arXiv subjects

Shuaiqi Zhang

Publications and source records attributed to Shuaiqi Zhang.

8 recordsLinked to original sources

arXiv2025

Sub-diffusive Black-Scholes model and Girsanov transform for sub-diffusions

We propose a novel Black-Scholes model under which the stock price processes are modeled by stochastic differential equations driven by sub-diffusions. The new framework can capture the less financial activity phenomenon during the bear markets while having the classical Black- Scholes model as its special case. The sub-diffusive spot market is arbitrage-free but is in general incomplete. We investigate the pricing for European-style contingent claims under this new model. For this, we study the Girsanov transform for sub-diffusions and use it to find risk-neutral probability measures for the new Black-Scholes model. Finally, we derive the explicit formula for the price of European call options and show that it can be determined by a partial differential equation (PDE) involving a fractional derivative in time, which we coin a time-fractional Black-Scholes PDE.

math.PR↗

arXiv2025

The Optimal Control Problem of Fully Coupled FBSDEs Driven by Sub-diffusion with Applications

This paper is devoted to an optimal control problem of fully coupled forward-backward stochastic differential equations driven by sub-diffusion, whose solutions are not Markov processes. The stochastic maximum principle is obtained, where the control domain may not be convex and the diffusion term is independent of the control variable. Additionally, problem with state constraint is researched by using Ekeland's variational principle. The theoretical results obtained are applied to a cash management optimization problem in bear market, and the optimal strategy is derived.

math.OC↗

arXiv2023

Stochastic maximum principle for sub-diffusions and its applications

In this paper, we study optimal stochastic control problems for stochastic systems driven by non-Markov sub-diffusion $B_{L_t}$, which have the mixed features of deterministic and stochastic controls. Here $B_t$ is the standard Brownian motion on $R$, and $L_t:= \inf\{r>0: S_r>t\}, \quad t\geq 0,$ is the inverse of a subordinator $S_t$ with drift $κ>0$ that is independent of $B_t$. We obtain stochastic maximum principles (SMP) for these systems using both convex and spiking variational methods, depending on whether the convex domain is convex or not. To derive SMP, we first establish a martingale representation theorem for sub-diffusions $B_{L_t}$, and then use it to derive the existence and uniqueness result for the solutions of backward stochastic differential equations (BSDEs) driven by sub-diffusions, which may be of independent interest. We also derive sufficient SMPs. Application to a linear quadratic system is given to illustrate the main results of this paper.

math.PR↗

arXiv2023

Fully coupled forward-backward stochastic differential equations driven by sub-diffusions

In this paper, we establish the existence and uniqueness of fully coupled forward-backward stochastic differential equations (FBSDEs in short) driven by anomalous sub-diffusions $B_{L_t}$ under suitable monotonicity conditions on the coefficients. Here $B$ is a Brownian motion on $\bf R$ and $L_t:= \inf\{r>0: S_r>t\}$, $t\geq 0,$ is the inverse of a subordinator $S$ with drift $κ>0$ that is independent of $B$. Various a priori estimates on the solutions of the FBSDEs are also presented.

math.PR↗

arXiv2020

A stochastic maximum principle for partially observed stochastic control systems with delay

This paper deals with partially-observed optimal control problems for the state governed by stochastic differential equation with delay. We develop a stochastic maximum principle for this kind of optimal control problems using a variational method and a filtering technique. Also, we establish a sufficient condition without assumption of the concavity. Two examples shed light on the theoretical results are established in the paper. In particular, in the example of an optimal investment problem with delay, its numerical simulation shows the effect of delay via a discretization technique for forward-backward stochastic differential equations (FBSDEs) with delay and anticipate terms.

math.OC↗

arXiv2016

Partial Information Differential Games for Mean-Field SDEs

This paper is concerned with non-zero sum differential games of mean-field stochastic differential equations with partial information and convex control domain. First, applying the classical convex variations, we obtain stochastic maximum principle for Nash equilibrium points. Subsequently, under additional assumptions, verification theorem for Nash equilibrium points is also derived. Finally, as an application, a linear quadratic example is discussed. The unique Nash equilibrium point is represented in a feedback form of not only the optimal filtering but also expected value of the system state, throughout the solutions of the Riccati equations.

math.OC↗

arXiv2016

A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance

In this article, we concern a kind of partially observed non-zero sum stochastic differential game based on forward and backward stochastic differential equations (FBSDEs). It is required that each player has his own observation equation, and the corresponding open-loop Nash equilibrium control is required to adapted to the filtration that the observation process generated. To find this open-loop Nash equilibrium point, we prove the maximum principle as a necessary condition of the existence of this point, and give a verification theorem as a sufficient condition to verify it is the real open-loop Nash equilibrium point. Combined this with reality, a financial investment problem is raised. We can obtain the explicit observable investment strategy by using stochastic filtering theory and the results above.

math.OC↗

arXiv2014

Optimal Dividend Payments for the Piecewise-Deterministic Poisson Risk Model

This paper considers the optimal dividend payment problem in piecewise-deterministic compound Poisson risk models. The objective is to maximize the expected discounted dividend payout up to the time of ruin. We provide a comparative study in this general framework of both restricted and unrestricted payment schemes, which were only previously treated separately in certain special cases of risk models in the literature. In the case of restricted payment scheme, the value function is shown to be a classical solution of the corresponding HJB equation, which in turn leads to an optimal restricted payment policy known as the threshold strategy. In the case of unrestricted payment scheme, by solving the associated integro-differential quasi-variational inequality, we obtain the value function as well as an optimal unrestricted dividend payment scheme known as the barrier strategy. When claim sizes are exponentially distributed, we provide easily verifiable conditions under which the threshold and barrier strategies are optimal restricted and unrestricted dividend payment policies, respectively. The main results are illustrated with several examples, including a new example concerning regressive growth rates.

math.OC↗