), Princeton, USA. Google Scholar [2] S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Rev. Finan. Stud., 23 (2010), 2970-3016.  doi: 10.1093/rfs/hhq028.  Google Scholar [3] T. Björk, A. Murgoci and X. Y. Zhou, Mean-variance portfolio optimization with state-dependent risk aversion, Math. Finance, 24 (2014), 1-24.  doi: 10.1111/j.1467-9965.2011.00515.x.  Google Scholar [4] C. Borell, Monotonicity properties of optimal investment strategies for log-Brownian asset prices, Math. Finance, 17 (2007), 143-153.  doi: 10.1111/j.1467-9965.2007.00297.x.  Google Scholar [5] Ł. Delong, Optimal investment for insurance company with exponential utility and wealth-dependent risk aversion coefficient, Math. Method. Oper. Res., 89 (2019), 73-113.  doi: 10.1007/s00186-019-00659-9.  Google Scholar [6] Ł. Delong and P. Imkeller P, Backward stochastic differential equations with time delayed generators-results and counterexamples, Ann. Appl. Probab., 20 (2010), 1512-1536.  doi: 10.1214/09-AAP663.  Google Scholar [7] Ł. Delong and P. Imkeller, On Malliavin's differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures, Stochastic Process. Appl., 120 (2010), 1748-1775.  doi: 10.1016/j.spa.2010.05.001.  Google Scholar [8] Y. Dong and R. Sircar, Time-Inconsistent Portfolio Investment Problems, In: Crisan D., Hambly B., Zariphopoulou T. (eds), Stochastic Analysis and Applications 2014,239-281. Springer Proceedings in Mathematics Statistics, vol 100. 2014, Springer, Cham. doi: 10.1007/978-3-319-11292-3_9.  Google Scholar [9] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.  Google Scholar [10] R. J. Elliott and T. K. Siu, Risk-based indifference pricing under a stochastic volatility model, Commun. Stoch. Anal., 4 (2010), 51-73.  doi: 10.31390/cosa.4.1.05.  Google Scholar [11] R. J. Elliott and T. K. Siu, A BSDE approach to a risk-based optimal investment of an insurer, Automatica, 47 (2011), 253-261.  doi: 10.1016/j.automatica.2010.10.032.  Google Scholar [12] V. Henderson and D. Hobson, Utility indifference pricing-an overview, In Volume on Indifference Pricing, (ed. R. Carmona), (2004), Princeton University Press. Google Scholar [13] E. Karni, Risk aversion for state-dependent utility functions: Measurement and applications, Int. Econ. Review, 24 (1983), 637-647.  doi: 10.2307/2648791.  Google Scholar [14] S. Klöppel and M. Schweizer, Dynamic Utility Indifference Valuation via Convex Risk Measures, Nccr Finrisk working paper No. 209, (2005), ETH Zürich. Google Scholar [15] S. Klöppel and M. Schweizer, Dynamic indifference valuation via convex risk measures, Math. Finance, 17 (2007), 599-627.  doi: 10.1111/j.1467-9965.2007.00317.x.  Google Scholar [16] D. Kramkov and M. Sirbu, Sensitivity analysis of utility based prices and risk- tolerance wealth processes, Ann. Appl. Probab., 16 (2006), 2140-2194.  doi: 10.1214/105051606000000529.  Google Scholar [17] D. Kramkov and M. Sirbu, Asymptotic analysis of utility-based hedging strategies for small number of contingent claims, Stochastic Process Appl., 117 (2007), 1606-1620.  doi: 10.1016/j.spa.2007.04.014.  Google Scholar [18] J. P. Lepeltier and J. San Martin., Backward stochastic differential equations with continuous coefficient, Statist. Probab. Lett. 32 (1997), 425–430. doi: 10.1016/S0167-7152(96)00103-4.  Google Scholar [19] J. Li and Q. Wei, Optimal control problem of fully coupled FBSDEs and viscosity solutions of Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim. 52 (2014), 1622–1662. doi: 10.1137/100816778.  Google Scholar [20] B. Øksendal and A. Sulem, Risk indifference pricing in jump diffusion markets, Math. Finance 19 (2009), 619–637. doi: 10.1111/j.1467-9965.2009.00382.x.  Google Scholar [21] B. Øksendal and A. Sulem, Portfolio optimization under model uncertainty and BSDE games, Quant. Finance 11 (2011), 1665–1674. doi: 10.1080/14697688.2011.615219.  Google Scholar [22] E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), 55–61. doi: 10.1016/0167-6911(90)90082-6.  Google Scholar [23] E. Rosazza Gianin, Risk measures via g-expectations, Insurance Math. Econom., 39 (2006), 19–34. doi: 10.1016/j.insmatheco.2006.01.002.  Google Scholar [24] H. Wang, J. Sun and J. Yong, Recursive utility processes, dynamic risk measures and quadratic backward stochastic Volterra integral equations, to appear in Appl. Math. Optim. (2020). doi: 10.1007/s00245-019-09641-7.  Google Scholar [25] T. Wang and J. Yong, Comparison theorems for some backward stochastic Volterra integral equations, Stochastic Process. Appl. 125 (2015), 1756–1798. doi: 10.1016/j.spa.2014.11.013.  Google Scholar [26] T. Wang and H. Zhang, Optimal control problems of forward-backward stochastic Volterra integral equations with closed control regions, SIAM J. Control Optim., 55 (2017), 2574-2602.  doi: 10.1137/16M1059801.  Google Scholar [27] T. Wang and J. Yong, Backward stochastic Volterra integral equations–representaton of adpated solutions, Stochastic Process. Appl., 129 (2019), 4926–4964. doi: 10.1016/j.spa.2018.12.016.  Google Scholar [28] Z. Wu and Z. Yu, Probabilistic interpretation for a system of quasilinear parabolic partial differential equations combined with algebra equations, Stochastic Process. Appl. 124 (2014), 3921–3947. doi: 10.1016/j.spa.2014.07.013.  Google Scholar [29] J. Xia, Risk aversion and portfolio selection in a continuous-time model, SIAM J. Control Optim, 49 (2011), 1916–1937. doi: 10.1137/10080871X.  Google Scholar [30] M. Xu, Risk measure pricing and hedging in incomplete markets, Ann. Financ. 2 (2006), 51–71. doi: 10.1007/s10436-005-0023-x.  Google Scholar [31] J. Yong, Continuous-time dynamic risk measures by backward stochastic Volterra integral equations, Appl. Anal., 86 (2007), 1429–1442. doi: 10.1080/00036810701697328.  Google Scholar [32] J. Yong, Well-posedness and regularity of backward stochastic Volterra integral equations, Probab. Theory. Related Fields, 142 (2008), 21–77. doi: 10.1007/s00440-007-0098-6.  Google Scholar
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