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Sixian Jin

Publications and source records attributed to Sixian Jin.

10 recordsLinked to original sources
arXiv2026

Optimal Selling of Defaultable Assets using the Distribution Builder

We consider the problem of when it is best to sell a risky asset in the framework of the \textit{distribution builder} approach under the consideration of potential ruin. This approach allows investors to express their preferences directly as a desired target distribution without first specifying a risk aversion or utility function. Mathematically, the problem is closely related to the Skorokhod embedding problem, where the goal is to attain a given distribution by stopping a diffusion process. We work in a general framework of one-dimensional diffusion processes and extend existing results to include the possibility of ruin. We first provide a full characterization of the set of distributions that can be attained before ruin occurs. Then, we formulate two optimization problems that tackle the issue of what to do if the originally specified distribution is either not attainable or not optimal: finding the attainable distribution closest to an unattainable distribution and selecting an optimal attainable distribution under first-order stochastic dominance constraints if the originally specified distribution is attainable, but not optimal. We show existence and uniqueness of solutions to these constrained convex optimization problems in an extended $f$-divergence framework, provide an analytic characterization of solutions and give numerical examples.

math.OC↗
arXiv2026

Long-Time Behaviors of Stochastic Linear-Quadratic Optimal Control Problems

This paper investigates the asymptotic behavior of the solution to a linear-quadratic stochastic optimal control problems. The so-called probability cell problem is introduced the first time. It serves as the probability interpretation of the well-known cell problem in the homogenization of Hamilton-Jacobi equations. By establishing a connection between this problem and the ergodic cost problem, we reveal the turnpike properties of the linear-quadratic stochastic optimal control problems from various perspectives.

math.OC↗
arXiv2025

The Stationary Behavior of Reflecting Coupled Brownian Motions with Applications to Shortest Remaining Processing Time Queues

With the objective of characterizing the stationary behavior of the scaling limit for shortest remaining processing time (SRPT) queues with a heavy-tailed processing time distribution, as obtained in Banerjee, Budhiraja, and Puha (BBP, 2022), we study reflecting coupled Brownian motions (RCBM) $(W_t(a), a, t \geq 0)$. These RCBM arise by regulating coupled Brownian motions (CBM) $(χ_t(a), a,t \geq 0)$ to remain nonnegative. Here, for $t\geq 0$, $χ_t(0)=0$ and $χ_t(a):=w(a)+σB_t-μ(a)t$ for $a>0$, $w(\cdot)$ is a suitable initial condition, $σ$ is a positive constant, $B$ is a standard Brownian motion, and $μ(\cdot)$ is an unbounded, positive, strictly decreasing drift function. In the context of the BBP (2022) scaling limit, the drift function is determined by the model parameters, and, for each $a\geq 0$, $W_{\cdot}(a)$ represents the scaling limit of the amount of work in the system of size $a$ or less. Thus, for the BBP (2022) scaling limit, the time $t$ values of the RCBM describe the random distribution of the size of the remaining work in the system at time $t$. Our principal results characterize the stationary distribution of the RCBM in terms of a maximum process $M_*(\cdot)$ associated with CBM starting from zero. We obtain an explicit representation for the finite-dimensional distributions of $M_*(\cdot)$ and a simple formula for its covariance. We further show that the RCBM converge in distribution to $M_*(\cdot)$ as time $t$ approaches infinity. From this, we deduce the stationary behavior of the BBP (2022) scaling limit, including obtaining an integral expression for the stationary queue length in terms of the associated maximum process. While its distribution appears somewhat complex, we compute the mean and variance explicitly, and we connect with the work of Lin, Wierman, and Zwart (2011) to offer an illustration of Little's Law.

math.PR↗
arXiv2020

Strong approximation of time-changed stochastic differential equations involving drifts with random and non-random integrators

The rates of strong convergence for various approximation schemes are investigated for a class of stochastic differential equations (SDEs) which involve a random time change given by an inverse subordinator. SDEs to be considered are unique in two different aspects: i) they contain two drift terms, one driven by the random time change and the other driven by a regular, non-random time variable; ii) the standard Lipschitz assumption is replaced by that with a time-varying Lipschitz bound. The difficulty imposed by the first aspect is overcome via an approach that is significantly different from a well-known method based on the so-called duality principle. On the other hand, the second aspect requires the establishment of a criterion for the existence of exponential moments of functions of the random time change.

math.PR↗
arXiv2020

Pricing the zero-coupon bond of the extended Cox-Ingersoll-Ross model using Malliavin calculus

In this paper, we price the zero-coupon bond of the extended Cox-Ingersoll-Ross model by a Dyson type formula established in one of the authors' paper Jin, Peng and Schelllhorn (2016) using Malliavin calculus. This formula provides a fast convergent series to represent the bond price, and it depends on the given drift and volatility of the interest rate process but not the instantaneous forward rate used in Maghsoodi (1996). This expression can be also regarded as a new solution to a class of Reccati equations with time-dependent coefficients.

math.PR↗
arXiv2020

Strong approximation of stochastic differential equations driven by a time-changed Brownian motion with time-space-dependent coefficients

The rate of strong convergence is investigated for an approximation scheme for a class of stochastic differential equations driven by a time-changed Brownian motion, where the random time changes $(E_t)_{t\ge 0}$ considered include the inverses of stable and tempered stable subordinators as well as their mixtures. Unlike those in the work of Jum and Kobayashi (2016), the coefficients of the stochastic differential equations discussed in this paper depend on the regular time variable $t$ rather than the time change $E_t$. This alteration makes it difficult to apply the method used in that paper. To overcome this difficulty, we utilize a Gronwall-type inequality involving a stochastic driver to control the moment of the error process. Moreover, in order to guarantee that an ultimately derived error bound is finite, we establish a useful criterion for the existence of exponential moments of powers of the random time change.

math.PR↗
arXiv2016

Estimation of the Pointwise Hölder Exponent of Hidden Multifractional Brownian Motion Using Wavelet Coefficients

We propose a wavelet-based approach to construct consistent estimators of the pointwise Hölder exponent of a multifractional Brownian motion, in the case where this underlying process is not directly observed. The relative merits of our estimator are discussed, and we introduce an application to the problem of estimating the functional parameter of a nonlinear model.

math.PR↗
arXiv2015

A Representation Theorem for Smooth Brownian Martingales - New Example

We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends on the second order Malliavin derivative operator evaluated along a "frozen path". The exponential operator can be expanded explicitly to a series representation, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result can be proven independently by two different methods. In the first method, one constructs a time-evolution equation, by passage to the limit of a special case of a backward Taylor expansion of an approximating discrete time martingale. The exponential formula is a solution of the time-evolution equation, but we emphasize in our article that the time-evolution equation is a separate result of independent interest. In the second method, which we only highlight in this article, we use the property of denseness of exponential functions. We provide several applications of the exponential formula, and briefly highlight numerical applications of the backward Taylor expansion.

math.PR↗
arXiv2015

Fractional Hida Malliavin Derivatives and Series Representations of Fractional Conditional Expectations

We represent fractional conditional expectations of a functional of fractional Brownian motion as a convergent series in L^2 space. When the target random variable is some function of a discrete trajectory of fractional Brownian motion, we obtain a backward Taylor series representation; when the target functional is generated by a continuous fractional filtration, the series representation is obtained by applying a "frozen path" operator and an exponential operator to the functional. Three examples are provided to show that our representation gives useful series expansions of ordinary expectations of target random variables.

math.PR↗